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 finite and has a unique maximal element . Then is a top element of .
Let be with , , and no other nontrivial ordering. Then and are both maximal in (but of course not tops).