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
- ^ a b Stanley, Richard (1984), "Quotients of Peck posets", Order, 1 (1): 29–34, doi: 10.1007/BF00396271, MR 0745587, S2CID 14857863.
- ^ a b c d Handbook of discrete and combinatorial mathematics, by Kenneth H. Rosen, John G. Michaels
- ^ Graham, R. L. (June 1978), "Maximum antichains in the partition lattice" (PDF), The Mathematical Intelligencer, 1 (2): 84–86, doi: 10.1007/BF03023067, MR 0505555, S2CID 120190991
.svg){kind=small} | This combinatorics-related article is a stub. You can help Wikipedia by expanding it. |