ThmDex – An index of mathematical definitions, results, and conjectures.
Set of symbols
Alphabet
Deduction system
Theory
Zermelo-Fraenkel set theory
Set
Binary cartesian set product
Binary relation
Binary endorelation
Preordering relation
Partial ordering relation
Partially ordered set
Minimal element
Minimum element
Definition D1821
Map minimum
Formulation 0
Let $P = (Y, {\preceq})$ be a D1103: Partially ordered set.
Let $f : X \to Y$ be a D18: Map.
A D2218: Set element $m \in f(X)$ is a minimum of $f$ with respect to $P$ if and only if \begin{equation} \forall \, y \in f(X) : m \preceq y \end{equation}
Formulation 1
Let $P = (Y, {\preceq})$ be a D1103: Partially ordered set.
Let $f : X \to Y$ be a D18: Map.
A D2218: Set element $m \in f(X)$ is a minimum of $f$ with respect to $P$ if and only if \begin{equation} \forall \, x \in X : m \preceq f(x) \end{equation}