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o3 = P

o3 : Poset</code></pre>
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          <p>It is unnecessary to pass the ground set if every vertex of the poset is a member of at least one relation in $R$.</p>
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              <pre><code class="language-macaulay2">i4 : poset {{1,2},{2,3},{3,4}}

o4 = Relation Matrix: | 1 1 1 1 |
                      | 0 1 1 1 |
                      | 0 0 1 1 |
                      | 0 0 0 1 |

o4 : Poset</code></pre>
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          <p>Sometimes it is easier to create a poset by passing the ground set and a binary function which compares elements in the ground set.</p>
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              <pre><code class="language-macaulay2">i5 : cmp = (a,b) -> b % a == 0;</code></pre>
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              <pre><code class="language-macaulay2">i6 : G = toList(1..10);</code></pre>
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              <pre><code class="language-macaulay2">i7 : P = poset(G, cmp)

o7 = P

o7 : Poset</code></pre>
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          <p>And, in other cases, it may be easy to find all relations in the poset.  In this case, if the matrix encoding all the relations is passed to the poset method, then it is not necessary to compute the transitive closure.  However, it should be noted that the method makes no checks on either the relations or the matrix in this case.</p>
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              <pre><code class="language-macaulay2">i8 : S = QQ[x,y,z];</code></pre>
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              <pre><code class="language-macaulay2">i9 : G = {x^2, x*y, z^2, x^2*y*z, x*y*z^3, x^2*y^2*z^3};</code></pre>
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              <pre><code class="language-macaulay2">i10 : R = flatten for g in G list for h in G list if h %g == 0 then {g,h} else continue;</code></pre>
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              <pre><code class="language-macaulay2">i11 : M = matrix apply(G, g -> apply(G, h -> if h %g == 0 then 1 else 0));

6       6
o11 : Matrix ZZ  &lt;-- ZZ</code></pre>
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              <pre><code class="language-macaulay2">i12 : P = poset(G, R, M)

o12 = P

o12 : Poset</code></pre>
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          <p>In the previous example the vertices of the poset were <a title="the class of all ring elements handled by the engine" href="../../Macaulay2Doc/html/___Ring__Element.html">RingElement</a>s.  In fact, the Posets package does not require the vertices to be of any particular type.  However, this also means when the package makes calls to external methods, it sometimes must relabel the vertices (usually to the index of the vertex in $G$).</p>
          <p>The option AntisymmetryStrategy determines how the method checks the input relations for antisymmetry (a necessary condition for a poset).  The default, &quot;rank&quot;, checks that the RelationMatrix has maximal rank.  The choice &quot;digraph&quot; uses the <a title="determines whether a graph is cyclic" href="../../Graphs/html/_is__Cyclic.html">isCyclic</a> method on the <a href="../../Graphs/html/___Digraph.html">Digraph</a> of the given relations.  Last, the choice &quot;none&quot; skips the check all together.</p>
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        <h2>Caveat</h2>
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          <p>Care should be taken when using the AntisymmetryStrategy option.  If &quot;none&quot; is specified but the relations are cyclic, then the returned Poset will not actually be a Poset.  In this case, the behavior of the package is unknown.</p>
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        <h2>See also</h2>
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          <li><span><a title="a class for partially ordered sets (posets)" href="___Poset.html">Poset</a> -- a class for partially ordered sets (posets)</span></li>
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          <h2>Ways to use <span class="tt">poset</span>:</h2>
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            <li><kbd>poset(List)</kbd></li>
            <li><kbd>poset(List,Function)</kbd></li>
            <li><kbd>poset(List,List)</kbd></li>
            <li><kbd>poset(List,List,Matrix)</kbd></li>
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          <h2>For the programmer</h2>
          <p>The object <a title="creates a new Poset object" href="_poset.html">poset</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>
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          <p>The source of this document is in <span class="tt">Posets.m2:2122:0</span>.</p>
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