nLab
maximal element

Maximal elements

Idea

An element of a poset (or proset) is maximal if no other (inequivalent) element is greater. A maximum must be maximal, and a maximal element of a toset must be a maximum. However, it’s easy to find posets with maximal elements that aren't maxima, or even with a unique maximal element that isn't a maximum. The existence of a maximal element is often given by Zorn's lemma.

Definition

Let PP be a preordered set and xx an element of PP. Then xx is maximal in PP if, whenever xyx \leq y in PP, we have yxy \leq x. Dually, xx is minimal in PP if, whenever yxy \leq x in PP, we have xyx \leq y.

Properties

Elementary properties

If PP has a top element, then this is the unique (up to equivalence) maximal element of PP.

Suppose that PP is totally ordered. Then a maximal element of PP is the same as a top element of PP.

Suppose that PP is finite and has a unique maximal element xx. Then xx is a top element of PP.

Deep properties

According to Zorn's Lemma, if every totally ordered subset of PP has an upper bound in PP, then PP has a maximal element.

Examples

Let PP be {a,b,c}\{a,b,c\} with aba \leq b, aca \leq c, and no other nontrivial ordering. Then bb and cc are both maximal in PP (but of course not tops).

Let PP be the disjoint union of \mathbb{N} (the poset of natural numbers) and a singleton {a}. Then aa is the unique maximal element of PP but still not a top.

Created on February 20, 2012 11:33:11 by Toby Bartels (98.23.156.44)