ANTI FUZZY IDEALS IN WEAK BCC-ALGEBRAS

Abstract

Abstract. We characterize the main types of anti fuzzy ideals in weakBCC-algebras. 1. IntroductionBCC-algebras (called also BIK + -algebras) are an algebraic model of BIK -logic, i.e., implicational logic based on modus ponens and some axioms schemecontaining the combinators B, I, and K. Weak BCC-algebras (called also BZ-algebras) have the same partial order as BCC-algebras and BCK-algebras butdo not have a minimal element. Many mathematicians studied various typesof algebras such as BCI-algebras, B-algebras, implication algebras, G-algebras,Hilbert algebras and do on. All these algebras have one distinguished element,satisfy some common identities and have a similar partial order. In fact, allthese algebras are a generalization or a special case of weak BCC-algebras. So,results obtained for weak BCC-algebras are, in some sense, fundamental forthese algebras, especially for BCC/BCH/BCI/BCK-algebras.A very important role in the theory of such algebras plays ideals. In BCK-algebras ideals are induced by partial order or by homomorphisms. All idealsdetermine congruences. In BCC-algebras there are congruences which are notdetermined by ideals [3]. Moreover, in BCC-algebras relations determined byideals (in the same way as in BCK-algebras) are not congruences, in general.So, in BCC-algebras the new concept of ideals should br introduced. Similarlyin weak BCC-algebras.2. PreliminariesIn this section, we give basic de nitions and facts on weak BCC-algebras.De nition 1. A weak BCC-algebra Xis an abstract algebra (X;;0) of type(2;0) satisfying the following axioms(i) ((xy) (zy)) (xz) = 0,(ii) xx= 0,