Definition:Minimal/Ordered Set

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Definition

Let $\left({S, \preceq}\right)$ be a poset.

Let $T \subseteq S$ be a subset of $S$.

An element $x \in T$ is a minimal element of $T$ iff:

$\forall y \in T: y \preceq x \implies x = y$

That is, the only element of $T$ that $x$ succeeds or is equal to is itself.

Alternatively, this can be put as:

$x \in T$ is a minimal element of $T$ iff:

$\neg \exists y \in T: y \prec x$

where $y \prec x$ denotes that $y \preceq x \land y \ne x$.

Compare this definition with that for a smallest element.

An element $x \in T$ is the smallest of $T$ iff:

$\forall y \in T: x \preceq y$

That is, $x$ is comparable to, and precedes, or is equal to, every $y \in S$.