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
Maximal element
Definition D668
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}
Children
Map maximum
Set upper bound
Results
Maximum element is unique