On annihilators of BCK-algebras. (English) Zbl 0847.06009

Abujabal, H. A. S.; Obaid, M. A.; Aslam, M.; Thaheem, A. B.

Czech. Math. J. 45, No. 4, 727-735 (1995).

Let \(X\) be a commutative BCK-algebra and \(A\) an ideal of \(X\). To any subset \(B\) of \(X\) we associate the set \((A : B) = \{x \in X : x \wedge B \subseteq A\}\), where \(x \wedge B = \{x \wedge y : y \in B\}\). We show that \((A : B)\) is an ideal of \(X\) and define it as the generalized annihilator of \(B\) (relative to \(A\)). If \(A = \{0\}\), then \((A : B)\) coincides with the usual annihilator of \(B\). These and some other properties of generalized annihilators are contained in Section 3 of this paper. Section 4 contains some applications of generalized annihilators in quotient BCK-algebras and in the theory of prime ideals of BCK-algebras. Using the technique of generalized annihilators, we show that the quotient BCK-algebra of an involutory BCK-algebra is again an involutory BCK-algebra. We also obtain a characterization of prime ideals: A categorical ideal \(A\) is prime if and only if \((A : B) = A\). Section 2 contains some preliminary material for the development of our results.

MSC:

06F35

BCK-algebras, BCI-algebras

Keywords:

commutative BCK-algebra; ideal; generalized annihilator; quotient; prime ideals; involutory BCK-algebra

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