Definition:Minimal Element/Definition 2

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

$\neg \exists y \in T: y \mathrel {\RR^\ne} x$

where $y \mathrel {\RR^\ne} x$ denotes that $y \mathrel \RR x$ but $y \ne x$.

Also defined as

Most treatments of the concept of a minimal element restrict the definition of the relation $\RR$ to the requirement that it be an ordering.

However, this is not strictly required, and this more general definition as used on $\mathsf{Pr} \infty \mathsf{fWiki}$ is of far more use.

Also see