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<!DOCTYPE html> <html lang="en"> <head> <title>monoid(...,Join=>...) -- specify how to handle degrees in the coefficient ring</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="_rings.html">rings</a> » <a href="_polynomial_springs.html">polynomial rings</a> » <a href="_monomial_sporderings.html">monomial orderings</a> » <a title="make or retrieve a monoid" href="_monoid.html">monoid</a> » <a title="specify how to handle degrees in the coefficient ring" href="_monoid_lp..._cm__Join_eq_gt..._rp.html">monoid(...,Join=>...)</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="_monoid_lp..._cm__Monomial__Order_eq_gt..._rp.html">next</a> | <a href="_monoid_lp..._cm__Heft_eq_gt..._rp.html">previous</a> | <a href="_monoid_lp..._cm__Monomial__Order_eq_gt..._rp.html">forward</a> | <a href="_monoid_lp..._cm__Heft_eq_gt..._rp.html">backward</a> | <a href="_monoid.html">up</a> | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>monoid(...,Join=>...) -- specify how to handle degrees in the coefficient ring</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">monoid[a,b,c, Join => P]</code></dd> </dl> </li> </ul> <div> <h2>Description</h2> <div> <p>The <span class="tt">Join</span> option specifies whether the degrees in the new monoid ring will be obtained by joining the degrees in the coefficient with the degrees in the monoid.</p> <p>By default, (multi)degrees are concatenated when forming polynomial rings over polynomial rings, as can be seen by examining the corresponding flattened monoid, which displays information about all of the variables.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : QQ[x][y] o1 = QQ[x][y] o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : oo.FlatMonoid o2 = monoid[y, x, Degrees => {{1}, {0}}, Heft => {2:1}, MonomialOrder => {MonomialSize => 32}] {0} {1} {GRevLex => {1} } {Position => Up } {GRevLex => {1} } o2 : GeneralOrderedMonoid</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : QQ[x][y][z] o3 = QQ[x][y][z] o3 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : oo.FlatMonoid o4 = monoid[z, y, x, Degrees => {{1}, {0}, {0}}, Heft => {3:1}, MonomialOrder => {MonomialSize => 32}] {0} {1} {0} {GRevLex => {1} } {0} {0} {1} {Position => Up } {2:(GRevLex => {1})} o4 : GeneralOrderedMonoid</code></pre> </td> </tr> </table> <div> <p>That behavior can be overridden with the <a title="specify how to handle degrees in the coefficient ring" href="_monoid_lp..._cm__Join_eq_gt..._rp.html">monoid(...,Join=>...)</a> option.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : QQ[x][y, Join => false] o5 = QQ[x][y] o5 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : oo.FlatMonoid o6 = monoid[y, x, Degrees => {2:1}, Heft => {1}, Join => false, MonomialOrder => {MonomialSize => 32}] {GRevLex => {1} } {Position => Up } {GRevLex => {1} } o6 : GeneralOrderedMonoid</code></pre> </td> </tr> </table> <div> <p>This option may also be used when creating a new ring from an existing ring, creating a tensor product ring, or symmetric algebra.</p> </div> </div> <div> <h2>See also</h2> <ul> <li><span><a title="make or retrieve a monoid" href="_monoid.html">monoid</a> -- make or retrieve a monoid</span></li> <li><span><a title="make a copy of a ring, with some features changed" href="_new__Ring.html">newRing</a> -- make a copy of a ring, with some features changed</span></li> <li><span><a title="tensor product of monoids" href="_tensor_lp__Monoid_cm__Monoid_rp.html">tensor(Ring,Ring)</a> -- tensor product of monoids</span></li> <li><span><a title="the symmetric algebra of a module" href="_symmetric__Algebra.html">symmetricAlgebra</a> -- the symmetric algebra of a module</span></li> </ul> </div> <div> <div> <h2>Functions with optional argument named <span class="tt">Join</span>:</h2> <ul> <li><span><a title="specify how to handle degrees in the coefficient ring" href="_monoid_lp..._cm__Join_eq_gt..._rp.html">monoid(...,Join=>...)</a> -- specify how to handle degrees in the coefficient ring</span></li> <li><span><kbd>newRing(...,Join=>...)</kbd> -- see <span><a title="make a copy of a ring, with some features changed" href="_new__Ring.html">newRing</a> -- make a copy of a ring, with some features changed</span></span></li> <li><span><kbd>symmetricAlgebra(...,Join=>...)</kbd> -- see <span><a title="the symmetric algebra of a module" href="_symmetric__Algebra.html">symmetricAlgebra</a> -- the symmetric algebra of a module</span></span></li> <li><span><kbd>tensor(Monoid,Monoid,Join=>...)</kbd> -- see <span><a title="tensor product of monoids" href="_tensor_lp__Monoid_cm__Monoid_rp.html">tensor(Monoid,Monoid)</a> -- tensor product of monoids</span></span></li> <li><span><kbd>tensor(Ring,Ring,Join=>...)</kbd> -- see <span><a title="tensor product of monoids" href="_tensor_lp__Monoid_cm__Monoid_rp.html">tensor(Monoid,Monoid)</a> -- tensor product of monoids</span></span></li> </ul> </div> <div> <h2>Further information</h2> <ul> <li><span>Default value: <span class="tt">null</span></span></li> <li><span>Function: <span><a title="make or retrieve a monoid" href="_monoid.html">monoid</a> -- make or retrieve a monoid</span></span></li> <li><span>Option key: <span><a title="an optional argument" href="___Join.html">Join</a> -- an optional argument</span></span></li> </ul> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Macaulay2Doc/functions/monoid-doc.m2:563:0</span>.</p> </div> </div> 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