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:
 * $$\not \exists y \in T: y \prec x$$

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".

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 $$x$$.