I~F~RMAT~~~ SCIENCES 45,51-59 (1988) 51 A zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Preserwtion Theorem for Fuzzy Number Theory GIANGIACOMO GERLA ~i~Jarti~ento ~ute~ut~~a ed Applimzioni, Universitri di Napoh, via Merzocunnone 8, Communicated by Azriel Rosenfeld ABSTRACT A precise formulation of Zadeh’s extension principle for fuzzy number theory is given. Moreover, it is proved that if an equational property p = q holds for the real numbers and the variables occurring in the expressions p and 4 are distinct, then this property holds for the fuzzy numbers as well. 1. INTRODUCTION As is well known, there are some properties of the real numbers, such as associativity and commutativity, which hold for the fuzzy numbers. On the other hand, there are other basic properties, such as distributivity or the existence of an inverse, which do not hold for the fuzzy numbers [l, 31. The main purpose of this paper is to examine this situation in a general way. Among other things, we prove that if a property is expressed by an equation p = 4, where p and 4 are terms whose variables are distinct, then this property is preserved. We also examine the question of an exact interpretation of an expression representing a function in fuzzy number theory, that is, the question of an unambiguous application of Zadeh’s extension principle [S]. 2. PRELIMINARIES If X is a subset of the real number set R, we write VX and AX to denote the least upper bound and the greatest lower bound of X, respectively. If x= (Xl,..., x,~) is finite, we use also the notations xi v * * * v x, and xi ,& . . . A x,. By R* we denote the set of the normal jii.zy numbers (for short, fuzzy numbers), i.e. the set of the functions r: R -+ [O,l] such that T(X) =l for OElsevier Science Publishing Co., Inc. 1988 52 Vanderbih Ave., New York, NY 10017