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<!DOCTYPE html> <html lang="en"> <head> <title>^ -- a binary operator, usually used for powers</title> <meta content="text/html; charset=utf-8" http-equiv="Content-Type"> <link type="text/css" rel="stylesheet" href="../../../../Macaulay2/Style/doc.css"> <link rel="stylesheet" href="../../../../Macaulay2/Style/katex/katex.min.css"> <script defer="defer" src="../../../../Macaulay2/Style/katex/katex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/auto-render.min.js"></script> <script> var macros = { "\\break": "\\\\", "\\ZZ": "\\mathbb{Z}", "\\NN": "\\mathbb{N}", "\\QQ": "\\mathbb{Q}", "\\RR": "\\mathbb{R}", "\\CC": "\\mathbb{C}", "\\PP": "\\mathbb{P}" }, delimiters = [ { left: "$$", right: "$$", display: true}, { left: "\\[", right: "\\]", display: true}, { left: "$", right: "$", display: false}, { left: "\\(", right: "\\)", display: false} ], ignoredTags = [ "kbd", "var", "samp", "script", "noscript", "style", "textarea", "pre", "code", "option" ]; document.addEventListener("DOMContentLoaded", function() { renderMathInElement(document.body, { delimiters: delimiters, macros: macros, ignoredTags: ignoredTags, trust: true }); }); </script> <style>.katex { font-size: 1em; }</style> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/copy-tex.min.js"></script> <script defer="defer" src="../../../../Macaulay2/Style/katex/contrib/render-a11y-string.min.js"></script> <script src="../../../../Macaulay2/Style/prism.js"></script> <script>var current_version = '1.25.06';</script> <script src="../../../../Macaulay2/Style/version-select.js"></script> <link type="image/x-icon" rel="icon" href="../../../../Macaulay2/Style/icon.gif"> </head> <body> <div id="buttons"> <div> <a href="https://macaulay2.com/">Macaulay2</a> <span id="version-select-container"></span> » <a title="Macaulay2 documentation" href="index.html">Documentation </a> <br><a href="_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="Macaulay2 documentation" href="index.html">Macaulay2Doc</a> » <a href="___The_sp__Macaulay2_splanguage.html">The Macaulay2 language</a> » <a href="_operators.html">operators</a> » <a title="a binary operator, usually used for powers" href="_%5E.html">^</a></span> </div> <div class="right"> <form method="get" action="https://www.google.com/search"> <input placeholder="Search" type="text" name="q" value=""> <input type="hidden" name="q" value="site:macaulay2.com/doc"> </form> <a href="__am.html">next</a> | <a href="__pc.html">previous</a> | <a href="__am.html">forward</a> | <a href="__pc.html">backward</a> | <a href="_operators.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>^ -- a binary operator, usually used for powers</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">x ^ y</code></dd> </dl> </li> </ul> <div> <h2>Description</h2> <p></p> This operator is used for exponentiation, making free modules and sheaves, for shifting complexes left or right, for projection maps involving direct sums, and for making nets. </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">symbol ^</span>:</h2> <ul> <li><span><a title="bracket power of an ideal" href="___Ideal_sp%5E_sp__Array.html">Ideal ^ Array</a> -- bracket power of an ideal</span></li> <li><span><kbd>MonomialIdeal ^ Array</kbd> -- see <span><a title="bracket power of an ideal" href="___Ideal_sp%5E_sp__Array.html">Ideal ^ Array</a> -- bracket power of an ideal</span></span></li> <li><span><a title="power of an ideal" href="___Ideal_sp%5E_sp__Z__Z.html">Ideal ^ ZZ</a> -- power of an ideal</span></li> <li><span><kbd>MonomialIdeal ^ ZZ</kbd> -- see <span><a title="power of an ideal" href="___Ideal_sp%5E_sp__Z__Z.html">Ideal ^ ZZ</a> -- power of an ideal</span></span></li> <li><span><kbd>Number ^ Ring</kbd> -- see <span><a title="lift to another ring" href="_lift.html">lift</a> -- lift to another ring</span></span></li> <li><span><kbd>Number ^ RingFamily</kbd> -- see <span><a title="lift to another ring" href="_lift.html">lift</a> -- lift to another ring</span></span></li> <li><span><span class="tt">List ^ List</span> (missing documentation)<!--tag: (^,List,List)--> </span></li> <li><span><a title="component of map corresponding to summand of target" href="___Matrix_sp%5E_sp__Array.html">Matrix ^ Array</a> -- component of map corresponding to summand of target</span></li> <li><span><a title="select rows" href="___Matrix_sp%5E_sp__List.html">Matrix ^ List</a> -- select rows</span></li> <li><span><kbd>MutableMatrix ^ List</kbd> -- see <span><a title="select rows" href="___Matrix_sp%5E_sp__List.html">Matrix ^ List</a> -- select rows</span></span></li> <li><span><kbd>Vector ^ List</kbd> -- see <span><a title="select rows" href="___Matrix_sp%5E_sp__List.html">Matrix ^ List</a> -- select rows</span></span></li> <li><span><a title="power" href="___Matrix_sp%5E_sp__Z__Z.html">Matrix ^ ZZ</a> -- power</span></li> <li><span><kbd>MutableMatrix ^ ZZ</kbd> -- see <span><a title="power" href="___Matrix_sp%5E_sp__Z__Z.html">Matrix ^ ZZ</a> -- power</span></span></li> <li><span><a title="projection onto summand" href="___Module_sp%5E_sp__Array.html">Module ^ Array</a> -- projection onto summand</span></li> <li><span><a title="projection onto summand" href="___Module_sp%5E_sp__List.html">Module ^ List</a> -- projection onto summand</span></li> <li><span><a title="direct sum" href="___Module_sp%5E_sp__Z__Z.html">Module ^ ZZ</a> -- direct sum</span></li> <li><span><a title="raise a net or string" href="___Net_sp%5E_sp__Z__Z.html">Net ^ ZZ</a> -- raise a net or string</span></li> <li><span><kbd>String ^ ZZ</kbd> -- see <span><a title="raise a net or string" href="___Net_sp%5E_sp__Z__Z.html">Net ^ ZZ</a> -- raise a net or string</span></span></li> <li><span><a title="construct a chain complex with prescribed Betti table" href="../../OldChainComplexes/html/___Ring_sp%5E_sp__Betti__Tally.html">Ring ^ BettiTally</a></span></li> <li><span><a title="make a free module" href="___Ring_sp%5E_sp__List.html">Ring ^ List</a> -- make a free module</span></li> <li><span><kbd>RingFamily ^ List</kbd> -- see <span><a title="make a free module" href="___Ring_sp%5E_sp__List.html">Ring ^ List</a> -- make a free module</span></span></li> <li><span><a title="make a free module" href="___Ring_sp%5E_sp__Z__Z.html">Ring ^ ZZ</a> -- make a free module</span></li> <li><span><kbd>RingFamily ^ ZZ</kbd> -- see <span><a title="make a free module" href="___Ring_sp%5E_sp__Z__Z.html">Ring ^ ZZ</a> -- make a free module</span></span></li> <li><span><kbd>MonoidElement ^ ZZ</kbd> -- see <span><a title="power" href="___Ring__Element_sp%5E_sp__Z__Z.html">RingElement ^ ZZ</a> -- power</span></span></li> <li><span><a title="power" href="___Ring__Element_sp%5E_sp__Z__Z.html">RingElement ^ ZZ</a> -- power</span></li> <li><span><a title="power" href="___Ring__Map_sp%5E_sp__Z__Z.html">RingMap ^ ZZ</a> -- power</span></li> <li><span><kbd>ScriptedFunctor ^ Thing</kbd> -- see <span><a title="the class of all scripted functors" href="___Scripted__Functor.html">ScriptedFunctor</a> -- the class of all scripted functors</span></span></li> <li><span><a title="vertically stacked copies of a string" href="___String_sp%5E_sp__Sequence.html">String ^ Sequence</a> -- vertically stacked copies of a string</span></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="a binary operator, usually used for powers" href="_%5E.html">^</a> is <span>a <a title="the class of all keywords" href="___Keyword.html">keyword</a></span>.</p> <div> <p>This operator may be used as a binary operator in an expression like <span class="tt">x^y</span>. The user may install <a href="_binary_spmethods.html">binary methods</a> for handling such expressions with code such as</p> <pre> X ^ Y := (x,y) -> ...</pre> <p>where <span class="tt">X</span> is the class of <span class="tt">x</span> and <span class="tt">Y</span> is the class of <span class="tt">y</span>.</p> </div> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Macaulay2Doc/operators/caret.m2:38:0</span>.</p> </div> </div> </div> </body> </html>