Generalization of some fuzzy ideals in BCK-algebras

A BCK-algebra which was introduced by K. Iseki and S. Tanaka (1978) is an important class of logical algebra which originated from two different methods: set theory and classical and non-classical propositional calculus. Clearly a BCK-algebra is a generalization of the notion of sets, with set substraction as the only fundamental non nullary operation and the notion of implication algebra. The concept of fuzzy sets introduced by L.A. Zadeh (1965) was applied to BCK-algebras by O.G. Xi (1991). Since then, many researchers have investigated various properties of this algebra. One of the main problem in fuzzy mathematics is how to carry out the ordinary concepts for the fuzzy case. The difficulty lies in how to pick out the rational generalization from a large number of available approaches. Fuzzy ideal is different from the ordinary ideal in the sense that one can not say which BCK-algebra element either belongs or does not belong to the fuzzy ideal under consideration. We discuss the notion of n-fold positive implicative ideals, n-fold commutative ideals and n-fold implicative ideals as a natural generalization of fuzzy positive implicative ideals, fuzzy commutative ideals and fuzzy implicative ideals in BCK-algebra. Then, using the notion of fuzzy point, we give some characterizations of fuzzy n-fold positive implicative ideals, fuzzy n-fold commutative ideals, fuzzy n-fold implicative ideals and establish some relation among them