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.
Let be a preordered set and an element of . Then is maximal in if, whenever in , we have . Dually, is minimal in if, whenever in , we have .
If has a top element, then this is the unique (up to equivalence) maximal element of .
Suppose that is totally ordered. Then a maximal element of is the same as a top element of .
Suppose that is finite and has a unique maximal element . Then is a top element of .
According to Zorn's Lemma, if every totally ordered subset of has an upper bound in , then has a maximal element.
Let be with , , and no other nontrivial ordering. Then and are both maximal in (but of course not tops).
Let be the disjoint union of (the poset of natural numbers) and a singleton {a}. Then is the unique maximal element of but still not a top.
Created on February 20, 2012 at 11:29:08. See the history of this page for a list of all contributions to it.