ThmDex – An index of mathematical definitions, results, and conjectures.
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Definition D667
Minimum element

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 $$\forall \, x \in X : (m, x) \in {\preceq}$$

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 $$\forall \, x \in X : m \preceq x$$

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 $$m \preceq X$$
Children
 ▶ Map minimum ▶ Set lower bound
Results
 ▶ Antitonicity of minimum ▶ Minimal element is minimum element in ordered set ▶ Minimum element is unique