Definition:Smallest Element/Subset
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Definition
Let $\struct {S, \preceq}$ be an ordered set.
Let $T \subseteq S$ be a subset of $S$.
An element $x \in T$ is the smallest element of $T$ if and only if:
- $\forall y \in T: x \preceq \restriction_T y$
where $\preceq \restriction_T$ denotes the restriction of $\preceq$ to $T$.
Also see
Sources
- 1967: Garrett Birkhoff: Lattice Theory (3rd ed.): $\S\text I.3$
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 3$: Relations