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 seen instead of "maximal element".

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 $$x$$.