Fuzzy Sets Algebra and Complementary Operations

Abstract

In this paper, the mathematical structure of a fuzzy sets algebra, which is a distributive, complete lattice, containing the power set of X, the universe of discourse, is investigated. A complementary operation N is definedsuch as (1) N(A)=A^c for each crisp set A, where A^c stands for the usual complement of A, (2) contrapositional, and (3) N^3=N (quasi involutive). It is shown that there are only three reasonable complementary operations : (1) one-to-oneoperations--Zadeh's complement, Sugeno's λ-complements, (2) an intuitionistic complement, and (3) an exclusion complement.