Definition:Maximal
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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:
- $\forall X \in \TT: T \subseteq X \implies T = X$