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<head><title>Poset -- a class for partially ordered sets (posets)</title>
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<div><h1>Poset -- a class for partially ordered sets (posets)</h1>
<div class="single"><h2>Description</h2>
<div><div>This class is a type of HashTable which represents finite posets. It consists of a ground set, a set of sequences (a,b) representing relationships where a is less than or equal to b, a matrix encoding these relations.</div>
<table class="examples"><tr><td><pre>i1 : G = {a,b,c,d,e}; -- the ground set</pre>
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<tr><td><pre>i2 : R = {(a,b),(b,c),(a,c),(a,d),(d,e)}; --a set of sequences representing relations that "generate" all other relations</pre>
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<tr><td><pre>i3 : P = poset (G,R) -- the matrix encoding these relations is computed by calling this function
o3 = Poset{cache => CacheTable }
GroundSet => {a, b, c, d, e}
RelationMatrix => | 1 1 1 1 1 |
| 0 1 1 0 0 |
| 0 0 1 0 0 |
| 0 0 0 1 1 |
| 0 0 0 0 1 |
Relations => {(a, b), (b, c), (a, c), (a, d), (d, e)}
o3 : Poset</pre>
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<div class="single"><h2>See also</h2>
<ul><li><span><a href="_poset.html" title="creating a poset">poset</a> -- creating a poset</span></li>
<li><span><a href="___Ground__Set.html" title="underlying set of a poset">GroundSet</a> -- underlying set of a poset</span></li>
<li><span><a href="___Relations.html" title="a set of relations in the poset that generates all other relations">Relations</a> -- a set of relations in the poset that generates all other relations</span></li>
<li><span><a href="___Relation__Matrix.html" title="the matrix expressing all of the relations between elements in a Poset">RelationMatrix</a> -- the matrix expressing all of the relations between elements in a Poset</span></li>
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<div class="waystouse"><h2>Functions and methods returning an object of class Poset :</h2>
<ul><li><span>lcmLattice(Ideal), see <span><a href="_lcm__Lattice.html" title="returns the LCM lattice of an ideal">lcmLattice</a> -- returns the LCM lattice of an ideal</span></span></li>
<li><span>lcmLattice(MonomialIdeal), see <span><a href="_lcm__Lattice.html" title="returns the LCM lattice of an ideal">lcmLattice</a> -- returns the LCM lattice of an ideal</span></span></li>
</ul>
<h2>Methods that use an object of class Poset :</h2>
<ul><li><span>adjacencyMatrix(Poset), see <span><a href="_adjacency__Matrix.html" title="returns adjacency matrix of a directed graph">adjacencyMatrix</a> -- returns adjacency matrix of a directed graph</span></span></li>
<li><span>compare(Poset,Thing,Thing), see <span><a href="_compare.html" title="returns boolean value for whether an element is less than another">compare</a> -- returns boolean value for whether an element is less than another</span></span></li>
<li><span>filter(Poset,Thing), see <span><a href="_filter.html" title="returns a principal filter generated by the given element">filter</a> -- returns a principal filter generated by the given element</span></span></li>
<li><span>isLattice(Poset), see <span><a href="_is__Lattice.html" title="determines if a poset is a lattice">isLattice</a> -- determines if a poset is a lattice</span></span></li>
<li><span>joinExists(Poset,Thing,Thing), see <span><a href="_join__Exists.html" title="determines if the join exists">joinExists</a> -- determines if the join exists</span></span></li>
<li><span>meetExists(Poset,Thing,Thing), see <span><a href="_meet__Exists.html" title="determines if the meet exists">meetExists</a> -- determines if the meet exists</span></span></li>
<li><span>orderIdeal(Poset,Thing), see <span><a href="_order__Ideal.html" title="returns a principal order ideal generated by the given element">orderIdeal</a> -- returns a principal order ideal generated by the given element</span></span></li>
<li><span>posetJoin(Poset,Thing,Thing), see <span><a href="_poset__Join.html" title="returns the join of two elements">posetJoin</a> -- returns the join of two elements</span></span></li>
<li><span>posetMeet(Poset,Thing,Thing), see <span><a href="_poset__Meet.html" title="returns the meet of two elements">posetMeet</a> -- returns the meet of two elements</span></span></li>
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<div class="waystouse"><h2>For the programmer</h2>
<p>The object <a href="___Poset.html" title="a class for partially ordered sets (posets)">Poset</a> is <span>a <a href="../../Macaulay2Doc/html/___Type.html">type</a></span>, with ancestor classes <a href="../../Macaulay2Doc/html/___Hash__Table.html" title="the class of all hash tables">HashTable</a> < <a href="../../Macaulay2Doc/html/___Thing.html" title="the class of all things">Thing</a>.</p>
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