partially ordered set
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English
[edit]![](http://upload.wikimedia.org/wikipedia/commons/thumb/7/76/Divisibility_poset.svg/220px-Divisibility_poset.svg.png)
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 7-10 we shall consider random fields over some subsets T of the partially ordered set TM.
- 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
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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