# Definition:Minimal

## Definition

### Relation

Let $\left({S, \mathcal R}\right)$ be a relational structure.

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

An element $x \in T$ is an $\mathcal R$-minimal element of $T$ if and only if:

$\forall y \in T: y \not \mathrel {\mathcal R} x$

### Ordered Set

Let $\left({S, \preceq}\right)$ be an ordered set.

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

An element $x \in T$ is a minimal element of $T$ if and only if:

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

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

### Minimal Set

Let $S$ be a set.

Let $\powerset S$ be the power set of $S$.

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

Let $\struct {\mathcal T, \subseteq}$ be the ordered set formed on $\mathcal T$ by $\subseteq$ considered as an ordering.

Then $T \in \mathcal T$ is a minimal set of $\mathcal T$ if and only if $T$ is a minimal element of $\struct {\mathcal T, \subseteq}$.

That is:

$\forall X \in \mathcal T: X \subseteq T \implies X = T$

### Mapping

Let $f: S \to \R$ be a real-valued function.

Let $f$ be bounded below by a 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 $x$.