Definition:Minimal

Ordered Set
Let $$\left({S; \preceq}\right)$$ be a poset.

An element $$x \in S$$ is minimal iff:

$$y \preceq x \Longrightarrow x = y$$

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

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 $$x$$.