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 seen instead of minimal element.

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