Sperner property of a partially ordered set
In the mathematics branch of order theory, a graded partially ordered set is said to have the Sperner property (and hence called Sperner poset), if no antichain within it is larger than the largest rank level (the set of elements of the same rank) in the poset. [1] Equivalently, it is the property that some rank level is a maximum antichain (since a rank level is an antichain).[2]
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, MR0745587.
- ^ a b c d Handbook of discrete and combinatorial mathematics, by Kenneth H. Rosen, John G. Michaels
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