How to Tell When a Product of Two Partially Ordered Spaces Has a Certain Property?

Authors

Francisco Zapata, The University of Texas at El Paso
Olga Kosheleva, The University of Texas at El PasoFollow
Karen Villaverde, New Mexico State University - Main CampusFollow

Publication Date

8-2011

Comments

Technical Report: UTEP-CS-11-22a

To appear in Journal of Uncertain Systems, 2012, Vol. 6, No. 2.

Abstract

In this paper, we describe how checking whether a givenproperty F is true for a product A₁ X A₂ of partiallyordered spaces can be reduced to checking several relatedproperties of the original spaces A_i.

This result can be useful in the analysis of propertiesof intervals [a,b] = {x: a <= x <= b}over general partially ordered spaces -- such as the spaceof all vectors with component-wise order or the set of allfunctions with component-wise ordering f <= g <-->for all x (f(x) <= g(x)). When we consider sets of pairs ofsuch objects A₁ X A₂, it is natural to define the orderon this set in terms of orders in A₁ and A₂ -- this is, e.g.,how ordering and intervals are defined on the set R² of all2-D vectors.

This result can also be useful in the analysis of orderedspaces describing different degrees of certainty in expert knowledge.