P-sets

Abstract

Starting from a poset P and a set A, we introduce P-sets as a natural generalization of Ω-sets. A P-set on A is defined by adding to A a special map from A² to P, which generalizes the classical equality relation. We prove that P-sets on A are naturally obtained from centralized closure systems in a family of all weak equivalences on A. Moreover, for every P-set there is a canonical representation in which the used centralized closure system replaces the poset P. Further, we present a classification of all P-sets by the family of cuts, where A and P are fixed. Different P-sets may have equal collections of cut sets. Necessary and sufficient conditions under which this happens are presented; i.e., all P-sets are classified according to the equality of collections of cut sets.