Definition:Maximal

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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 $x \preceq y \land x \ne y$.

Let $S$ be a set.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.

Let $\mathcal T \subseteq \mathcal P \left({S}\right)$ be a subset of $\mathcal P \left({S}\right)$.

Let $\left({\mathcal T, \subseteq}\right)$ be the poset formed on $\mathcal T$ by $\subseteq$ considered as an ordering.

Then $T \in \mathcal T$ is a maximal set of $\mathcal T$ iff $T$ is a maximal element of $\left({\mathcal T, \subseteq}\right)$.

That is:

$\forall X \in \mathcal T: T \subseteq X \implies T = X$

Let $f$ be a mapping defined on a subset of the real numbers $S \subseteq \R$.

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 $x$.