▾ 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
▾ Maximal element
Maximum 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 maximum element in $P$ if and only if \begin{equation} \forall \, x \in X : (x, m) \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 maximum element in $P$ if and only if \begin{equation} \forall \, x \in X : x \preceq m \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 maximum element in $P$ if and only if \begin{equation} X \preceq m \end{equation}
Dual definition
» Minimum element
Also known as
Greatest element, Orderwise one, Top element
Child definitions
» D1822: Map maximum
» D296: Set upper bound
Results
» R1077: Maximum element is unique