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Concept
Antichain
Summary
In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable.
The size of the largest antichain in a partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (totally ordered subsets) into which the set can be partitioned. Dually, the height of the partially ordered set (the length of its longest chain) equals by Mirsky's theorem the minimum number of antichains into which the set can be partitioned.
The family of all antichains in a finite partially ordered set can be given join and meet operations, making them into a distributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families
and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting the number of antichains of a finite partially ordered set is #P-complete.
Let be a partially ordered set. Two elements and of a partially ordered set are called comparable if If two elements are not comparable, they are called incomparable; that is, and are incomparable if neither
A chain in is a subset in which each pair of elements is comparable; that is, is totally ordered. An antichain in is a subset of in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in
(However, some authors use the term "antichain" to mean strong antichain, a subset such that there is no element of the poset smaller than two distinct elements of the antichain.)
A maximal antichain is an antichain that is not a proper subset of any other antichain. A maximum antichain is an antichain that has cardinality at least as large as every other antichain. The of a partially ordered set is the cardinality of a maximum antichain.
Official source
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