PlanetMath: antichain

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

antichain

(Definition)

A subset of a poset 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 is one which is maximal.

In particular, if 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.

"antichain" is owned by Henry.

(view preamble)

See Also: tree (set theoretic), Aronszajn tree

Also defines:

antichain, maximal antichain

Log in to rate this entry.
(view current ratings)

Cross-references: level, branch, intersect, tree, comparable, poset, subset
There are 17 references to this entry.

This is version 3 of antichain, born on 2002-07-26, modified 2002-08-18.
Object id is 3212, canonical name is Antichain.
Accessed 6176 times total.

Classification:

AMS MSC:	05C05 (Combinatorics :: Graph theory :: Trees)
	03E05 (Mathematical logic and foundations :: Set theory :: Other combinatorial set theory)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact