Definition:Maximal/Ordered Set

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Definition

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

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

An element $x \in T$ is a maximal element of $T$ iff:

$x \preceq y \implies x = y$

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

Alternatively, this can be put as:

$x \in T$ is a maximal element of $T$ iff:

$\neg \exists y \in T: x \prec y$

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

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, $x$.