MI-algebras: A new framework for arithmetics of (extensional) fuzzy numbers

Distinct so far existing arithmetics of fuzzy numbers, usually stemming from the Zadeh’s extensional principle, do not preserve some of the important properties of the standard arithmetics of real numbers. Although one cannot expect that a generalization of the standard arithmetic will preserve precisely all its properties, the most important properties should be preserved at least in a weakened form. We present a novel framework of arithmetics of extensional fuzzy numbers that preserves more or less all the important (algebraic) properties of the arithmetic of real numbers and thus, seems to be an important seed for further investigations on this topic. The suggested approach to arithmetics of extensional fuzzy numbers is demonstrated on many examples and besides the algebraic properties, it is also shown that it carries some desirable practical properties. The investigation leads to novel algebraic structures – MI-algebras (MImonoids, MI-groups, MI-fields) – that abstract the discussed properties. The main idea of these structures is based on a set of “pseudoidentities” that complements the only commonly used identity element in a monoid/group structure. These pseudoidentities are elements that generalize the identity-like behavior and that allow us to weaken the standard form of algebraic properties that are highly desirable to be preserved at leats in the weakened form. Appropriate properties of suggested MI-structures are introduced and demonstrated again on a plenty of practical examples.