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.
The lattice of partitions of a finite set typically lacks the Sperner property.
[3] 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 strongly Sperner poset is a graded poset which is k-Sperner for all values of k up to the largest rank value.