Some fuzzy concepts of BCI, BCK and MV-algebras

We show that in an MV-algebra Z, for each of the listed properties and its fuzzy analogue: implicative, prime, essential, weakly essential, and maximal, the following are equivalent: (i) the fuzzy ideal v has the fuzzy property, (ii) the level ideal Zv has the property, (iii) the fuzzy ideal χZ(v) has the fuzzy property. It is shown that if a non-constant fuzzy ideal v of Z is fuzzy weakly essential and fuzzy prime, then it is either fuzzy essential or fuzzy weakly self-reflexive. This means that if a proper ideal I of Z is prime and satisfies I perpendicular contains B(Z), then either I perpendicular = {0} or I perpendicular perpendicular = I. We establish a precise one-to-one correspondence between the set of fuzzy closed ideals of a quasi-commutative BCI-algebra and its set of fuzzy congruences, giving also a one-to-one correspondence between its set of closed ideals and its congruences.