partially ordered set
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English[edit]
Noun[edit]
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 consisting of a set and a partial order in . As usual, when the meaning is clear, we may suppress the notation of "" and speak of the partially ordered set .
 The ordered fields defined earlier are easily seen to be examples of partially ordered sets.
 1994, I. V. Evstigneev, P. E. Greenwood, Markov Fields over Countable Partially Ordered Sets: Extrema and Splitting, American Mathematical Society, page 35,
 In sections 710 we shall consider random fields over some subsets T of the partially ordered set T_{M}.
 2000, David Arnold, Abelian Groups and Representations of Finite Partially Ordered Sets, Springer, page 45,
 The invention of a derivative of a finite partially ordered set by Nazarova and Roiter in the late 1960s or early 1970s was a seminal event in the subject of representations of finite partially ordered sets (see [Simson 92]).
 1959 [D. Van Nostrand], Edward James McShane, Truman Arthur Botts, Real Analysis, 2005, Dover, page 28,
Usage notes[edit]
 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[edit]
 (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[edit]
 (order theory): category
Hyponyms[edit]
 (order theory): lattice, totally ordered set
Translations[edit]
set having a specified partial order


See also[edit]
Further reading[edit]
 Complete partial order on Wikipedia.Wikipedia
 Hasse diagram on Wikipedia.Wikipedia
 Lattice (order) on Wikipedia.Wikipedia
 Order theory on Wikipedia.Wikipedia