Characterizations of N-Ary Fuzzy Set Operations Which Induce Homomorphic Random Set Operations
Abstract
The fledgling discipline of fuzzy set theory has now grown to encompass well over two thousand papers. The thrust of this paper is to expand earlier unary and binary homomorphic relations between fuzzy and random set operations, by obtaining systematic characterizations for three classes of n-ary fuzzy set operations, which under the mapping for any fuzzy subset A of X, yield homomorphic images, i.e., ordinary set operations applicable to the random sets S(A): Binary operations are presented in Theorem 3, unary operations, in Theorem 4, and n-ary operations in Theorem 6: Theorem 5 demonstrates the general case, Corollary 1 exhibits characterizations for statistically independent image random sets, and Corollary 2 is concerned with a certain simplified subclass of continuous operations. In the case of binary fuzzy set operations leading to homomorphic random set compositions the characterizing structure for the membership function is a simple sum of at most four terms, each term being also an elementary combination of the individual component membership functions and/ or a functional form that satisfies a constraint equation.
Document Details
- Document Type
- Technical Report
- Publication Date
- Jan 01, 1982
- Accession Number
- ADA241215
Entities
People
- I. R. Goodman