Definition:Maximal

Ordered Set
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.

In the context of numbers, the terms greatest or highest are often informally used for maximal.

The term maximum is frequently used, particularly when $S$ is a set of numbers.

Alternative Definition
Some sources define the greatest element or the maximum element as:

$x \in T$ is the maximum element of $\left({T, \preceq}\right)$ iff:
 * $\forall y \in T: y \preceq x$

but this limits the concept to sets where such an $x$ is comparable to all $y \in T$.

Mapping
Let $f$ be a mapping defined on a poset $\left({S, \preceq}\right)$.

Let $f$ be bounded above by a supremum $B$.

It may or may not be the case that $\exists x \in S: f \left({x}\right) = B$.

If such a value exists, it is called the maximal value or maximum of $f$ on $S$, and that this maximum is attained at $x$.

Also see

 * Minimal

Sources for Ordered Set

 * : $\S 23$
 * : $\S 3.1$
 * : $\S 7$
 * : $\S 2.7$

Sources for Mapping

 * : $\S 7.13$