MathML scripted elements

We continue the review of basic math notations and focus on building MathML elements with scripts.

Prerequisites:	Basic software installed , basic knowledge of working with files , and HTML basics (study Introduction to HTML .)
Objective:	To get familiar with basic MathML elements producing scripted elements.

Subscripts and superscripts

Similarly to what we saw in the previous article , the <msub > , <msup > and <msubsup > have a special structure expecting exactly two elements (for <msub > , <msup > ) or three elements (for <msubsup > ):

html

Below is the rendering of the above example in your browser.

You should notice that:

The second child of the <msub > element is attached as a subscript of its first child.
The second child of the <msup > element is attached as a superscript of its first child.
The second and third children of the <msubsup > element are respectively attached as a subscript and superscript of its first child.
The text inside scripts is scaled down.

Note: The MathML elements <msub > and <msup > are different from the HTML elements <sub > and <sup > . They allow authors to provide arbitrary MathML subtrees as scripts, not just text.

Underscripts and overscripts

The <munder > , <mover > and <munderover > elements are very similar except that they are used to attach underscripts and overscripts. Instead of giving details, we will let you figure out their definitions yourself with the following exercise.

Active learning: Recognize under/over scripts

In the following example, try to guess the names of the mystery elements (written as question marks) and click the button to reveal the solution:

                                        
                                            
                                                
                                                    <
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    code
                                                
                                                >
                                            
                                            &lt;
                                            
                                                
                                                    <
                                                    span
                                                
                                                >
                                            
                                            ????????
                                            
                                                
                                                    </
                                                    span
                                                
                                                >
                                            
                                            &gt;
                                            
                                                
                                                    </
                                                    code
                                                
                                                >
                                            
                                            element with exactly two children
(child1, child2):

                                            
                                                
                                                    <
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mtext
                                                
                                                >
                                            
                                            child1
                                            
                                                
                                                    </
                                                    mtext
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mtext
                                                
                                                >
                                            
                                            child2
                                            
                                                
                                                    </
                                                    mtext
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    code
                                                
                                                >
                                            
                                            &lt;
                                            
                                                
                                                    <
                                                    span
                                                
                                                >
                                            
                                            ????????
                                            
                                                
                                                    </
                                                    span
                                                
                                                >
                                            
                                            &gt;
                                            
                                                
                                                    </
                                                    code
                                                
                                                >
                                            
                                            element with exactly three children
(child1, child2 and child3):

                                            
                                                
                                                    <
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mtext
                                                
                                                >
                                            
                                            child1
                                            
                                                
                                                    </
                                                    mtext
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mtext
                                                
                                                >
                                            
                                            child2
                                            
                                                
                                                    </
                                                    mtext
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mtext
                                                
                                                >
                                            
                                            child3
                                            
                                                
                                                    </
                                                    mtext
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    code
                                                
                                                >
                                            
                                            &lt;
                                            
                                                
                                                    <
                                                    span
                                                
                                                >
                                            
                                            ????????
                                            
                                                
                                                    </
                                                    span
                                                
                                                >
                                            
                                            &gt;
                                            
                                                
                                                    </
                                                    code
                                                
                                                >
                                            
                                            element with exactly two children
(child1, child2):

                                            
                                                
                                                    <
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mtext
                                                
                                                >
                                            
                                            child1
                                            
                                                
                                                    </
                                                    mtext
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mtext
                                                
                                                >
                                            
                                            child2
                                            
                                                
                                                    </
                                                    mtext
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    p
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    button"
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    showSolution"
                                                
                                                value
                                                
                                                    =
                                                    "
                                                    Show solution"
                                                
                                                />
                                            
                                            
                                                
                                                    </
                                                    p
                                                
                                                >

                                        
                                            document.
                                            getElementById
                                            (
                                            "showSolution"
                                            )
                                            .
                                            addEventListener
                                            (
                                            "click"
                                            ,
                                            (
                                            )
                                            =>
                                            {
                                            const
                                            maths =
                                            Array.
                                            from
                                            (
                                            document.
                                            getElementsByTagName
                                            (
                                            "math"
                                            )
                                            )
                                            ;
                                            Array.
                                            from
                                            (
                                            document.
                                            getElementsByTagName
                                            (
                                            "span"
                                            )
                                            )
                                            .
                                            forEach
                                            (
                                            (
                                            
                                                span,
                                                index
                                            
                                            )
                                            =>
                                            {
                                            span.
                                            textContent =
                                            maths[
                                            index]
                                            .
                                            firstElementChild.
                                            tagName;
                                            }
                                            )
                                            ;
                                            }
                                            )
                                            ;

Active learning: Recognize scripted elements

The following MathML formula contains a more complex expression, nesting fractions, roots and scripts. Try to guess the elements laid out with scripted elements <msub > , <msup > , <msubsup > , <munder > , <mover > , <munderover > . Each time you click such an element, it is highlighted and a confirmation message is displayed. Finally, read the MathML source to verify whether that corresponds to your expectation.

                                        
                                            
                                                <!
                                                doctype
                                                html
                                                >
                                            
                                            
                                                
                                                    <
                                                    html
                                                
                                                lang
                                                
                                                    =
                                                    "
                                                    en-US"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    meta
                                                
                                                charset
                                                
                                                    =
                                                    "
                                                    utf-8"
                                                
                                                />
                                            
                                            
                                                
                                                    <
                                                    title
                                                
                                                >
                                            
                                            My page with scripted elements
                                            
                                                
                                                    </
                                                    title
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    link
                                                
                                                rel
                                                
                                                    =
                                                    "
                                                    stylesheet"
                                                
                                                href
                                                
                                                    =
                                                    "
                                                    https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css"
                                                
                                                />
                                            
                                            
                                                
                                                    </
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    body
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    math
                                                
                                                display
                                                
                                                    =
                                                    "
                                                    block"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            β
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            ⎵
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            |
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            α
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            →
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            |
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msup
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            s
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msup
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            −
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            ∑
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            =
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            1
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            n
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            a
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            −
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msubsup
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            K
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            0
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msubsup
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    input
                                                
                                                type
                                                
                                                    =
                                                    "
                                                    button"
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    clearOutput"
                                                
                                                value
                                                
                                                    =
                                                    "
                                                    Reset"
                                                
                                                />
                                            
                                            
                                                
                                                    <
                                                    div
                                                
                                                id
                                                
                                                    =
                                                    "
                                                    output"
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    div
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    body
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    html
                                                
                                                >

                                        
                                            const
                                            scriptedElements =
                                            Array.
                                            from
                                            (
                                            document.
                                            querySelectorAll
                                            (
                                            "msub, msup, msubsup, munder, mover, munderover"
                                            )
                                            ,
                                            )
                                            ;
                                            const
                                            outputDiv =
                                            document.
                                            getElementById
                                            (
                                            "output"
                                            )
                                            ;
                                            function
                                            clearHighlight
                                            (
                                            )
                                            {
                                            scriptedElements.
                                            forEach
                                            (
                                            (
                                            scripted
                                            )
                                            =>
                                            {
                                            scripted.
                                            classList.
                                            remove
                                            (
                                            "highlight"
                                            )
                                            ;
                                            }
                                            )
                                            ;
                                            }
                                            scriptedElements.
                                            forEach
                                            (
                                            (
                                            scripted
                                            )
                                            =>
                                            {
                                            scripted.
                                            addEventListener
                                            (
                                            "click"
                                            ,
                                            (
                                            )
                                            =>
                                            {
                                            clearHighlight
                                            (
                                            )
                                            ;
                                            scripted.
                                            classList.
                                            add
                                            (
                                            "highlight"
                                            )
                                            ;
                                            outputDiv.
                                            insertAdjacentHTML
                                            (
                                            "beforeend"
                                            ,
                                            
                                                `
                                                <p ><strong >You clicked an <code >&lt;
                                                
                                                    ${
                                                    scripted.
                                                    tagName}
                                                
                                                &gt;</code >element.</strong ></p >
                                                `
                                            
                                            ,
                                            )
                                            ;
                                            }
                                            )
                                            ;
                                            }
                                            )
                                            ;
                                            document.
                                            getElementById
                                            (
                                            "clearOutput"
                                            )
                                            .
                                            addEventListener
                                            (
                                            "click"
                                            ,
                                            (
                                            )
                                            =>
                                            {
                                            clearHighlight
                                            (
                                            )
                                            ;
                                            outputDiv.
                                            innerHTML =
                                            ""
                                            ;
                                            }
                                            )
                                            ;

More operator properties

We have previously seen some properties of the <mo > element namely stretching in the vertical direction and spacing. Now that scripted elements are available, we can extend that list. We will do that by tweaking our previous example .

Stretching in horizontal direction

Let's first perform the substitutions $β ≔ z_{1} + z_{2}$ and $α ≔ v_{1} + v_{2}$ :

                                        
                                            
                                                <!
                                                doctype
                                                html
                                                >
                                            
                                            
                                                
                                                    <
                                                    html
                                                
                                                lang
                                                
                                                    =
                                                    "
                                                    en-US"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    meta
                                                
                                                charset
                                                
                                                    =
                                                    "
                                                    utf-8"
                                                
                                                />
                                            
                                            
                                                
                                                    <
                                                    title
                                                
                                                >
                                            
                                            My page with horizontal stretchy operators
                                            
                                                
                                                    </
                                                    title
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    link
                                                
                                                rel
                                                
                                                    =
                                                    "
                                                    stylesheet"
                                                
                                                href
                                                
                                                    =
                                                    "
                                                    https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css"
                                                
                                                />
                                            
                                            
                                                
                                                    </
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    body
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    math
                                                
                                                display
                                                
                                                    =
                                                    "
                                                    block"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            z
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            1
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            z
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            2
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            ⎵
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            |
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            v
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            1
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            v
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            2
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            →
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            |
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msup
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            s
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msup
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            −
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            ∑
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            =
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            1
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            n
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            a
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            −
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msubsup
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            K
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            0
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msubsup
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    body
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    html
                                                
                                                >

We now realize that the bottom bracket "⎵" and the rightward arrow "→" stretch horizontally to cover the width of the substituted values. Recall that some vertical operators can stretch to cover the height of non-stretchy siblings inside an <mrow > . Similarly some horizontal operators can stretch to cover the width of non-stretchy siblings in an <munder > , <mover > or <munderover > element.

Note: Stretching can happen for any child of the <munder > , <mover > or <munderover > element, not just the underscript or overscript.

Large operator and limits

So far our example has actually been rendered with the display="block" attribute. Let's look at the same example, as rendered without that attribute:

                                        
                                            
                                                <!
                                                doctype
                                                html
                                                >
                                            
                                            
                                                
                                                    <
                                                    html
                                                
                                                lang
                                                
                                                    =
                                                    "
                                                    en-US"
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    meta
                                                
                                                charset
                                                
                                                    =
                                                    "
                                                    utf-8"
                                                
                                                />
                                            
                                            
                                                
                                                    <
                                                    title
                                                
                                                >
                                            
                                            My page with moved limits and small largeop
                                            
                                                
                                                    </
                                                    title
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    link
                                                
                                                rel
                                                
                                                    =
                                                    "
                                                    stylesheet"
                                                
                                                href
                                                
                                                    =
                                                    "
                                                    https://fred-wang.github.io/MathFonts/LatinModern/mathfonts.css"
                                                
                                                />
                                            
                                            
                                                
                                                    </
                                                    head
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    body
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            z
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            1
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            z
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            2
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            ⎵
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munder
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mroot
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            |
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            v
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            1
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            +
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            v
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            2
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            →
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            |
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msup
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            s
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            3
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msup
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mfrac
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            −
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            ∑
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            =
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            1
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            n
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    munderover
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            a
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msub
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mo
                                                
                                                >
                                            
                                            −
                                            
                                                
                                                    </
                                                    mo
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    msubsup
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            K
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mn
                                                
                                                >
                                            
                                            0
                                            
                                                
                                                    </
                                                    mn
                                                
                                                >
                                            
                                            
                                                
                                                    <
                                                    mi
                                                
                                                >
                                            
                                            i
                                            
                                                
                                                    </
                                                    mi
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msubsup
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    msqrt
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    mrow
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    math
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    body
                                                
                                                >
                                            
                                            
                                                
                                                    </
                                                    html
                                                
                                                >

As expected, the formula is no longer centered and the rendering is modified to minimized the height. Focusing on the summation symbol, one can notice that the sigma is drawn smaller and that the scripts of the <munderover > element are now attached as a subscript and a superscript! This is due to two properties of the "∑" operator:

largeop : The operator is drawn with a bigger glyph if the <math > tag has a display="block" attribute.
movablelimits : The underscripts and overscripts attached to the operator are respectively rendered as subscripts and superscripts if the <math > tag does not have the display="block" attribute.

Note: The largeop property is actually unrelated to scripts, even though operators having this property are typically scripted. The movablelimits property is taken into account for <munder > and <mover > elements too.

Summary

In this article, we've finished reviewing basic layout introducing elements <msub > , <msup > , <msubsup > , <munder > , <mover > , <munderover > for subscripts, superscripts, underscripts and overscripts. Using these elements, we were able to briefly introduce new properties of the <mo > element. In the next article, we will continue focus on tabular layout .