partially ordered set

Noun

Meaning

(set theory, order theory, loosely) A set that has a given, elsewhere specified partial order.

(set theory, order theory, formally) The ordered pair comprising a set and its partial order.

Synonyms

Examples

An abstract polyhedron is a certain kind of partially ordered set (poset) of elements, such that adjacencies, or connections, between elements of the set correspond to adjacencies between elements (faces, edges, etc.) of a polyhedron. Source: Internet

Definition A partially ordered set (poset) P is said to satisfy the ascending chain condition (ACC) if every strictly ascending sequence of elements eventually terminates. Source: Internet

Empty chain as boundary case In the formulation of Zorn's lemma above, the partially ordered set P is not explicitly required to be non-empty. Source: Internet

Examples * The cofinality of a partially ordered set with greatest element is 1 as the set consisting only of the greatest element is cofinal (and must be contained in every other cofinal subset). Source: Internet

For instance, a lattice is a partially ordered set in which all finite subsets have both a supremum and an infimum, and a complete lattice is a partially ordered set in which all subsets have both a supremum and an infimum. Source: Internet

In mathematics, a complete lattice is a partially ordered set in which all subsets have both a supremum (join) and an infimum (meet). Source: Internet

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