Maximal and minimal elements

A maximal element has no strictly greater element; a minimal element has no strictly smaller one. Either can be nonunique, unlike a greatest or least element. These notions are defined for arbitrary partially ordered sets.

An element $m$ of a partially ordered set is maximal when $m\le x$ implies $x=m$. A poset can have several maximal elements and need not have a greatest element.

An element is maximal when there is no strictly greater element of the partially ordered set. This is different from being a greatest element, which must dominate every other element.

An element $m$ of a partially ordered set is minimal when $x\le m$ implies $x=m$.