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appendixc.xml
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<?xml version="1.0" encoding="utf-8"?><!DOCTYPE html
SYSTEM "mathml.dtd">
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head>
<meta http-equiv="Content-Type" content="application/xhtml+xml; charset=utf-8" />
<title>Operator Dictionary</title>
<meta content="width=device-width, initial-scale=1, shrink-to-fit=no" name="viewport" />
<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;
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border-bottom-style: double;
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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;
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.attention { border-style: solid;
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.attribute-Name { background: #F9F5C0; }
.method-Name { background: #C0C0F9; }
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</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><script type="text/javascript" src="sorttable.js"></script></head>
<body>
<h1><a id="oper-dict"></a>C Operator Dictionary (Non-Normative)</h1>
<!-- TOP NAVIGATION BAR -->
<nav id="toc">
<ol class="toc">
<li><a href="Overview.xml"><span class="secno"></span> Mathematical Markup Language (MathML) Version 4.0</a></li>
<li><a href="appendixb.xml"><span class="secno">B </span> Media Types Registrations</a></li>
<li><a href="appendixd.xml"><span class="secno">D </span> Glossary</a></li>
<li><a href="appendixc.xml"><span class="secno">C </span> Operator Dictionary (Non-Normative)</a>
<ol class="toc">
<li><a href="appendixc.xml#oper-dict.index"><span class="secno">C.1 </span> Indexing of the operator dictionary</a>
</li>
<li><a href="appendixc.xml#oper-dict.format"><span class="secno">C.2 </span> Format of operator dictionary entries</a>
</li>
<li><a href="appendixc.xml#oper-dict.space"><span class="secno">C.3 </span> Notes on lspace and
rspace attributes</a>
</li>
<li><a href="appendixc.xml#oper-dict.entries-table"><span class="secno">C.4 </span> Operator dictionary entries</a>
</li>
</ol>
</li>
</ol>
</nav>
<div class="div1">
<p>The following table gives the suggested dictionary of rendering
properties for operators, fences, separators, and accents in MathML, all of
which are represented by <code>mo</code> elements. For brevity,
all such elements will be called simply "operators" in this
Appendix.</p>
<div class="div2">
<h2><a id="oper-dict.index"></a>C.1 Indexing of the operator dictionary</h2>
<p>Note that the dictionary is indexed not just by the element
content, but by the element content and <code>form</code> attribute
value, together. Operators with more than one possible form have more
than one entry. The MathML specification describes how the renderer
chooses ("infers") which form to use when no <code>form</code>
attribute is given; see <a href="chapter3.xml#presm.formdefval">Section 3.2.5.7.2 Default value of the form attribute</a>. </p>
<p>Having made that choice, or with the <code>form</code> attribute
explicitly specified in the <code><mo></code> element's start
tag, the MathML renderer uses the remaining attributes from the
dictionary entry for the appropriate single form of that operator,
ignoring the entries for the other possible forms.</p>
<p>In the table below, all non-ASCII characters are represented by
XML-style hexadecimal numeric character references.
The choice of markup (character data, numeric character reference or named entity
reference)
for a given character in MathML has no effect on its rendering.</p>
</div>
<div class="div2">
<h2><a id="oper-dict.format"></a>C.2 Format of operator dictionary entries</h2>
<p>Each row of the table is indexed as described above by the both the character
(given by code point and Unicode Name) and the value of the form attribute.
The fourth column gives the <b>priority</b> which as described in <a href="chapter3.xml#presm.mrow">Section 3.3.1 Horizontally Group Sub-Expressions
<code><mrow></code></a>),
is significant for the proper grouping of sub-expressions using <code><mrow></code>.
The rule described
there is especially relevant to the automatic generation of MathML by
conversion from other formats for displayed mathematics, such as T<span style="vertical-align:-.4ex; margin-left:-.15ex; margin-right:-.15ex">E</span>X,
which do not always specify how sub-expressions nest.</p>
<p>The length valued attributes such as <code>lspace</code> are given explicitly in the following columns.
Boolean valued attributes such as <code>stretchy</code> are specified together in the final <b>Properties</b>
column by listing the attribute name if its value should be set to <b>true</b> by the dictionary.
Finally some rarely used non-boolean properties are shown in the Properties column
together with a value,
for example <code>linebreakstyle</code>=after.</p>
<p>Any attribute not listed for some entry has its default value,
which is given in parentheses in the table of attributes in <a href="chapter3.xml#presm.mo">Section 3.2.5 Operator, Fence, Separator or Accent
<code><mo></code></a>.</p>
<table>
<tbody>
<tr>
<th>(</th>
<th>(</th>
<th class="uname">left parenthesis</th>
<th>prefix</th>
<td>1</td>
<td>0</td>
<td>0</td>
<td></td>
<td>fence, stretchy</td>
</tr>
</tbody>
</table>
<p>could be expressed as an <code>mo</code> element by:
</p>
<table>
<tr>
<td>
<pre class="mathml">
<mo form="prefix" fence="true" stretchy="true" lspace="0em" rspace="0em"> ( </mo>
</pre>
</td>
<td class="mathml-render" valign="middle"><math xmlns="http://www.w3.org/1998/Math/MathML">
<mo form="prefix" fence="true" stretchy="true" lspace="0em" rspace="0em"> ( </mo>
</math></td>
</tr>
</table>
<p>
(note the whitespace added around the content for readability;
Such whitespace will be ignored by a MathML system,
as described in <a href="chapter2.xml#fund.collapse">Section 2.1.7 Collapsing Whitespace in Input</a>.</p>
<p>This entry means that, for MathML renderers which use this
suggested operator dictionary, giving the element
<code><mo form="prefix"> ( </mo></code>
alone, or simply
<code><mo> ( </mo></code>
in a position for which
<code>form="prefix"</code>
would be inferred (see below), is equivalent to giving the element
with all attributes as shown above.</p>
<p>In some versions of this specification,
the rows of the table may be reordered by clicking on a heading in the top row,
to cause the table to be ordered by that column.</p>
</div>
<div class="div2">
<h2><a id="oper-dict.space"></a>C.3 Notes on <code>lspace</code> and
<code>rspace</code> attributes</h2>
<p>The values for <code>lspace</code> and <code>rspace</code> given here range from 0 to 7,
they are given numerically in order to save space in the table,
the values should be taken as referring to the named mathematical spaces, as follows.</p>
<table>
<thead>
<tr>
<th>Table Entry</th>
<th>Named Space</th>
<th>Default Length</th>
</tr>
</thead>
<tbody>
<tr>
<td>0</td>
<td></td>
<td>0 em</td>
</tr>
<tr>
<td>1</td>
<td>veryverythinmathspace</td>
<td>1/18 em</td>
</tr>
<tr>
<td>2</td>
<td>verythinmathspace</td>
<td>2/18 em</td>
</tr>
<tr>
<td>3</td>
<td>thinmathspace</td>
<td>3/18 em</td>
</tr>
<tr>
<td>4</td>
<td>mediummathspace</td>
<td>4/18 em</td>
</tr>
<tr>
<td>5</td>
<td>thickmathspace</td>
<td>5/18 em</td>
</tr>
<tr>
<td>6</td>
<td>verythickmathspace</td>
<td>6/18 em</td>
</tr>
<tr>
<td>7</td>
<td>veryverythickmathspace</td>
<td>7/18 em</td>
</tr>
</tbody>
</table>
<p>
For the invisible operators whose content is <code>&InvisibleTimes;</code> or <code>&ApplyFunction;</code>,
it is suggested that MathML renderers choose spacing in a context-sensitive
way (which is an exception to the static values given in the following
table). For <code><mo>&ApplyFunction;</mo></code>, the total
spacing ("lspace"+"rspace") in
expressions such as "sin <var>x</var>" (where the right operand
doesn't start with a fence) should be greater than zero; for
<code><mo>&InvisibleTimes;</mo></code>, the total spacing
should be greater than zero when both operands (or the nearest tokens on
either side, if on the baseline) are identifiers displayed in a non-slanted
font (i.e.. under the suggested rules, when both operands are
multi-character identifiers).</p>
<p>Some renderers may wish to use no spacing for most operators
appearing in scripts (i.e. when <code>scriptlevel</code> is greater
than 0; see <a href="chapter3.xml#presm.mstyle">Section 3.3.4 Style Change <code><mstyle></code></a>), as is the case in T<span style="vertical-align:-.4ex; margin-left:-.15ex; margin-right:-.15ex">E</span>X.</p>
</div>
<div class="div2">
<h2><a id="oper-dict.entries-table"></a>C.4 Operator dictionary entries</h2>
<table class="sortable">
<thead>
<tr>
<th>Character</th>
<th>Glyph</th>
<th>Name</th>
<th>form</th>
<th>priority</th>
<th>lspace</th>
<th>rspace</th>
<th>Properties</th>
</tr>
</thead>
<tbody>
<tr>
<th abbr="8216">&#x2018;</th>
<th>‘</th>
<th class="uname">left single quotation mark</th>
<th>prefix</th>
<td>10</td>
<td>0</td>
<td>0</td>
<td>fence</td>
</tr>
<tr>
<th abbr="8217">&#x2019;</th>
<th>’</th>
<th class="uname">right single quotation mark</th>
<th>postfix</th>
<td>10</td>
<td>0</td>
<td>0</td>
<td>fence</td>
</tr>
<tr>
<th abbr="8220">&#x201C;</th>
<th>“</th>
<th class="uname">left double quotation mark</th>
<th>prefix</th>
<td>10</td>
<td>0</td>
<td>0</td>
<td>fence</td>
</tr>
<tr>
<th abbr="8221">&#x201D;</th>
<th>”</th>
<th class="uname">right double quotation mark</th>
<th>postfix</th>
<td>10</td>
<td>0</td>
<td>0</td>
<td>fence</td>
</tr>
<tr>
<th abbr="40">(</th>
<th>(</th>
<th class="uname">left parenthesis</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="41">)</th>
<th>)</th>
<th class="uname">right parenthesis</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="91">[</th>
<th>[</th>
<th class="uname">left square bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="93">]</th>
<th>]</th>
<th class="uname">right square bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="123">{</th>
<th>{</th>
<th class="uname">left curly bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="124">|</th>
<th>|</th>
<th class="uname">vertical line</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="124">|</th>
<th>|</th>
<th class="uname">vertical line</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="124">||</th>
<th>||</th>
<th class="uname">multiple character operator: ||</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="124">||</th>
<th>||</th>
<th class="uname">multiple character operator: ||</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="124">|||</th>
<th>|||</th>
<th class="uname">multiple character operator: |||</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="124">|||</th>
<th>|||</th>
<th class="uname">multiple character operator: |||</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="125">}</th>
<th>}</th>
<th class="uname">right curly bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="8214">&#x2016;</th>
<th>‖</th>
<th class="uname">double vertical line</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy</td>
</tr>
<tr>
<th abbr="8214">&#x2016;</th>
<th>‖</th>
<th class="uname">double vertical line</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy</td>
</tr>
<tr>
<th abbr="8968">&#x2308;</th>
<th>⌈</th>
<th class="uname">left ceiling</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="8969">&#x2309;</th>
<th>⌉</th>
<th class="uname">right ceiling</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="8970">&#x230A;</th>
<th>⌊</th>
<th class="uname">left floor</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="8971">&#x230B;</th>
<th>⌋</th>
<th class="uname">right floor</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="9001">&#x2329;</th>
<th>〈</th>
<th class="uname">left-pointing angle bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="9002">&#x232A;</th>
<th>〉</th>
<th class="uname">right-pointing angle bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10098">&#x2772;</th>
<th>❲</th>
<th class="uname">light left tortoise shell bracket ornament</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10099">&#x2773;</th>
<th>❳</th>
<th class="uname">light right tortoise shell bracket ornament</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10214">&#x27E6;</th>
<th>⟦</th>
<th class="uname">mathematical left white square bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10215">&#x27E7;</th>
<th>⟧</th>
<th class="uname">mathematical right white square bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10216">&#x27E8;</th>
<th>⟨</th>
<th class="uname">mathematical left angle bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10217">&#x27E9;</th>
<th>⟩</th>
<th class="uname">mathematical right angle bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10218">&#x27EA;</th>
<th>⟪</th>
<th class="uname">mathematical left double angle bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10219">&#x27EB;</th>
<th>⟫</th>
<th class="uname">mathematical right double angle bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10220">&#x27EC;</th>
<th>⟬</th>
<th class="uname">mathematical left white tortoise shell bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10221">&#x27ED;</th>
<th>⟭</th>
<th class="uname">mathematical right white tortoise shell bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10222">&#x27EE;</th>
<th>⟮</th>
<th class="uname">mathematical left flattened parenthesis</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10223">&#x27EF;</th>
<th>⟯</th>
<th class="uname">mathematical right flattened parenthesis</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10624">&#x2980;</th>
<th>⦀</th>
<th class="uname">triple vertical bar delimiter</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy</td>
</tr>
<tr>
<th abbr="10624">&#x2980;</th>
<th>⦀</th>
<th class="uname">triple vertical bar delimiter</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy</td>
</tr>
<tr>
<th abbr="10627">&#x2983;</th>
<th>⦃</th>
<th class="uname">left white curly bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10628">&#x2984;</th>
<th>⦄</th>
<th class="uname">right white curly bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10629">&#x2985;</th>
<th>⦅</th>
<th class="uname">left white parenthesis</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10630">&#x2986;</th>
<th>⦆</th>
<th class="uname">right white parenthesis</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10631">&#x2987;</th>
<th>⦇</th>
<th class="uname">z notation left image bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10632">&#x2988;</th>
<th>⦈</th>
<th class="uname">z notation right image bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10633">&#x2989;</th>
<th>⦉</th>
<th class="uname">z notation left binding bracket</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10634">&#x298A;</th>
<th>⦊</th>
<th class="uname">z notation right binding bracket</th>
<th>postfix</th>
<td>20</td>
<td>0</td>
<td>0</td>
<td>fence, stretchy, symmetric</td>
</tr>
<tr>
<th abbr="10635">&#x298B;</th>
<th>⦋</th>
<th class="uname">left square bracket with underbar</th>
<th>prefix</th>
<td>20</td>
<td>0</td>
<td>0</td>