Definition:Maximal/Element

Definition
Let $\left({S, \preceq}\right)$ be a poset.

An element $x \in S$ is maximal iff:


 * $x \preceq y \implies x = y$

That is, the only element of $S$ that $x$ precedes or is equal to is itself.

Comparison with Greatest Element
Compare this definition with that for a greatest element.

An element $x \in S$ is the greatest iff:


 * $\forall y \in S: y \preceq x$

That is, every $y \in S$ is comparable to $x$, and precedes, or is equal to, $x$.

Note that when a poset is in fact a totally ordered set, the terms maximal element and greatest element are equivalent.

Also see

 * Minimal element