antichain

A subset $A$ of a poset $(P,<_{P})$ is an antichain if no two elements are comparable. That is, if $a,b\in A$ then $a\nless_{P}b$ and $b\nless_{P}a$ .

A maximal antichain of $T$ is one which is maximal.

In particular, if $(P,<_{P})$ is a tree then the maximal antichains are exactly those antichains which intersect every branch, and if the tree is splitting then every level is a maximal antichain.

Title	antichain
Canonical name	Antichain
Date of creation	2013-03-22 12:52:25
Last modified on	2013-03-22 12:52:25
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	6
Author	Henry (455)
Entry type	Definition
Classification	msc 05C05
Classification	msc 03E05
Related topic	TreeSetTheoretic
Related topic	Aronszajn
Defines	antichain
Defines	maximal antichain