In the 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 (the set 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.
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]