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 $\struct {S, \preceq}$ 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:
- $\forall X \in \mathcal T: X \subseteq T \implies X = T$
Also see
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: minimal
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: minimal