Definition:Maximal Element/Definition 2

Definition
Let $\left({S, \preceq}\right)$ be an ordered set.

Let $T \subseteq S$ be a subset of $S$.

An element $x \in T$ is a maximal element of $T$ :
 * $\neg \exists y \in T: x \prec y$

where $x \prec y$ denotes that $x \preceq y \land x \ne y$.

That is, $x$ has no strict successor.

Also see

 * Equivalence of Definitions of Maximal Element