ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D300
Supremum element

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
Let $\mathsf{UB} = \mathsf{UB}_P(E)$ be the D552: Set of upper bounds of $E \subseteq X$ with respect to $P$.
A D2218: Set element $x_0 \in X$ is a supremum of $E$ with respect to $P$ if and only if
 (1) $x_0 \in \mathsf{UB}$ (2) $\forall \, x \in \mathsf{UB} : (x_0, x) \in {\preceq}$

Let $P = (X, {\preceq})$ be a D1103: Partially ordered set.
Let $\mathsf{UB} = \mathsf{UB}_P(E)$ be the D552: Set of upper bounds of $E \subseteq X$ with respect to $P$.
A D2218: Set element $x_0 \in X$ is a supremum of $E$ with respect to $P$ if and only if
 (1) $x_0 \in \mathsf{UB}$ (2) $\forall \, x \in \mathsf{UB} : x_0 \preceq x$