Powerset Operator Foundations For Poslat Fuzzy Set Theories And Topologies

Abstract

This chapter summarizes those powerset operator foundations of all mathematical and fuzzy set disciplines in which the operations of taking the image and preimage of (fuzzy) subsets play a fundamental role; such disciplines include algebra, measure theory and analysis, and topology. We first outline such foundations for the fixed-basis case—where the lattice of membership values or basis is fixed for objects in a particular category, and then extend these foundations to the variable-basis case—where the basis is allowed to vary from object to object within a particular category. Such foundations underlie almost all chapters of this volume. Additional applications include justifications for the Zadeh Extension Principle [19] and characterizations of fuzzy associative memories in the sense of Kosko [9]. Full proofs of all results, along with additional material, are found in Rodabaugh [16]—no proofs are repeated even when a result below extends its counterpart of [16]; some results are also found in Manes [11] and Rodabaugh [14, 15].