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

Abstract

The existing arithmetics of fuzzy numbers, usually stemming from the α-cut arithmetic, do not preserve some of the important properties of the standard arithmetics of real numbers. Although one cannot expect a generalization of standard arithmetic to perfectly preserve all its properties, the most important properties should be preserved at least in a weakened form. We present a novel framework for arithmetics of extensional fuzzy numbers that more or less preserves 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 by many examples, and, besides its algebraic properties, it is also shown to carry some desirable practical properties. The investigation leads to novel algebraic structures – MI-algebras (MI-monoids, 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 allow us to weaken the standard forms of algebraic properties that are highly desirable to preserve at least in a weakened form. Appropriate properties of suggested MI-structures are introduced and demonstrated by a number of practical examples.