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<!DOCTYPE html> <html lang="en"> <head> <title>monoid(...,Degrees=>...) -- specify the degrees of the variables</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 the degrees of the variables" href="_monoid_lp..._cm__Degrees_eq_gt..._rp.html">monoid(...,Degrees=>...)</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__Degree__Lift_eq_gt..._rp.html">next</a> | <a href="_monoid_lp..._cm__Weights_eq_gt..._rp.html">previous</a> | <a href="_monoid_lp..._cm__Degree__Lift_eq_gt..._rp.html">forward</a> | <a href="_monoid_lp..._cm__Weights_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(...,Degrees=>...) -- specify the degrees of the variables</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">monoid[x,y,z, Degrees => {d1, d2, d3}]</code></dd> <dd><code class="language-macaulay2">monoid[x,y,z, DegreeRank => r]</code></dd> <dd><code class="language-macaulay2">monoid[x,y,z, DegreeGroup => ZZ^2 ++ coker matrix 3]</code></dd> </dl> </li> </ul> <div> <h2>Description</h2> <div> <p>The <span class="tt">Degrees</span> option specifies the degrees of the variables in the monoid. If provided, the value must be a list or sequence with as many entries (after <a title="remove subsequences from a sequence or list" href="_splice.html">splicing</a> and <a title="flatten a nested list or a matrix" href="_flatten.html">flattening</a>) as there are variables. Each degree is an integers or a list of integers. Degrees provided as integers will be converted into multidegrees of length 1.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : QQ[x,y,z, Degrees => {2:1, 2}] o1 = QQ[x..z] o1 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : degrees oo o2 = {{1}, {1}, {2}} o2 : List</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : QQ[x,y, Degrees => {{1,0}, {0,1}}] o3 = QQ[x..y] o3 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : degrees oo o4 = {{1, 0}, {0, 1}} o4 : List</code></pre> </td> </tr> </table> <div> <p>The <span class="tt">DegreeRank</span> option specifies the degree length of the monoid. If provided, the value must be an integer. If the <span class="tt">Degrees</span> option is not provided, the degrees of the variables are determined similar to this example.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : QQ[a..f, DegreeRank => 3] o5 = QQ[a..f] o5 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : transpose matrix degrees oo o6 = | 1 0 0 0 0 0 | | 0 1 0 0 0 0 | | 0 0 1 1 1 1 | 3 6 o6 : Matrix ZZ <-- ZZ</code></pre> </td> </tr> </table> <div> <p>Finally, the <span class="tt">DegreeGroup</span> option specifies the degree group of the monoid. If provided, the value must be a $\ZZ$-module. If the <span class="tt">Degrees</span> option is not provided, the degrees of the variables are determined similar to the previous example.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i7 : QQ[a..f, DegreeGroup => ZZ^2 ++ coker matrix 3] -- Warning: computations over rings with torsion grading groups are experimental o7 = QQ[a..f] o7 : PolynomialRing</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : degreeGroup oo o8 = cokernel | 0 | | 0 | | 3 | 3 o8 : ZZ-module, quotient of ZZ</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">Degrees</span>:</h2> <ul> <li><span><a title="specify the degrees of the variables" href="_monoid_lp..._cm__Degrees_eq_gt..._rp.html">monoid(...,Degrees=>...)</a> -- specify the degrees of the variables</span></li> <li><span><kbd>newRing(...,Degrees=>...)</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(...,Degrees=>...)</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,Degrees=>...)</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,Degrees=>...)</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="___Degrees.html">Degrees</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> </div> </body> </html>