# 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. ^ a b Stanley, Richard (1984), "Quotients of Peck posets", Order, 1 (1): 29–34, doi:10.1007/BF00396271, MR 0745587.
2. ^ a b c d Handbook of discrete and combinatorial mathematics, by Kenneth H. Rosen, John G. Michaels
3. ^ 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