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T-ideals in BZ-algebras and T-type BZ-algebras

In this paper, we introduce the notions of T-type BZ-algebra and T-ideal in BZ-algebra as the generalizations of one of BCI-algebras. We also introduce the notion of associative ideal in BZ-algebra. We investigate the properties and characterizations of them, and obtain the following main results: Let be a BZ-algebra, (1) X is a quasi-associative which implies that X is a T-type BZ-algebra, and the inverse is not true; (2) If X is a T-type BZ-algebra, then X has a homomorphism with an involution group. (3) An ideal I of X is an T-ideal if and only if (0*x)*x ∈I for any x ∈X. (4) X is a T-type BZ-algebra if and only if for each ideal I of X is a T-ideal or the zero ideal {0} of X is a T-ideal. (5) For any ideal I of X, the quotient algebra X over I (i.e. X/I) is a T-type BZ-algebra if and only if I is a T-ideal of X. (6) an ideal of BZ-algebra is associated if and only if it is anti-grouped ideal and T-ideal. (7) BZ-algebra X is associated if and only if each ideal of X (or its zero ideal) is an associative ideal. (8) For any ideal I of X, the quotient algebra X over I (i.e. X/I) is associated if and only if I is an associative ideal of X.