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<!DOCTYPE html> <html lang="en"> <head> <title>isAtomic -- determines if a lattice is atomic</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 lattice is atomic" href="_is__Atomic.html">isAtomic</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__Bounded.html">next</a> | <a href="_is__Antichain.html">previous</a> | <a href="_is__Bounded.html">forward</a> | <a href="_is__Antichain.html">backward</a> | up | <a href="master.html">index</a> | <a href="toc.html">toc</a> </div> </div> <hr> <div> <h1>isAtomic -- determines if a lattice is atomic</h1> <ul> <li> <dl class="element"> <dt>Usage: </dt> <dd><code class="language-macaulay2">i = isAtomic 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 lattice</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 atomic</span></li> </ul> </li> </ul> <div> <h2>Description</h2> <div> <p>The lattice $P$ is atomic if every non-minimal vertex of $P$ is the join of atoms of $P$. Equivalently, $P$ is atomic if every non-minimal, non-atom vertex of $P$ covers at least two vertices.</p> <p>The diamond poset is atomic. Also $n$ <a title="generates the boolean lattice on $n$ elements" href="_boolean__Lattice.html">booleanLattice</a>s are atomic.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i1 : P = poset {{1, 2}, {1, 3}, {1, 4}, {2, 5}, {3, 5}, {4, 5}};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i2 : isLattice P o2 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i3 : isAtomic P o3 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i4 : isAtomic booleanLattice 4 o4 = true</code></pre> </td> </tr> </table> <div> <p>The following lattice is non-atomic. Also, $n$ <a title="generates the chain poset on $n$ elements" href="_chain.html">chain</a>s are non-atomic, for $n \geq 3$.</p> </div> <table class="examples"> <tr> <td> <pre><code class="language-macaulay2">i5 : Q = poset {{1, 2}, {1, 3}, {2, 4}, {2, 5}, {3, 4}, {4, 6}, {5, 6}};</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i6 : isLattice Q o6 = true</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i7 : isAtomic Q o7 = false</code></pre> </td> </tr> <tr> <td> <pre><code class="language-macaulay2">i8 : isAtomic chain 5 o8 = false</code></pre> </td> </tr> </table> </div> <div> <h2>See also</h2> <ul> <li><span><a title="computes the list of elements covering the minimal elements of a poset" href="_atoms.html">atoms</a> -- computes the list of elements covering the minimal elements of a poset</span></li> <li><span><a title="determines if a poset is a lattice" href="_is__Lattice.html">isLattice</a> -- determines if a poset is a lattice</span></li> </ul> </div> <div> <div class="waystouse"> <h2>Ways to use <span class="tt">isAtomic</span>:</h2> <ul> <li><kbd>isAtomic(Poset)</kbd></li> </ul> </div> <div class="waystouse"> <h2>For the programmer</h2> <p>The object <a title="determines if a lattice is atomic" href="_is__Atomic.html">isAtomic</a> is <span>a <a title="a type of method function" href="../../Macaulay2Doc/html/___Method__Function.html">method function</a></span>.</p> </div> <hr> <div class="waystouse"> <p>The source of this document is in <span class="tt">Posets.m2:5542:0</span>.</p> </div> </div> </div> </body> </html>