Definition:Strictly Maximal Element

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

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

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

$\forall y \in T: x \not \mathrel \RR y$

Also known as

This strictly maximal relation is often referred to as a maximal relation in some expositions of this subject.

The appellation strictly maximal has been coined by $\mathsf{Pr} \infty \mathsf{fWiki}$ so as to distinguish between this and the more mainstream concept of a maximal element which does not preclude $x \mathrel \RR x$.

Also see