Some fuzzy concepts of BCI, BCK and MV-algebras

✍ Scribed by C.S. Hoo

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 602 KB
• Volume: 18
• Category: Article
• ISSN: 0888-613X
• DOI: 10.1016/s0888-613x(98)00004-8

✦ Synopsis

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 Z,. has the property, (iii) the fuzzy ideal Zz, 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 ~ C B(Z), then either 11 = {0} or I ±z = 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.