05C05-Antichain.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
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\pmcreated{2013-03-22 12:52:25}
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\pmtitle{antichain}
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\pmtype{Definition}
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\pmdefines{maximal antichain}

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\begin{document}
A subset $A$ of a poset $(P,<_P)$ is an \emph{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 \emph{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.
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