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chapter3.html
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<!DOCTYPE HTML>
<html lang="en">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
<title>Presentation Markup</title>
<meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no">
<style type="text/css">
/** Table of Contents taken from html 5.2 spec
stop flicker in toc **/
.toc a {
/* More spacing; use padding to make it part of the click target. */
padding-top: 0.1rem;
/* Larger, more consistently-sized click target */
display: block;
/* Reverse color scheme */
color: black;
border-color: #3980B5;
border-bottom-width: 3px !important;
margin-bottom: 0px !important;
}
.egmeta {
color:#5555AA;font-style:italic;font-family:serif;font-weight:bold;
}
div.graphic{margin-left:2em}
table.syntax {
font-size: 75%;
background-color: #DDDDDD;
border: thin solid;
}
table.syntax td {
border: solid thin;
}
table.syntax th {
text-align: left;
}
table.attributes td { padding-left:0.5em; padding-right:0.5em; border: solid thin; }
table.attributes td.attname { white-space:nowrap; vertical-align:top;}
table.attributes td.attdesc { background-color:#F0F0FF; padding-left:2em; padding-right:2em}
th.uname {font-size: 50%; text-align:left;}
code { font-family: monospace; }
div.constraint,
div.issue,
div.note,
div.notice { margin-left: 2em; }
li p { margin-top: 0.3em;
margin-bottom: 0.3em; }
div.exampleInner pre { margin-left: 1em;
margin-top: 0em; margin-bottom: 0em}
div.exampleOuter {border: 4px double gray;
margin: 0em; padding: 0em}
div.exampleInner { background-color: #d5dee3;
border-top-width: 4px;
border-top-style: double;
border-top-color: #d3d3d3;
border-bottom-width: 4px;
border-bottom-style: double;
border-bottom-color: #d3d3d3;
padding: 4px; margin: 0em }
div.exampleWrapper { margin: 4px }
div.exampleHeader { font-weight: bold;
margin: 4px}
a.mainindex {font-weight: bold;}
li.sitem {list-style-type: none;}
.error { color: red }
div.mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example {border:solid thin black;
padding: 0.5em;
margin: 0.5em 0 0.5em 0;
}
div.strict-mathml-example h5 {
margin-top: -0.3em;
margin-bottom: -0.5em;}
var.meta {background-color:green}
var.transmeta {background-color:red}
pre.mathml {padding: 0.5em;
background-color: #FFFFDD;}
pre.mathml-fragment {padding: 0.5em;
background-color: #FFFFDD;}
pre.strict-mathml {padding: 0.5em;
background-color: #FFFFDD;}
span.uname {color:#999900;font-size:75%;font-family:sans-serif;}
.minitoc { border-style: solid;
border-color: #0050B2;
border-width: 1px ;
padding: 0.3em;}
.attention { border-style: solid;
border-width: 1px ;
color: #5D0091;
background: #F9F5DE;
border-color: red;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.attribute-Name { background: #F9F5C0; }
.method-Name { background: #C0C0F9; }
.IDL-definition { border-style: solid;
border-width: 1px ;
color: #001000;
background: #E0FFE0;
border-color: #206020;
margin-left: 1em;
margin-right: 1em;
margin-top: 0.25em;
margin-bottom: 0.25em; }
.baseline {vertical-align: baseline}
#eqnoc1 {width: 10%}
#eqnoc2 {width: 80%; text-align: center; }
#eqnoc3 {width: 10%; text-align: right; }
div.div1 {margin-bottom: 1em;}
.h3style {
text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2;
font-size: 125%;
}
h4 { text-align: left;
font-family: sans-serif;
font-weight: normal;
color: #0050B2; }
h5 { text-align: left;
font-family: sans-serif;
font-weight: bold;
color: #0050B2; }
th {background: #E0FFE0;}
p, blockquote, h4 { font-family: sans-serif; }
dt, dd, dl, ul, li { font-family: sans-serif; }
pre, code { font-family: monospace }
a.termref {
text-decoration: none;
color: black;
}
.mathml-render {
font-family: serif;
font-size: 130%;
border: solid 4px green;
padding-left: 1em;
padding-right: 1em;
}
</style>
<link rel="stylesheet" type="text/css" href="https://www.w3.org/StyleSheets/TR/2016/W3C-ED.css"><script src="//www.w3.org/scripts/TR/2016/fixup.js"> </script></head>
<body>
<h1><a id="presm"></a>3 Presentation Markup</h1>
<!-- TOP NAVIGATION BAR -->
<nav id="toc">
<ol class="toc">
<li><a href="Overview.html"><span class="secno"></span> Mathematical Markup Language (MathML) Version 4.0</a></li>
<li><a href="chapter2.html"><span class="secno">2 </span> MathML Fundamentals</a></li>
<li><a href="chapter4.html"><span class="secno">4 </span> Content Markup</a></li>
<li><a href="chapter3.html"><span class="secno">3 </span> Presentation Markup</a>
<ol class="toc">
<li><a href="chapter3.html#presm.intro"><span class="secno">3.1 </span> Introduction</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.1"><span class="secno">3.1.1 </span> What Presentation Elements Represent</a>
</li>
<li><a href="chapter3.html#id.3.1.2"><span class="secno">3.1.2 </span> Terminology Used In This Chapter</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.2.1"><span class="secno">3.1.2.1 </span> Types of presentation elements</a>
</li>
<li><a href="chapter3.html#id.3.1.2.2"><span class="secno">3.1.2.2 </span> Terminology for other classes of elements and their relationships</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.reqarg"><span class="secno">3.1.3 </span> Required Arguments</a>
<ol class="toc">
<li><a href="chapter3.html#presm.inferredmrow"><span class="secno">3.1.3.1 </span> Inferred <code><mrow></code>s</a>
</li>
<li><a href="chapter3.html#id.3.1.3.2"><span class="secno">3.1.3.2 </span> Table of argument requirements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#id.3.1.4"><span class="secno">3.1.4 </span> Elements with Special Behaviors</a>
</li>
<li><a href="chapter3.html#presm.bidi"><span class="secno">3.1.5 </span> Directionality</a>
<ol class="toc">
<li><a href="chapter3.html#presm.bidi.math"><span class="secno">3.1.5.1 </span> Overall Directionality of Mathematics Formulas</a>
</li>
<li><a href="chapter3.html#presm.bidi.token"><span class="secno">3.1.5.2 </span> Bidirectional Layout in Token Elements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.scriptlevel"><span class="secno">3.1.6 </span> Displaystyle and Scriptlevel</a>
</li>
<li><a href="chapter3.html#presm.linebreaking"><span class="secno">3.1.7 </span> Linebreaking of Expressions</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.7.1"><span class="secno">3.1.7.1 </span> Control of Linebreaks</a>
</li>
<li><a href="chapter3.html#presm.lbalgorithm"><span class="secno">3.1.7.2 </span> Automatic Linebreaking Algorithm (Informative)</a>
</li>
<li><a href="chapter3.html#presm.inline.lbalgorithm"><span class="secno">3.1.7.3 </span> Linebreaking Algorithm for Inline Expressions (Informative)</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.warnfinetuning"><span class="secno">3.1.8 </span> Warning about fine-tuning of presentation</a>
<ol class="toc">
<li><a href="chapter3.html#presm.warntweaking"><span class="secno">3.1.8.1 </span> Warning: non-portability of tweaking</a>
</li>
<li><a href="chapter3.html#presm.warnspacing"><span class="secno">3.1.8.2 </span> Warning: spacing should not be used to convey meaning</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.summary"><span class="secno">3.1.9 </span> Summary of Presentation Elements</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.1.9.1"><span class="secno">3.1.9.1 </span> Token Elements</a>
</li>
<li><a href="chapter3.html#id.3.1.9.2"><span class="secno">3.1.9.2 </span> General Layout Schemata</a>
</li>
<li><a href="chapter3.html#id.3.1.9.3"><span class="secno">3.1.9.3 </span> Script and Limit Schemata</a>
</li>
<li><a href="chapter3.html#id.3.1.9.4"><span class="secno">3.1.9.4 </span> Tables and Matrices</a>
</li>
<li><a href="chapter3.html#id.3.1.9.5"><span class="secno">3.1.9.5 </span> Elementary Math Layout</a>
</li>
<li><a href="chapter3.html#id.3.1.9.6"><span class="secno">3.1.9.6 </span> Enlivening Expressions</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.presatt"><span class="secno">3.1.10 </span> Mathematics style attributes common to presentation elements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.tokel"><span class="secno">3.2 </span> Token Elements</a>
<ol class="toc">
<li><a href="chapter3.html#presm.tokenchars"><span class="secno">3.2.1 </span>
Token Element Content Characters, <code><mglyph/></code></a>
<ol class="toc">
<li><a href="chapter3.html#presm.symbolchars"><span class="secno">3.2.1.1 </span> Alphanumeric symbol characters</a>
</li>
<li><a href="chapter3.html#presm.mglyph"><span class="secno">3.2.1.2 </span> Using images to represent
symbols <code><mglyph/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.1.2.1"><span class="secno">3.2.1.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.1.2.2"><span class="secno">3.2.1.2.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.2.1.2.3"><span class="secno">3.2.1.2.3 </span> Example</a>
</li>
<li><a href="chapter3.html#presm.mglyph.deprecated"><span class="secno">3.2.1.2.4 </span> Deprecated Attributes</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.commatt"><span class="secno">3.2.2 </span> Mathematics style attributes common to token elements</a>
<ol class="toc">
<li><a href="chapter3.html#presm.deprecatt"><span class="secno">3.2.2.1 </span> Deprecated style attributes on token elements</a>
</li>
<li><a href="chapter3.html#presm.html.insertion"><span class="secno">3.2.2.2 </span> Embedding HTML in MathML</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mi"><span class="secno">3.2.3 </span> Identifier <code><mi></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.3.1"><span class="secno">3.2.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.3.2"><span class="secno">3.2.3.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#presm.mi.examples"><span class="secno">3.2.3.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mn"><span class="secno">3.2.4 </span> Number <code><mn></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.4.1"><span class="secno">3.2.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.4.2"><span class="secno">3.2.4.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#presm.mn.examples"><span class="secno">3.2.4.3 </span> Examples</a>
</li>
<li><a href="chapter3.html#presm.not.single.mn"><span class="secno">3.2.4.4 </span> Numbers that should not be written
using <code><mn></code> alone</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mo"><span class="secno">3.2.5 </span> Operator, Fence, Separator or Accent
<code><mo></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.5.1"><span class="secno">3.2.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mo.attrs"><span class="secno">3.2.5.2 </span> Attributes</a>
<ol class="toc">
<li><a href="chapter3.html#presm.mo.dict.attrs"><span class="secno">3.2.5.2.1 </span> Dictionary-based attributes</a>
</li>
<li><a href="chapter3.html#presm.lbattrs"><span class="secno">3.2.5.2.2 </span> Linebreaking attributes</a>
</li>
<li><a href="chapter3.html#presm.lbindent.attrs"><span class="secno">3.2.5.2.3 </span> Indentation attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#id.3.2.5.3"><span class="secno">3.2.5.3 </span> Examples with ordinary operators</a>
</li>
<li><a href="chapter3.html#id.3.2.5.4"><span class="secno">3.2.5.4 </span> Examples with fences and separators</a>
</li>
<li><a href="chapter3.html#presm.invisibleops"><span class="secno">3.2.5.5 </span> Invisible operators</a>
</li>
<li><a href="chapter3.html#id.3.2.5.6"><span class="secno">3.2.5.6 </span> Names for other special operators</a>
</li>
<li><a href="chapter3.html#id.3.2.5.7"><span class="secno">3.2.5.7 </span> Detailed rendering rules for <code><mo></code> elements</a>
<ol class="toc">
<li><a href="chapter3.html#presm.opdict"><span class="secno">3.2.5.7.1 </span> The operator dictionary</a>
</li>
<li><a href="chapter3.html#presm.formdefval"><span class="secno">3.2.5.7.2 </span> Default value of the form attribute</a>
</li>
<li><a href="chapter3.html#id.3.2.5.7.3"><span class="secno">3.2.5.7.3 </span> Exception for embellished operators</a>
</li>
<li><a href="chapter3.html#id.3.2.5.7.4"><span class="secno">3.2.5.7.4 </span> Rationale for definition of embellished operators</a>
</li>
<li><a href="chapter3.html#presm.opspacing"><span class="secno">3.2.5.7.5 </span> Spacing around an operator</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.op.stretch"><span class="secno">3.2.5.8 </span> Stretching of operators, fences and accents</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.5.8.1"><span class="secno">3.2.5.8.1 </span> Example of stretchy attributes</a>
</li>
<li><a href="chapter3.html#id.3.2.5.8.2"><span class="secno">3.2.5.8.2 </span> Vertical Stretching Rules</a>
</li>
<li><a href="chapter3.html#presm.horiz.stretch.rules"><span class="secno">3.2.5.8.3 </span> Horizontal Stretching Rules</a>
</li>
<li><a href="chapter3.html#id.3.2.5.8.4"><span class="secno">3.2.5.8.4 </span> Rules Common to both Vertical and Horizontal Stretching</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mo.linebreaks"><span class="secno">3.2.5.9 </span> Examples of Linebreaking</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mtext"><span class="secno">3.2.6 </span> Text <code><mtext></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.6.1"><span class="secno">3.2.6.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.6.2"><span class="secno">3.2.6.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.2.6.3"><span class="secno">3.2.6.3 </span> Examples</a>
</li>
<li><a href="chapter3.html#presm.mixtextmath"><span class="secno">3.2.6.4 </span> Mixing text and mathematics</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mspace"><span class="secno">3.2.7 </span> Space <code><mspace/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.7.1"><span class="secno">3.2.7.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.7.2"><span class="secno">3.2.7.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.2.7.3"><span class="secno">3.2.7.3 </span> Examples</a>
</li>
<li><a href="chapter3.html#id.3.2.7.4"><span class="secno">3.2.7.4 </span> Definition of space-like elements</a>
</li>
<li><a href="chapter3.html#id.3.2.7.5"><span class="secno">3.2.7.5 </span> Legal grouping of space-like elements</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.ms"><span class="secno">3.2.8 </span> String Literal <code><ms></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.2.8.1"><span class="secno">3.2.8.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.2.8.2"><span class="secno">3.2.8.2 </span> Attributes</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.genlayout"><span class="secno">3.3 </span> General Layout Schemata</a>
<ol class="toc">
<li><a href="chapter3.html#presm.mrow"><span class="secno">3.3.1 </span> Horizontally Group Sub-Expressions
<code><mrow></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.1.1"><span class="secno">3.3.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.1.2"><span class="secno">3.3.1.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.1.3"><span class="secno">3.3.1.3 </span> Proper grouping of sub-expressions using <code><mrow></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.1.3.1"><span class="secno">3.3.1.3.1 </span> <code><mrow></code> of one argument</a>
</li>
<li><a href="chapter3.html#id.3.3.1.3.2"><span class="secno">3.3.1.3.2 </span> Precise rule for proper grouping</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#id.3.3.1.4"><span class="secno">3.3.1.4 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mfrac"><span class="secno">3.3.2 </span> Fractions <code><mfrac></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.2.1"><span class="secno">3.3.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.2.2"><span class="secno">3.3.2.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.2.3"><span class="secno">3.3.2.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mroot"><span class="secno">3.3.3 </span> Radicals <code><msqrt></code>, <code><mroot></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.3.1"><span class="secno">3.3.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.3.2"><span class="secno">3.3.3.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mstyle"><span class="secno">3.3.4 </span> Style Change <code><mstyle></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.4.1"><span class="secno">3.3.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mstyle.attrs"><span class="secno">3.3.4.2 </span> Attributes</a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.4.2.1"><span class="secno">3.3.4.2.1 </span> Deprecated Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#id.3.3.4.3"><span class="secno">3.3.4.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.merror"><span class="secno">3.3.5 </span> Error Message <code><merror></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.5.1"><span class="secno">3.3.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.5.2"><span class="secno">3.3.5.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.5.3"><span class="secno">3.3.5.3 </span> Example</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mpadded"><span class="secno">3.3.6 </span> Adjust Space Around Content
<code><mpadded></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.6.1"><span class="secno">3.3.6.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mpaddedatt"><span class="secno">3.3.6.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.6.3"><span class="secno">3.3.6.3 </span> Meanings of size and position attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mphantom"><span class="secno">3.3.7 </span> Making Sub-Expressions Invisible <code><mphantom></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.7.1"><span class="secno">3.3.7.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.7.2"><span class="secno">3.3.7.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.7.3"><span class="secno">3.3.7.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mfenced"><span class="secno">3.3.8 </span> Expression Inside Pair of Fences
<code><mfenced></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.8.1"><span class="secno">3.3.8.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.8.2"><span class="secno">3.3.8.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.8.3"><span class="secno">3.3.8.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.menclose"><span class="secno">3.3.9 </span> Enclose Expression Inside Notation
<code><menclose></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.3.9.1"><span class="secno">3.3.9.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.3.9.2"><span class="secno">3.3.9.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.3.9.3"><span class="secno">3.3.9.3 </span> Examples</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.scrlim"><span class="secno">3.4 </span> Script and Limit Schemata</a>
<ol class="toc">
<li><a href="chapter3.html#presm.msub"><span class="secno">3.4.1 </span> Subscript <code><msub></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.1.1"><span class="secno">3.4.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.1.2"><span class="secno">3.4.1.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.msup"><span class="secno">3.4.2 </span> Superscript <code><msup></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.2.1"><span class="secno">3.4.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.2.2"><span class="secno">3.4.2.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.msubsup"><span class="secno">3.4.3 </span> Subscript-superscript Pair <code><msubsup></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.3.1"><span class="secno">3.4.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.subsupatt"><span class="secno">3.4.3.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.3.3"><span class="secno">3.4.3.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.munder"><span class="secno">3.4.4 </span> Underscript <code><munder></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.4.1"><span class="secno">3.4.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.4.2"><span class="secno">3.4.4.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.4.3"><span class="secno">3.4.4.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mover"><span class="secno">3.4.5 </span> Overscript <code><mover></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.5.1"><span class="secno">3.4.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.5.2"><span class="secno">3.4.5.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.5.3"><span class="secno">3.4.5.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.munderover"><span class="secno">3.4.6 </span> Underscript-overscript Pair
<code><munderover></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.6.1"><span class="secno">3.4.6.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.6.2"><span class="secno">3.4.6.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.6.3"><span class="secno">3.4.6.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mmultiscripts"><span class="secno">3.4.7 </span> Prescripts and Tensor Indices
<code><mmultiscripts></code>,
<code><mprescripts/></code>,
<code><none/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.4.7.1"><span class="secno">3.4.7.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.4.7.2"><span class="secno">3.4.7.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.4.7.3"><span class="secno">3.4.7.3 </span> Examples</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.tabmat"><span class="secno">3.5 </span> Tabular Math</a>
<ol class="toc">
<li><a href="chapter3.html#presm.mtable"><span class="secno">3.5.1 </span> Table or Matrix
<code><mtable></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.1.1"><span class="secno">3.5.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mtable.attrs"><span class="secno">3.5.1.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.1.3"><span class="secno">3.5.1.3 </span> Examples</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mtr"><span class="secno">3.5.2 </span> Row in Table or Matrix <code><mtr></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.2.1"><span class="secno">3.5.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.5.2.2"><span class="secno">3.5.2.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mlabeledtr"><span class="secno">3.5.3 </span> Labeled Row in Table or Matrix
<code><mlabeledtr></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.3.1"><span class="secno">3.5.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.5.3.2"><span class="secno">3.5.3.2 </span> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.3.3"><span class="secno">3.5.3.3 </span> Equation Numbering</a>
</li>
<li><a href="chapter3.html#id.3.5.3.4"><span class="secno">3.5.3.4 </span> Example</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mtd"><span class="secno">3.5.4 </span> Entry in Table or Matrix <code><mtd></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.4.1"><span class="secno">3.5.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mtdatts"><span class="secno">3.5.4.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.malign"><span class="secno">3.5.5 </span> Alignment Markers
<code><maligngroup/></code>, <code><malignmark/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.5.5.1"><span class="secno">3.5.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.5.5.2"><span class="secno">3.5.5.2 </span> Specifying alignment groups</a>
</li>
<li><a href="chapter3.html#id.3.5.5.3"><span class="secno">3.5.5.3 </span> Table cells that are not divided into alignment groups</a>
</li>
<li><a href="chapter3.html#id.3.5.5.4"><span class="secno">3.5.5.4 </span> Specifying alignment points using <code><malignmark/></code></a>
</li>
<li><a href="chapter3.html#id.3.5.5.5"><span class="secno">3.5.5.5 </span> <code><malignmark/></code> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.5.6"><span class="secno">3.5.5.6 </span> <code><maligngroup/></code> Attributes</a>
</li>
<li><a href="chapter3.html#id.3.5.5.7"><span class="secno">3.5.5.7 </span> Inheritance of groupalign values</a>
</li>
<li><a href="chapter3.html#id.3.5.5.8"><span class="secno">3.5.5.8 </span> MathML representation of an alignment example</a>
</li>
<li><a href="chapter3.html#id.3.5.5.9"><span class="secno">3.5.5.9 </span> Further details of alignment elements</a>
</li>
<li><a href="chapter3.html#id.3.5.5.10"><span class="secno">3.5.5.10 </span> A simple alignment algorithm</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.elementary"><span class="secno">3.6 </span> Elementary Math</a>
<ol class="toc">
<li><a href="chapter3.html#presm.mstack"><span class="secno">3.6.1 </span> Stacks of Characters <code><mstack></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.1.1"><span class="secno">3.6.1.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.6.1.2"><span class="secno">3.6.1.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mlongdiv"><span class="secno">3.6.2 </span> Long Division <code><mlongdiv></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.2.1"><span class="secno">3.6.2.1 </span> Description</a>
</li>
<li><a href="chapter3.html#presm.mlongdiv.attrs"><span class="secno">3.6.2.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.msgroup"><span class="secno">3.6.3 </span> Group Rows with Similiar Positions <code><msgroup></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.3.1"><span class="secno">3.6.3.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.6.3.2"><span class="secno">3.6.3.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.msrow"><span class="secno">3.6.4 </span> Rows in Elementary Math <code><msrow></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.4.1"><span class="secno">3.6.4.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.6.4.2"><span class="secno">3.6.4.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mscarries"><span class="secno">3.6.5 </span> Carries, Borrows, and Crossouts <code><mscarries></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.5.1"><span class="secno">3.6.5.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.6.5.2"><span class="secno">3.6.5.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.mscarry"><span class="secno">3.6.6 </span> A Single Carry <code><mscarry></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.6.1"><span class="secno">3.6.6.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.6.6.2"><span class="secno">3.6.6.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.msline"><span class="secno">3.6.7 </span> Horizontal Line <code><msline/></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.6.7.1"><span class="secno">3.6.7.1 </span> Description</a>
</li>
<li><a href="chapter3.html#id.3.6.7.2"><span class="secno">3.6.7.2 </span> Attributes</a>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.elemmath.examples"><span class="secno">3.6.8 </span> Elementary Math Examples</a>
<ol class="toc">
<li><a href="chapter3.html#presm.addsub"><span class="secno">3.6.8.1 </span> Addition and Subtraction</a>
</li>
<li><a href="chapter3.html#presm.mult"><span class="secno">3.6.8.2 </span> Multiplication</a>
</li>
<li><a href="chapter3.html#presm.mlongdiv.ex"><span class="secno">3.6.8.3 </span> Long Division</a>
</li>
<li><a href="chapter3.html#presm.repeatdec"><span class="secno">3.6.8.4 </span> Repeating decimal</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.enliven"><span class="secno">3.7 </span> Enlivening Expressions</a>
<ol class="toc">
<li><a href="chapter3.html#presm.maction"><span class="secno">3.7.1 </span> Bind Action to Sub-Expression
<code><maction></code></a>
<ol class="toc">
<li><a href="chapter3.html#id.3.7.1.1"><span class="secno">3.7.1.1 </span> Attributes</a>
</li>
</ol>
</li>
</ol>
</li>
<li><a href="chapter3.html#presm.semantics"><span class="secno">3.8 </span> Semantics and Presentation</a>
</li>
</ol>
</li>
</ol>
</nav>
<div class="div1">
<div class="div2">
<h2><a id="presm.intro"></a>3.1 Introduction</h2>
<p>This chapter specifies the "presentation" elements of
MathML, which can be used to describe the layout structure of mathematical
notation.</p>
<div class="div3">
<h3><a id="id.3.1.1"></a>3.1.1 What Presentation Elements Represent</h3>
<p>Presentation elements correspond to the "constructors"
of traditional mathematical notation — that is, to the basic
kinds of symbols and expression-building structures out of which any
particular piece of traditional mathematical notation is built.
Because of the importance of traditional visual notation, the
descriptions of the notational constructs the elements represent are
usually given here in visual terms. However, the elements are
medium-independent in the sense that they have been designed to
contain enough information for good spoken renderings as well. Some
attributes of these elements may make sense only for visual media, but
most attributes can be treated in an analogous way in audio as well
(for example, by a correspondence between time duration and horizontal
extent).</p>
<p>MathML presentation elements only suggest (i.e. do not require)
specific ways of rendering in order to allow for medium-dependent
rendering and for individual preferences of style. This specification
describes suggested visual rendering rules in some detail, but a
particular MathML renderer is free to use its own rules as long as its
renderings are intelligible.</p>
<p>The presentation elements are meant to express the syntactic
structure of mathematical notation in much the same way as titles, sections,
and paragraphs capture the higher-level syntactic structure of a
textual document. Because of this, a single row of identifiers and operators
will often be represented by multiple nested <code>mrow</code> elements rather than
a single <code>mrow</code>. For example, "<var>x</var> + <var>a</var> /
<var>b</var>" typically is represented as:
</p>
<pre class="mathml">
<mrow>
<mi> x </mi>
<mo> + </mo>
<mrow>
<mi> a </mi>
<mo> / </mo>
<mi> b </mi>
</mrow>
</mrow>
</pre>
<p>Similarly, superscripts are attached to the full expression constituting
their base rather than to the just preceding character. This
structure permits better-quality rendering of mathematics, especially when
details of the rendering environment, such as display widths, are not
known ahead of time to the document author. It also greatly eases automatic
interpretation of the represented mathematical structures.</p>
<p>Certain characters are used
to name identifiers or operators that in traditional notation render the
same as other symbols or usually rendered invisibly. For example, the entities
<code>&DifferentialD;</code>, <code>&ExponentialE;</code>, and
<code>&ImaginaryI;</code> denote notational symbols semantically distinct from visually
identical letters used as simple variables.
Likewise, the entities
<code>&InvisibleTimes;</code>,
<code>&ApplyFunction;</code>,
<code>&InvisibleComma;</code> and the character U+2064
(INVISIBLE PLUS) usually render invisibly but represent significant information.
These entities have distinct spoken renderings, may influence visual linebreaking
and spacing,
and may effect the evaluation or meaning of particular expressions.
Accordingly, authors should use these entities wherever they are applicable.
For instance, the expression represented visually as
"<var>f</var>(<var>x</var>)" would usually be spoken in English as
"<var>f</var> of <var>x</var>" rather than just
"<var>f</var> <var>x</var>". MathML conveys this meaning by using
the <code>&ApplyFunction;</code> operator after the
"<var>f</var>", which, in this case, can be aurally rendered as
"of".</p>
<p>The complete list of MathML entities is described in <a href="appendixg.html#Entities">[Entities]</a>.</p>
</div>
<div class="div3">
<h3><a id="id.3.1.2"></a>3.1.2 Terminology Used In This Chapter</h3>
<p>It is strongly recommended that, before reading the present
chapter, one read <a href="chapter2.html#fund.syntax">Section 2.1 MathML Syntax and Grammar</a> on MathML syntax and
grammar, which contains important information on MathML notations and
conventions. In particular, in this chapter it is assumed that the
reader has an understanding of basic XML terminology described in
<a href="chapter2.html#fund.xmlsyntax">Section 2.1.3 Children versus Arguments</a>, and the attribute value notations and
conventions described in <a href="chapter2.html#fund.attval">Section 2.1.5 MathML Attribute Values</a>.</p>
<p>The remainder of this section introduces MathML-specific
terminology and conventions used in this chapter.</p>
<div class="div4">
<h4><a id="id.3.1.2.1"></a>3.1.2.1 Types of presentation elements</h4>
<p>The presentation elements are divided into two classes.
<a href="#presm.tokel"><em>Token elements</em></a>
represent individual symbols, names, numbers, labels, etc.
<em>Layout schemata</em> build expressions out of parts and can have
only elements as content (except for whitespace, which they ignore).
These are subdivided into
<a href="#presm.genlayout">General Layout</a>,
<a href="#presm.scrlim">Script and Limit</a>,
<a href="#presm.tabmat">Tabular Math</a> and
<a href="#presm.elementary">Elementary Math</a> schemata.
There
are also a few empty elements used only in conjunction with certain layout
schemata.</p>
<p>All individual "symbols" in a mathematical expression should be
represented by MathML token elements. The primary MathML token element
types are identifiers (e.g. variables or function names), numbers, and
operators (including fences, such as parentheses, and separators, such
as commas). There are also token elements used to represent text or
whitespace that has more aesthetic than mathematical significance
and other elements representing "string literals" for compatibility with
computer algebra systems. Note that although a token element
represents a single meaningful "symbol" (name, number, label,
mathematical symbol, etc.), such symbols may be comprised of more than
one character. For example <code>sin</code> and <code>24</code> are
represented by the single tokens <code><mi>sin</mi></code>
and <code><mn>24</mn></code> respectively.</p>
<p>In traditional mathematical notation, expressions are recursively
constructed out of smaller expressions, and ultimately out of single
symbols, with the parts grouped and positioned using one of a small
set of notational structures, which can be thought of as "expression
constructors". In MathML, expressions are constructed in the same way,
with the layout schemata playing the role of the expression
constructors. The layout schemata specify the way in which
sub-expressions are built into larger expressions. The terminology
derives from the fact that each layout schema corresponds to a
different way of "laying out" its sub-expressions to form a larger
expression in traditional mathematical typesetting.</p>
</div>
<div class="div4">
<h4><a id="id.3.1.2.2"></a>3.1.2.2 Terminology for other classes of elements and their relationships</h4>
<p>The terminology used in this chapter for special classes of elements, and for
relationships between elements, is as follows: The <em>presentation elements</em> are
the MathML elements defined in this chapter.
These elements are listed in <a href="#presm.summary">Section 3.1.9 Summary of Presentation Elements</a>.
The <em>content elements</em> are the MathML elements defined
in <a href="chapter4.html">Chapter 4 Content Markup</a>.</p>
<p>A MathML <em>expression</em> is a single instance of any of the
presentation elements with the exception of the empty elements
<code>none</code> or <code>mprescripts</code>,
or is a single instance of any of the content elements which are allowed as
content of presentation elements (described in <a href="chapter5.html#mixing.cminpm">Section 5.3.2 Content Markup in Presentation Markup</a>).
A <em>sub-expression</em> of an expression
<var>E</var> is any MathML expression that is part of the content of
<var>E</var>, whether <em>directly</em> or <em>indirectly</em>,
i.e. whether it is a "child" of <var>E</var> or not.</p>
<p>Since layout schemata attach special meaning to the number and/or
positions of their children, a child of a layout schema is also called
an <em>argument</em> of that element. As a consequence of the
above definitions, the content of a layout schema consists exactly of
a sequence of zero or more elements that are its
arguments. </p>
</div>
</div>
<div class="div3">
<h3><a id="presm.reqarg"></a>3.1.3 Required Arguments</h3>
<p>Many of the elements described herein require a specific number of
arguments (always 1, 2, or 3). In the detailed descriptions of
element syntax given below, the number of required arguments is
implicitly indicated by giving names for the arguments at various
positions. A few elements have additional requirements on the number
or type of arguments, which are described with the individual
element. For example, some elements accept sequences of zero or more
arguments — that is, they are allowed to occur with no arguments
at all.</p>
<p>Note that MathML elements encoding rendered space <em>do</em>
count as arguments of the elements in which they appear.
See <a href="#presm.mspace">Section 3.2.7 Space <code><mspace/></code></a> for a discussion of the proper use of such
space-like elements.</p>
<div class="div4">
<h4><a id="presm.inferredmrow"></a>3.1.3.1 Inferred <code><mrow></code>s</h4>
<p>The elements listed in the following table as requiring 1*
argument (<code>msqrt</code>, <code>mstyle</code>, <code>merror</code>,
<code>mpadded</code>, <code>mphantom</code>, <code>menclose</code>,
<code>mtd</code>, <code>mscarry</code>,
and <code>math</code>)
conceptually accept a single argument,
but actually accept any number of children.
If the number of children is 0 or is more than 1, they treat their contents
as a single <em>inferred</em> <code>mrow</code> formed from all their children,
and treat this <code>mrow</code> as the argument.
</p>
<p>For example,
</p>
<pre class="mathml-fragment">
<mtd>
</mtd>
</pre>
<p>
is treated as if it were
</p>
<pre class="mathml-fragment">
<mtd>
<mrow>
</mrow>
</mtd>
</pre>
<p>
and
</p>
<pre class="mathml">
<msqrt>