One Hat Cyber Team

One Hat Cyber Team

Your IP : 216.73.216.14

Server IP : 194.44.31.54

Server : Linux zen.imath.kiev.ua 4.18.0-553.77.1.el8_10.x86_64 #1 SMP Fri Oct 3 14:30:23 UTC 2025 x86_64

Server Software : Apache/2.4.37 (Rocky Linux) OpenSSL/1.1.1k

PHP Version : 5.6.40

Buat File | Buat Folder

Dir : ~/usr/share/doc/Macaulay2/Posets/html/

Edit File: _is__Eulerian_lp__Poset_rp.html

<!DOCTYPE html>
<html lang="en">
  <head>
    <title>isEulerian(Poset) -- determines if a ranked poset is Eulerian</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="determines if a ranked poset is Eulerian" href="_is__Eulerian_lp__Poset_rp.html">isEulerian(Poset)</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="_is__Geometric.html">next</a> | <a href="_is__Distributive.html">previous</a> | <a href="_is__Geometric.html">forward</a> | <a href="_is__Distributive.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a>      </div>
    </div>
    <hr>
    <div>
      <h1>isEulerian(Poset) -- determines if a ranked poset is Eulerian</h1>
      <ul>
        <li><span>Function: <a title="determines if a graph or digraph is Eulerian" href="../../Graphs/html/_is__Eulerian.html">isEulerian</a></span></li>
        <li>
          <dl class="element">
            <dt>Usage: </dt>
            <dd><code class="language-macaulay2">i = isEulerian P</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>, a ranked poset</span></li>
          </ul>
        </li>
        <li>Outputs:          <ul>
            <li><span><span class="tt">i</span>, <span>a <a title="the class of boolean values" href="../../Macaulay2Doc/html/___Boolean.html">Boolean value</a></span>, whether $P$ is Eulerian</span></li>
          </ul>
        </li>
      </ul>
      <div>
        <h2>Description</h2>
        <div>
          <p>The poset $P$ is Eulerian if every non-trivial <a title="computes the subposet contained between two points" href="_closed__Interval.html">closedInterval</a> of $P$ has an equal number of vertices of even and odd rank.</p>
          <p>The $n$ <a title="generates the chain poset on $n$ elements" href="_chain.html">chain</a> is non-Eulerian for $n \geq 3$.</p>
        </div>
        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i1 : isEulerian chain 10

o1 = false</code></pre>
            </td>
          </tr>
        </table>
        <div>
          <p>The <a title="generates the face poset of a simplicial complex" href="_face__Poset.html">facePoset</a> of the <a title="make an abstract simplicial complex from a list of faces" href="../../SimplicialComplexes/html/_simplicial__Complex_lp__List_rp.html">simplicialComplex</a> of an $n$ cycle is Eulerian.</p>
        </div>
        <table class="examples">
          <tr>
            <td>
              <pre><code class="language-macaulay2">i2 : n = 10;</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i3 : R = QQ[x_0..x_(n-1)];</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i4 : F = facePoset simplicialComplex apply(n, i -> x_i * x_((i+1)%n));</code></pre>
            </td>
          </tr>
          <tr>
            <td>
              <pre><code class="language-macaulay2">i5 : isEulerian F

o5 = true</code></pre>
            </td>
          </tr>
        </table>
      </div>
      <div>
        <h2>See also</h2>
        <ul>
          <li><span><a title="computes the subposet contained between two points" href="_closed__Interval.html">closedInterval</a> -- computes the subposet contained between two points</span></li>
          <li><span><a title="determines if a poset is ranked" href="_is__Ranked.html">isRanked</a> -- determines if a poset is ranked</span></li>
          <li><span><a title="computes the Moebius function at every pair of elements of a poset" href="_moebius__Function.html">moebiusFunction</a> -- computes the Moebius function at every pair of elements of a poset</span></li>
        </ul>
      </div>
      <div>
        <div class="waystouse">
          <h2>Ways to use this method:</h2>
          <ul>
            <li><span><a title="determines if a ranked poset is Eulerian" href="_is__Eulerian_lp__Poset_rp.html">isEulerian(Poset)</a> -- determines if a ranked poset is Eulerian</span></li>
          </ul>
        </div>
        <hr>
        <div class="waystouse">
          <p>The source of this document is in <span class="tt">Posets.m2:5719:0</span>.</p>
        </div>
      </div>
    </div>
  </body>

</html>