Definition:Minimal Element/Definition 2
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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:
- $\neg \exists y \in T: y \mathrel {\RR^\ne} x$
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
- Results about minimal elements can be found here.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S\text I.3$
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.5$: Relations
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): maximal
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): maximal