Definition:Minimal Element/Definition 1
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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 minimal element (under $\RR$) of $T$ if and only if:
- $\forall y \in T: y \mathrel \RR x \implies x = y$
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
- Results about minimal elements can be found here.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1968: A.N. Kolmogorov and S.V. Fomin: Introductory Real Analysis ... (previous) ... (next): $\S 3.1$: Partially ordered sets
- 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): Chapter $1$: Sets and mappings: $\S 1.5$: Ordered Sets
- 1996: Winfried Just and Martin Weese: Discovering Modern Set Theory. I: The Basics ... (previous) ... (next): Part $1$: Not Entirely Naive Set Theory: Chapter $2$: Partial Order Relations