▾ 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
Formulation 0
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) $X \neq \emptyset$
A D2218: Set element $m \in X$ is a minimum element in $P$ if and only if \begin{equation} \forall \, x \in X : (m, x) \in {\preceq} \end{equation}
Formulation 1
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) $X \neq \emptyset$
A D2218: Set element $m \in X$ is a minimum element in $P$ if and only if \begin{equation} \forall \, x \in X : m \preceq x \end{equation}
Formulation 2
Let $P = (X, {\preceq})$ be a D1103: Partially ordered set such that
(i) $X \neq \emptyset$
A D2218: Set element $m \in X$ is a minimum element in $P$ if and only if \begin{equation} m \preceq X \end{equation}
Dual definition
» Maximum element
Also known as
Least element, Bottom element, Orderwise zero
Child definitions
» D1821: Map minimum
» D297: Set lower bound
Results
» R1083: Minimal element is minimum element in ordered set
» R1120: Antitonicity of minimum
» R1076: Minimum element is unique