partial order
English edit
Noun edit
partial order (plural partial orders)
- (set theory, order theory) (informal) An ordering of the elements of a collection that behaves like that of the natural numbers by size, except that some elements may not be comparable (if all elements are comparable, it is called a total order); (formal) a binary relation that is reflexive, antisymmetric, and transitive.
- 1986, Kenneth R. Goodearl, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, Softcover reprint 2010, page xxi,
- A partial order on a set X is any reflexive, antisymmetric, transitive relation on X. In most cases, partial orders are denoted ≤.
- 1999, Paul A. S. Ward, An Online Algorithm for Dimension-Bound Analysis, Patrick Amestoy, P. Berger, M. Daydé, I. Duff, V. Frayssé, L. Giraud, D. Ruiz (editors), Euro-Par ’99 Parallel Processing: 5th International Euro-Par Conference, Proceedings, Springer, LNCS 1685, page 144,
- The vector-clock size necessary to characterize causality in a distributed computation is bounded by the dimension of the partial order induced by that computation.
- 2008, David Eppstein, Jean-Claude Falmagne, Sergei Ovchinnikov, Media Theory: Interdisciplinary Applied Mathematics, Springer, page 7:
- Consider an arbitrary finite set S. The family of all strict partial orders (asymmetric, transitive, cf. 1.8.3, p. 14) on S enjoys a remarkable property: any partial order P can be linked to any other partial order P’ by a sequence of steps each of which consists of changing the order either by adding one ordered pair of elements of S (imposing an ordering between two previously-incomparable elements) or by removing one ordered pair (causing two previously related elements to become incomparable), without ever leaving the family .
- 1986, Kenneth R. Goodearl, Partially Ordered Abelian Groups with Interpolation, American Mathematical Society, Softcover reprint 2010, page xxi,
Synonyms edit
Hypernyms edit
Hyponyms edit
Related terms edit
Translations edit
binary relation that is reflexive, antisymmetric, and transitive
|
References edit
- B. Dushnik and E. W. Miller, Partially Ordered Sets, Amer. J. Math. 63 (1941), 600-610.
Further reading edit
- Partially ordered set on Wikipedia.Wikipedia
- Complete partial order on Wikipedia.Wikipedia