Sperner property of a partially ordered set

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In order-theoretic mathematics, a graded partially ordered set is said to have the Sperner property (and hence is called a Sperner poset), if no antichain within it is larger than the largest rank level (one of the sets of elements of the same rank) in the poset.[1] Since every rank level is itself an antichain, the Sperner property is equivalently the property that some rank level is a maximum antichain.[2] The Sperner property and Sperner posets are named after Emanuel Sperner, who proved Sperner's theorem stating that the family of all subsets 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.[3]

Variations[edit]

A k-Sperner poset is a graded poset in which no union of k antichains is larger than the union of the k largest rank levels,[1] or, equivalently, the poset has a maximum k-family consisting of k rank levels.[2]

A strict Sperner poset is a graded poset in which all maximum antichains are rank levels.[2]

A strongly Sperner poset is a graded poset which is k-Sperner for all values of 'k up to the largest rank value.[2]

References[edit]