Sperner property of a partially ordered set

<p>In <a href="/facts/Order_theory/gn6fM3oJ">order-theoretic mathematics</a>, a <a href="/facts/Graded_poset/SAUdJ7op">graded</a> <a href="/facts/Partially_ordered_set/qb4a3FAX">partially ordered set</a> is said to have the Sperner property (and hence is called a Sperner poset), if no <a href="/facts/Antichain/rPuqkKJt">antichain</a> within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset. Since every rank level is itself an antichain, the Sperner property is equivalently the property that some rank level is a maximum antichain. The Sperner property and Sperner posets are named after <a href="/facts/Emanuel_Sperner/XYgfttqG">Emanuel Sperner</a>, who proved <a href="/facts/Sperner%2527s_theorem/uAzReTnz">Sperner's theorem</a> stating that the <a href="/facts/Powerset/FVuf8PDD">family of all subsets</a> of a finite set (partially ordered by set inclusion) has this property.  The lattice of partitions of a finite set typically lacks the Sperner property.
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Sperner property of a partially ordered set open-in-new

Sperner property of a partially ordered set