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

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
Full Text:

### References:

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