Definition:Strictly Minimal Element

Definition

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 minimal element (under $\RR$) of $T$ if and only if:

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

Also known as

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

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

Also see

• Definition:Minimal Element

• Definition:Strictly Maximal Element
• Definition:Maximal Element

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