SOME HYPER K-ALGEBRAIC STRUCTURES INDUCED BY MAX-MIN GENERAL FUZZY AUTOMATA

Abstract

We present some connections between the max-min general fuzzy automaton theory and the hyper structure theory. First, we introduce a hyper BCK-algebra induced by a max-min general fuzzy automaton. Then, we study the properties of this hyper BCK-algebra. Particularly, some theorems and results for hyper BCK-algebra are proved. For example, it is shown that this structure consists of different types of (positive implicative) commutative hyper K-ideals. As a generalization, we extend the definition of this hyper BCK-algebra to a bounded hyper K-algebra and obtain relative results. Automata are the prime examples of general computational systems over dis- crete spaces. The incorporation of fuzzy logic into automata theory resulted in fuzzy automata which can handle continuous spaces. In particular, the concept of membership assignment, output mapping, multi-membership resolution, and the concept of acceptance for fuzzy automata were developed in general fuzzy automata. The concept of BCK-algebra has originated on two different ways. One mo- tivation is based on set theory. In set theory, there are three most elementary and fundamental operations introduced by L. Kantorovic and E. Livenson. These fundamental operations are union, intersection and the set difference. Another mo- tivation is taken from classical and non-classical propositional calculi. There are some systems which contain the only implication functor among the logical func- tors. These examples are systems of positive implicational calculus, weak positive implicational calculus by A. Church and BCK-systems by C.As. Meredith. Corsini and Leoreanu (7) presented some connections between a deterministic finite automaton and the hyper algebraic structure theory. Now, here we intend to find some relationships between max-min general fuzzy automata and the hyper K-algebraic structures. So, first we introduce a hyper BCK-algebra induced by max-min general fuzzy automata. Then, we discuss the properties of this BCK- algebra and obtain some useful results. A fuzzy finite-state automaton (FFA) is a six-tuple denoted by ˜