Involutory and invertible fuzzy BCK-algebras

The concept of a fuzzy annihilator in a commutative BCK-algebra will be introduced and then we investigate some basic properties. Using this notion, we define an involutory (resp, invertible) fuzzy ideal. We prove that (i) every bounded implicative BCK-algebra is an involutory fuzzy BCK-algebra, and (ii) every categorical commutative BCK-algebra is an invertible fuzzy BCK-algebra. Let X be an involutory and invertible fuzzy BCK-algebra and let FI(X) denote the set of all fuzzy ideals of X. We show that (iii) (FI(X), boolean OR, boolean AND) is a distributive lattice, and (iv) (FI(X), iota, boolean OR, boolean AND, *) is a quasi-Boolean algebra. (C) 2001 Elsevier Science B.V. All rights reserved.