Definition:Minimal Element/Comparison with Smallest Element
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Definition
Compare the definition of a minimal element with that of a smallest element.
Consider the ordered set $\struct {S, \preceq}$ such that $T \subseteq S$.
An element $x \in T$ is the smallest element of $T$ if and only if:
- $\forall y \in T: x \preceq y$
That is, $x$ is comparable with, and precedes, or is equal to, every $y \in T$.
An element $x \in T$ is a minimal element of $T$ if and only if:
- $y \preceq x \implies x = y$
That is, $x$ precedes, or is equal to, every $y \in T$ which is comparable with $x$.
If all elements are comparable with $x$, then such a minimal element is indeed the smallest element.
Note that when an ordered set is in fact a totally ordered set, the terms minimal element and smallest element are equivalent.
Sources
- 1960: Paul R. Halmos: Naive Set Theory ... (previous) ... (next): $\S 14$: Order