Sperner property of a partially ordered set
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
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
- ↑ 1.0 1.1 Stanley, Richard (1984), "Quotients of Peck posets", Order 1 (1): 29–34, doi:10.1007/BF00396271.
- ↑ 2.0 2.1 2.2 2.3 Handbook of discrete and combinatorial mathematics, by Kenneth H. Rosen, John G. Michaels
- ↑ "Maximum antichains in the partition lattice", The Mathematical Intelligencer 1 (2): 84–86, June 1978, doi:10.1007/BF03023067, https://www.math.ucsd.edu/~ronspubs/78_14_antichains.pdf
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