<?xml version="1.0" encoding="utf-8" ?> <!-- for emacs: -*- coding: utf-8 -*- -->
<!-- Apache may like this line in the file .htaccess: AddCharset utf-8 .html -->
<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.1 plus MathML 2.0 plus SVG 1.1//EN" "http://www.w3.org/2002/04/xhtml-math-svg/xhtml-math-svg-flat.dtd" >
<html xmlns="http://www.w3.org/1999/xhtml" xml:lang="en">
<head><title>operators</title>
<link rel="stylesheet" type="text/css" href="../../../../Macaulay2/Style/doc.css"/>
</head>
<body>
<table class="buttons">
<tr>
<td><div><a href="__eq.html">next</a> | <a href="___Mutable__Hash__Table.html">previous</a> | <a href="_conditional_spexecution.html">forward</a> | <a href="_hash_sptables.html">backward</a> | <a href="___The_sp__Macaulay2_splanguage.html">up</a> | <a href="index.html">top</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> | <a href="http://www.math.uiuc.edu/Macaulay2/">Macaulay2 web site</a></div>
</td>
</tr>
</table>
<div><a href="index.html" title="">Macaulay2Doc</a> > <a href="___The_sp__Macaulay2_splanguage.html" title="">The Macaulay2 language</a> > <a href="_operators.html" title="">operators</a></div>
<hr/>
<div><h1>operators</h1>
<div><p>Here we present all of the unary and binary operators in the language. They are members of the class <a href="___Keyword.html" title="the class of all keywords">Keyword</a>. The binary operators are <a href="_!_eq.html" title="inequality">!=</a> <a href="__sh.html" title="length, or access to elements">#</a> <a href="__sh_qu.html" title="check for presence of elements">#?</a> <a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> <a href="__am.html" title="a binary operator">&</a> <a href="__st.html" title="a binary operator, usually used for multiplication">*</a> <a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a> <a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> <a href="__pl_pl.html" title="a binary operator, usually used for direct sum">++</a> <a href="__cm.html" title="the comma, used for separating entries in a list or sequence">,</a> <a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> <a href="_-_gt.html" title="make a function">-></a> <a href="_..html" title="access to elements whose key is a symbol">.</a> <a href="_...html" title="a binary operator, used for sequences of consecutive items">..</a> <a href="_.._lt.html" title="a binary operator, used for sequences of consecutive items, not including the endpoint">..<</a> <a href="_._qu.html" title="check for presence of elements whose key is a symbol">.?</a> <a href="__sl.html" title="a binary operator, usually used for division">/</a> <a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> <a href="__co.html" title="a binary operator, uses include repetition; ideal quotients">:</a> <a href="__co_eq.html" title="assignment of method or new local variable">:=</a> <a href="__se.html" title="statement separator">;</a> <a href="__lt.html" title="less than"><</a> <a href="__lt-.html" title="assignment with left side evaluated"><-</a> <a href="__lt_lt.html" title="a binary operator (file output, ...)"><<</a> <a href="__lt_eq.html" title="less than or equal"><=</a> <a href="__lt_eq_eq.html" title="a unary and binary operator"><==</a> <a href="__lt_eq_eq_eq.html" title="a unary and binary operator"><===</a> <a href="__lt_eq_eq_gt.html" title="a binary operator"><==></a> <a href="__eq.html" title="assignment">=</a> <a href="__eq!_eq.html" title="strict inequality">=!=</a> <a href="__eq_eq.html" title="equality">==</a> <a href="___Thing_sp_eq_eq_eq_sp__Thing.html" title="strict equality">===</a> <a href="__eq_eq_eq_gt.html" title="a binary operator">===></a> <a href="__eq_eq_gt.html" title="a binary operator">==></a> <a href="__eq_gt.html" title="produce an Option">=></a> <a href="__gt.html" title="greater than">></a> <a href="__gt_eq.html" title="greater than or equal">>=</a> <a href="__gt_gt.html" title="a binary operator, uses include bit shifting, or attaching optional inputs to functions">>></a> <a href="__qu.html" title="comparison operator">?</a> <a href="__at.html" title="a binary operator">@</a> <a href="__at_at.html" title="a binary operator">@@</a> <a href="__bs.html" title="a binary operator">\</a> <a href="__bs_bs.html" title="a binary operator">\\</a> <a href="_^.html" title="a binary operator, usually used for powers">^</a> <a href="_^_st_st.html" title="a binary operator, usually used for tensor or Cartesian power">^**</a> <a href="_^^.html" title="a binary operator">^^</a> <a href="__us.html" title="a binary operator, used for subscripting and access to elements">_</a> <a href="__vb.html" title="a binary operator, often used for horizontal concatenation">|</a> <a href="__vb-.html" title="a binary operator">|-</a> <a href="__vb_vb.html" title="a binary operator, often used for vertical concatenation">||</a> <a href="_and.html" title="conjunction">and</a> <a href="_or.html" title="disjunction">or</a> <a href="___S__P__A__C__E.html" title="blank operator; often used for function application, making polynomial rings">SPACE</a> , the unary prefix operators are <a href="__sh.html" title="length, or access to elements">#</a> <a href="__st.html" title="a binary operator, usually used for multiplication">*</a> <a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> <a href="__cm.html" title="the comma, used for separating entries in a list or sequence">,</a> <a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> <a href="__lt.html" title="less than"><</a> <a href="__lt_lt.html" title="a binary operator (file output, ...)"><<</a> <a href="__lt_eq.html" title="less than or equal"><=</a> <a href="__lt_eq_eq.html" title="a unary and binary operator"><==</a> <a href="__lt_eq_eq_eq.html" title="a unary and binary operator"><===</a> <a href="__gt.html" title="greater than">></a> <a href="__gt_eq.html" title="greater than or equal">>=</a> <a href="__qu.html" title="comparison operator">?</a> <a href="__vb-.html" title="a binary operator">|-</a> <a href="_not.html" title="negation">not</a> , and the unary postfix operators are <a href="_!.html" title="factorial">!</a> <a href="__lp_st_rp.html" title="a unary postfix operator, used for indicating a graded object">(*)</a> <a href="__cm.html" title="the comma, used for separating entries in a list or sequence">,</a> <a href="__se.html" title="statement separator">;</a> <a href="_^_st.html" title="a unary postfix operator, used for indicating pullback maps">^*</a> <a href="__us_st.html" title="a unary postfix operator, used for indicating pushforward maps">_*</a> <a href="_~.html" title="a unary postfix operator">~</a> .</p>
<p>Of those, the ones for which users may install new methods are the binary operators <a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> <a href="__am.html" title="a binary operator">&</a> <a href="__st.html" title="a binary operator, usually used for multiplication">*</a> <a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a> <a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> <a href="__pl_pl.html" title="a binary operator, usually used for direct sum">++</a> <a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> <a href="_...html" title="a binary operator, used for sequences of consecutive items">..</a> <a href="_.._lt.html" title="a binary operator, used for sequences of consecutive items, not including the endpoint">..<</a> <a href="__sl.html" title="a binary operator, usually used for division">/</a> <a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> <a href="__co.html" title="a binary operator, uses include repetition; ideal quotients">:</a> <a href="__lt_lt.html" title="a binary operator (file output, ...)"><<</a> <a href="__lt_eq_eq.html" title="a unary and binary operator"><==</a> <a href="__lt_eq_eq_eq.html" title="a unary and binary operator"><===</a> <a href="__lt_eq_eq_gt.html" title="a binary operator"><==></a> <a href="__eq_eq.html" title="equality">==</a> <a href="__eq_eq_eq_gt.html" title="a binary operator">===></a> <a href="__eq_eq_gt.html" title="a binary operator">==></a> <a href="__gt_gt.html" title="a binary operator, uses include bit shifting, or attaching optional inputs to functions">>></a> <a href="__qu.html" title="comparison operator">?</a> <a href="__at.html" title="a binary operator">@</a> <a href="__at_at.html" title="a binary operator">@@</a> <a href="__bs.html" title="a binary operator">\</a> <a href="__bs_bs.html" title="a binary operator">\\</a> <a href="_^.html" title="a binary operator, usually used for powers">^</a> <a href="_^_st_st.html" title="a binary operator, usually used for tensor or Cartesian power">^**</a> <a href="_^^.html" title="a binary operator">^^</a> <a href="__us.html" title="a binary operator, used for subscripting and access to elements">_</a> <a href="__vb.html" title="a binary operator, often used for horizontal concatenation">|</a> <a href="__vb-.html" title="a binary operator">|-</a> <a href="__vb_vb.html" title="a binary operator, often used for vertical concatenation">||</a> <a href="_and.html" title="conjunction">and</a> <a href="_or.html" title="disjunction">or</a> <a href="___S__P__A__C__E.html" title="blank operator; often used for function application, making polynomial rings">SPACE</a> , the unary prefix operators <a href="__st.html" title="a binary operator, usually used for multiplication">*</a> <a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> <a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> <a href="__lt.html" title="less than"><</a> <a href="__lt_lt.html" title="a binary operator (file output, ...)"><<</a> <a href="__lt_eq.html" title="less than or equal"><=</a> <a href="__lt_eq_eq.html" title="a unary and binary operator"><==</a> <a href="__lt_eq_eq_eq.html" title="a unary and binary operator"><===</a> <a href="__gt.html" title="greater than">></a> <a href="__gt_eq.html" title="greater than or equal">>=</a> <a href="__qu.html" title="comparison operator">?</a> <a href="__vb-.html" title="a binary operator">|-</a> <a href="_not.html" title="negation">not</a> , and the unary postfix operators <a href="_!.html" title="factorial">!</a> <a href="__lp_st_rp.html" title="a unary postfix operator, used for indicating a graded object">(*)</a> <a href="_^_st.html" title="a unary postfix operator, used for indicating pullback maps">^*</a> <a href="__us_st.html" title="a unary postfix operator, used for indicating pushforward maps">_*</a> <a href="_~.html" title="a unary postfix operator">~</a> .</p>
</div>
<div><h3>Menu</h3>
<h4>assignment</h4>
<ul><li><span><a href="__eq.html" title="assignment">=</a> -- assignment</span></li>
<li><span><a href="__co_eq.html" title="assignment of method or new local variable">:=</a> -- assignment of method or new local variable</span></li>
<li><span><a href="__lt-.html" title="assignment with left side evaluated"><-</a> -- assignment with left side evaluated</span></li>
</ul>
<h4>equality testing</h4>
<ul><li><span><a href="__eq_eq.html" title="equality">==</a> -- equality</span></li>
<li><span><a href="_!_eq.html" title="inequality">!=</a> -- inequality</span></li>
<li><span>===, see <span><a href="___Thing_sp_eq_eq_eq_sp__Thing.html" title="strict equality">Thing === Thing</a> -- strict equality</span></span></li>
<li><span><a href="__eq!_eq.html" title="strict inequality">=!=</a> -- strict inequality</span></li>
</ul>
<h4>comparison of objects</h4>
<ul><li><span><a href="__lt.html" title="less than"><</a> -- less than</span></li>
<li><span><a href="__lt_eq.html" title="less than or equal"><=</a> -- less than or equal</span></li>
<li><span><a href="__gt.html" title="greater than">></a> -- greater than</span></li>
<li><span><a href="__gt_eq.html" title="greater than or equal">>=</a> -- greater than or equal</span></li>
<li><span><a href="__qu.html" title="comparison operator">?</a> -- comparison operator</span></li>
</ul>
<h4>predicates</h4>
<ul><li><span><a href="_and.html" title="conjunction">and</a> -- conjunction</span></li>
<li><span><a href="_or.html" title="disjunction">or</a> -- disjunction</span></li>
<li><span><a href="_not.html" title="negation">not</a> -- negation</span></li>
</ul>
<h4>functions</h4>
<ul><li><span><a href="___S__P__A__C__E.html" title="blank operator; often used for function application, making polynomial rings">SPACE</a> -- blank operator; often used for function application, making polynomial rings</span></li>
<li><span><a href="_-_gt.html" title="make a function">-></a> -- make a function</span></li>
<li><span><a href="__at_at.html" title="a binary operator">@@</a> -- a binary operator</span></li>
<li><span><a href="__bs_bs.html" title="a binary operator">\\</a> -- a binary operator</span></li>
<li><span><a href="__bs.html" title="a binary operator">\</a> -- a binary operator</span></li>
</ul>
<h4>subscripting and object access</h4>
<ul><li><span><a href="__us.html" title="a binary operator, used for subscripting and access to elements">_</a> -- a binary operator, used for subscripting and access to elements</span></li>
<li><span><a href="_..html" title="access to elements whose key is a symbol">.</a> -- access to elements whose key is a symbol</span></li>
<li><span><a href="_._qu.html" title="check for presence of elements whose key is a symbol">.?</a> -- check for presence of elements whose key is a symbol</span></li>
<li><span><a href="__sh.html" title="length, or access to elements">#</a> -- length, or access to elements</span></li>
<li><span><a href="__sh_qu.html" title="check for presence of elements">#?</a> -- check for presence of elements</span></li>
</ul>
<h4>arithmetic operators</h4>
<ul><li><span><a href="_!.html" title="factorial">!</a> -- factorial</span></li>
<li><span><a href="__pl.html" title="a unary or binary operator, usually used for addition">+</a> -- a unary or binary operator, usually used for addition</span></li>
<li><span><a href="_-.html" title="a unary or binary operator, usually used for negation or subtraction">-</a> -- a unary or binary operator, usually used for negation or subtraction</span></li>
<li><span><a href="__st.html" title="a binary operator, usually used for multiplication">*</a> -- a binary operator, usually used for multiplication</span></li>
<li><span><a href="__sl.html" title="a binary operator, usually used for division">/</a> -- a binary operator, usually used for division</span></li>
<li><span><a href="__sl_sl.html" title="a binary operator, usually used for quotient">//</a> -- a binary operator, usually used for quotient</span></li>
<li><span><a href="__pc.html" title="a binary operator, usually used for remainder and reduction">%</a> -- a binary operator, usually used for remainder and reduction</span></li>
<li><span><a href="_^.html" title="a binary operator, usually used for powers">^</a> -- a binary operator, usually used for powers</span></li>
<li><span><a href="__am.html" title="a binary operator">&</a> -- a binary operator</span></li>
<li><span><a href="__pl_pl.html" title="a binary operator, usually used for direct sum">++</a> -- a binary operator, usually used for direct sum</span></li>
<li><span><a href="__st_st.html" title="a binary operator, usually used for tensor product or Cartesian product">**</a> -- a binary operator, usually used for tensor product or Cartesian product</span></li>
<li><span><a href="_^_st_st.html" title="a binary operator, usually used for tensor or Cartesian power">^**</a> -- a binary operator, usually used for tensor or Cartesian power</span></li>
<li><span><a href="_~.html" title="a unary postfix operator">~</a> -- a unary postfix operator</span></li>
<li><span><a href="__lp_st_rp.html" title="a unary postfix operator, used for indicating a graded object">(*)</a> -- a unary postfix operator, used for indicating a graded object</span></li>
<li><span><a href="__co.html" title="a binary operator, uses include repetition; ideal quotients">:</a> -- a binary operator, uses include repetition; ideal quotients</span></li>
</ul>
<h4>miscellaneous operators</h4>
<ul><li><span><a href="_...html" title="a binary operator, used for sequences of consecutive items">..</a> -- a binary operator, used for sequences of consecutive items</span></li>
<li><span><a href="_.._lt.html" title="a binary operator, used for sequences of consecutive items, not including the endpoint">..<</a> -- a binary operator, used for sequences of consecutive items, not including the endpoint</span></li>
<li><span><a href="__lt_lt.html" title="a binary operator (file output, ...)"><<</a> -- a binary operator (file output, ...)</span></li>
<li><span><a href="__gt_gt.html" title="a binary operator, uses include bit shifting, or attaching optional inputs to functions">>></a> -- a binary operator, uses include bit shifting, or attaching optional inputs to functions</span></li>
<li><span><a href="__vb.html" title="a binary operator, often used for horizontal concatenation">|</a> -- a binary operator, often used for horizontal concatenation</span></li>
<li><span><a href="__vb_vb.html" title="a binary operator, often used for vertical concatenation">||</a> -- a binary operator, often used for vertical concatenation</span></li>
<li><span><a href="__at.html" title="a binary operator">@</a> -- a binary operator</span></li>
<li><span><a href="_^^.html" title="a binary operator">^^</a> -- a binary operator</span></li>
<li><span><a href="__vb-.html" title="a binary operator">|-</a> -- a binary operator</span></li>
<li><span><a href="__lt_eq_eq_gt.html" title="a binary operator"><==></a> -- a binary operator</span></li>
<li><span><a href="__eq_eq_eq_gt.html" title="a binary operator">===></a> -- a binary operator</span></li>
<li><span><a href="__eq_eq_gt.html" title="a binary operator">==></a> -- a binary operator</span></li>
<li><span><a href="__lt_eq_eq_eq.html" title="a unary and binary operator"><===</a> -- a unary and binary operator</span></li>
<li><span><a href="__lt_eq_eq.html" title="a unary and binary operator"><==</a> -- a unary and binary operator</span></li>
</ul>
<h4>built in operators</h4>
<ul><li><span><a href="__eq_gt.html" title="produce an Option">=></a> -- produce an Option</span></li>
<li><span><a href="__cm.html" title="the comma, used for separating entries in a list or sequence">,</a> -- the comma, used for separating entries in a list or sequence</span></li>
<li><span><a href="__se.html" title="statement separator">;</a> -- statement separator</span></li>
</ul>
<h4>further information</h4>
<ul><li><span><a href="_precedence_spof_spoperators.html" title="">precedence of operators</a></span></li>
<li><span><a href="_operator__Attributes.html" title="">operatorAttributes</a></span></li>
</ul>
</div>
</div>
</body>
</html>
