# Definition:Maximal

## Definition

### Maximal Element of Set

Let $\struct {S, \RR}$ be a relational structure.

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

An element $x \in T$ is a maximal element (under $\RR$) of $T$ if and only if:

$x \mathrel \RR y \implies x = y$

### Maximal Set

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

Let $\TT \subseteq \powerset S$ be a subset of $\powerset S$.

Let $\struct {\TT, \subseteq}$ be the ordered set formed on $\TT$ by $\subseteq$ considered as an ordering.

Then $T \in \TT$ is a maximal set of $\TT$ if and only if $T$ is a maximal element of $\struct {\TT, \subseteq}$.

That is:

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