partial order

Definitions

(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.

Equivalents

Examples

“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.”

“Consider⟳ an arbitrary finite set⟳ S. The family #92;mathcal#123;P#125; 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 #92;mathcal#123;P#125;.”

CEFR level