Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, Part I: Fuzzy functions and their applications

Strong fuzzy functions and perfect fuzzy functions based on *-fuzzy equalities were studied, and applied to fuzzy control [Demirci, M. (2000) "Fuzzy functions and their applications", J. Math. Anal. Appl. 252, 495-517] and vague algebra [Demirci, M. (1999b) "Vague groups", J. Math. Anal. Appl. 230, 142-156; Demirci, M. (2002a) "Fundamentals of M-vague algebra and M-vague arithmetic operations", Int. J. Uncertainty Fuzziness Knowledge-Based Systems 10 (1), 25-75]. This paper introduces the general theory of strong (perfect) fuzzy functions on the basis of many-valued equivalence relations, and establishes the fundamental tools of this theory. Another aim of the present paper is to develop the modelling of input/output systems by strong (perfect) fuzzy functions, and is to propose the description of laws of nature by strong (perfect) fuzzy functions. The present paper also forms the rudimentary base of M-vague algebra on the basis of many-valued equivalence relations, presented in the cognate papers [Demirci, M. (2003d) "Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, Part II: Vague algebraic notions", Int. J. General Systems 32, 157-175; Demirci, M. (2003e) "Foundations of fuzzy functions and vague algebra based on many-valued equivalence relations, Part III: Constructions of vague algebraic notions and vague arithmetic operations", Int. J. General Systems 32, 177-201].