Arithmetics of extensional fuzzy numbers - part I: Introduction

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Up to our best knowledge, 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 classical (real) numbers. Obviously, although we cannot expect that a generalization of standard arithmetic will preserve precisely all its properties however, at least the most important ones should be preserved. 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 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.