Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part III: constructions of vague algebraic notions and vague arithmetic operations

Abstract

As a continuation of Demirci [(2003b) “Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part II: vague algebraic notions”, Int. J. Gen. Syst. 32, 157–175], starting from the algebraic concepts in the classical sense, this paper deals with the constructions of vague semigroups, vague monoids, vague groups, vague rings and vague fields on the basis of many-valued equivalence relations. Vague addition operations and vague multiplication operations on the basis of many-valued equivalence relations, which are naturally derived from vague algebraic notions in Demirci [(2003b) “Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, part II: vague algebraic notions”, Int. J. Gen. Syst. 32, 157–175], and their constructions from the arithmetic operations in the classical sense are also subjects of this paper.

Keywords:

• Fuzzy algebra, fuzzy group, vague group, vague arithmetic, fuzzy arithmetic, fuzzy function