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<!DOCTYPE html> <html lang="en"> <head> <title>hibiRing -- produces the Hibi ring of a poset</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="../../Macaulay2Doc/html/index.html">Documentation </a> <br><a href="../../Macaulay2Doc/html/_packages_spprovided_spwith_sp__Macaulay2.html">Packages</a> » <span><a title="a package for working with partially ordered sets" href="index.html">Posets</a> :: <a title="produces the Hibi ring of a poset" href="_hibi__Ring.html">hibiRing</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="_h__Polynomial.html">next</a> | <a href="_hibi__Ideal.html">previous</a> | <a href="_h__Polynomial.html">forward</a> | <a href="_hibi__Ideal.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>hibiRing -- produces the Hibi ring of a poset</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">H = hibiRing P</code></dd> <dd><code class="language-macaulay2">H = hibiRing(P, Strategy => "kernel")</code></dd> <dd><code class="language-macaulay2">H = hibiRing(P, Strategy => "4ti2")</code></dd> </dl> </li> <li>Inputs: <ul> <li><span><span class="tt">P</span>, <span>an instance of the type <a title="a class for partially ordered sets (posets)" href="___Poset.html">Poset</a></span>, </span></li> </ul> </li> <li><a href="../../Macaulay2Doc/html/_using_spfunctions_spwith_spoptional_spinputs.html">Optional inputs</a>: <ul> <li><span><span class="tt">CoefficientRing</span><span class="tt"> => </span><span>a <a title="the class of all rings" href="../../Macaulay2Doc/html/___Ring.html">ring</a></span>, <span>default value QQ</span>, which specifies the coefficient ring of the <a title="the class of all ordered monoid rings" href="../../Macaulay2Doc/html/___Polynomial__Ring.html">PolynomialRing</a> $H$</span></li> <li><span><span class="tt">Strategy</span><span class="tt"> => </span><span>a <a title="the class of all strings" href="../../Macaulay2Doc/html/___String.html">string</a></span>, <span>default value "kernel"</span>, which specifies whether to use Macaulay2's native <a title="kernel of a map" href="../../Macaulay2Doc/html/_kernel.html">kernel</a> method (Strategy => "kernel") or the package <a title="Interface for 4ti2" href="../../FourTiTwo/html/index.html">FourTiTwo</a> (Strategy => "4ti2")</span></li> </ul> </li> <li>Outputs: <ul> <li><span><span class="tt">H</span>, <span>a <a title="the class of all quotient rings" href="../../Macaulay2Doc/html/___Quotient__Ring.html">quotient ring</a></span>, the toric algebra which is isomorphic to the Hibi ring of $P$</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The Hibi ring of $P$ is a monomial algebra generated by the monomials which generate the Hibi ideal (<a title="produces the Hibi ideal of a poset" href="_hibi__Ideal.html">hibiIdeal</a>). That is, the monomials built in $2n$ variables $x_0, \ldots, x_{n-1}, y_0, \ldots, y_{n-1}$, where $n$ is the size of the ground set of $P$. The monomials are in bijection with order ideals in $P$. Let $I$ be an order ideal of $P$. Then the associated monomial is the product of the $x_i$ associated with members of $I$ and the $y_i$ associated with non-members of $I$.</p> <p>This method returns the toric quotient algebra isomorphic to the Hibi ring. The ideal is the ideal of Hibi relations. The generators of the <a title="the class of all ordered monoid rings" href="../../Macaulay2Doc/html/___Polynomial__Ring.html">PolynomialRing</a> $H$ is built over are of the form $t_I$ where $I$ is an order ideal of $P$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : hibiRing booleanLattice 2 QQ[t , t , t , t , t , t ] {} {0} {0, 1} {0, 1, 2} {0, 2} {0, 1, 2, 3} o1 = ---------------------------------------------------------- t t - t t {0} {0, 1, 2} {0, 1} {0, 2} o1 : QuotientRing</code></pre> </td> </tr> </table> <div> <p>The Hibi ring of the $n$ chain is just a polynomial ring in $n+1$ variables.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i2 : hibiRing chain 4 o2 = QQ[t , t , t , t , t ] {} {0} {0, 1} {0, 1, 2} {0, 1, 2, 3} o2 : PolynomialRing</code></pre> </td> </tr> </table> <div> <p>In some cases, it may be faster to use the <a title="Interface for 4ti2" href="../../FourTiTwo/html/index.html">FourTiTwo</a> method <a title="calculates a Groebner basis of the toric ideal I_A, given A; invokes "groebner" from 4ti2" href="../../FourTiTwo/html/_toric__Groebner.html">toricGroebner</a> to generate the Hibi relations. Using the Strategy "4ti2" tells the method to use this approach.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i3 : hibiRing(divisorPoset 6, Strategy => "4ti2") QQ[t , t , t , t , t , t ] {} {0} {0, 1} {0, 1, 2} {0, 2} {0, 1, 2, 3} o3 = ---------------------------------------------------------- - t t + t t {0} {0, 1, 2} {0, 1} {0, 2} o3 : QuotientRing</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="produces the Hibi ideal of a poset" href="_hibi__Ideal.html">hibiIdeal</a> -- produces the Hibi ideal of a poset</span></li> <li><span><a title="computes the elements below given elements in a poset" href="_order__Ideal.html">orderIdeal</a> -- computes the elements below given elements in a poset</span></li> <li><span><a title="computes the elements below a given element in a poset" href="_principal__Order__Ideal.html">principalOrderIdeal</a> -- computes the elements below a given element in a poset</span></li> <li><span><a title="produces the p-partition ring of a poset" href="_p__Partition__Ring.html">pPartitionRing</a> -- produces the p-partition ring of a poset</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">hibiRing</span>:</h2> <ul> <li><kbd>hibiRing(Poset)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="produces the Hibi ring of a poset" href="_hibi__Ring.html">hibiRing</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function__With__Options.html">method function with options</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Posets.m2:2390:0</span>.</p> </div> </div> </div> </body> </html>