Possibilistic Measures Taking Their Values in Spaces Given by Inclusion-Closed Fragments of Power-Sets

摘要

Fuzzy sets with non-numerical membership degrees, as well as the related possibilistic distributions and measures, have been developed mostly under the simplifying assumption that their membership or possibility degrees are taken from a complete lattice, so that all the supremum and infimum values to be processed are defined. In this paper the conditions imposed on the space in which possibilistic measures take their values are weakened in such a way that this space is defined by an inclusion-closed system of subsets of a space X, so that all subsets of sets from this system are in this system also incorporated. Let us note that the system of all finite subsets of an infinite space X (an incomplete lattice) or the system of all subsets of X the cardinality of which does not exceed a fixed positive integer are particular and intuitive examples of inclusion-closed systems of subsets of X. Some simple properties of such possibilistic measures are analyzed and compared with the properties of their standard versions taking values in complete lattices.