partially ordered set

== English == === Noun === partially ordered set (plural partially ordered sets) (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. 1959 [D. Van Nostrand], Edward James McShane, Truman Arthur Botts, Real Analysis, 2005, Dover, page 28, A partially ordered set means a pair ( P , ≻ ) {\displaystyle (P,\succ )} consisting of a set P {\displaystyle P} and a partial order ≻ {\displaystyle \succ } in P {\displaystyle P} . As usual, when the meaning is clear, we may suppress the notation of " ≻ {\displaystyle \succ } " and speak of the partially ordered set P {\displaystyle P} . The ordered fields defined earlier are easily seen to be examples of partially ordered sets. ==== Usage notes ==== The two senses are commonly used interchangeably, there rarely being a need to distinguish between them. The components of the ordered pair may be referred to separately as the ground set and partial order. ==== Synonyms ==== (set on which a partial order is defined): ground set, poset (ordered pair of set and partial order): poset See also Thesaurus:partially ordered set ==== Hypernyms ==== (order theory): category ==== Hyponyms ==== (order theory): lattice, totally ordered set ==== Translations ==== === See also === complete partial order partial order === Further reading === Complete partial order on Wikipedia.Wikipedia Hasse diagram on Wikipedia.Wikipedia Lattice (order) on Wikipedia.Wikipedia Order theory on Wikipedia.Wikipedia