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 $\mathcal T \subseteq \powerset S$ be a subset of $\powerset S$.

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


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


That is:

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


Also see