Definition:Minimal

Ordered Set
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:


 * $y \preceq x \implies x = y$

That is, the only element of $S$ 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 $y \preceq x \land y \ne x$.

In the context of numbers, the terms smallest, least or lowest are often informally used for minimal.

The term minimum is frequently is frequently used, particularly when $S$ is a set of numbers.

Alternative Definition
Some sources define the smallest element or the minimum element as:

$x \in T$ is the minimum element of $\left({T, \preceq}\right)$ iff:
 * $\forall y \in T: x \preceq y$

but this limits the concept to sets where such an $x$ is comparable to all $y \in T$.

Mapping
Let $f$ be a mapping defined on a poset $\left({S, \preceq}\right)$.

Let $f$ be bounded below by an infimum $B$.

It may or may not be the case that $\exists x \in S: f \left({x}\right) = B$.

If such a value exists, it is called the minimal value or minimum of $f$ on $S$, and that this minimum is attained at $x$.

Also see

 * Maximal

Sources for Ordered Set

 * : $\S 3.1$
 * : $\S 7$
 * : $\S 2.7$
 * : $\S 1.5$

Sources for Mapping

 * : $\S 7.13$