# zbMATH — the first resource for mathematics

On proper BCH-algebras. (English) Zbl 0583.03050
A BCI algebra can be characterized by the axioms: ((x*y)*(x*z))*$$(z*y)=0$$, $$(x*y)*z=(x*z)*y$$, $$x*x=0$$, which correspond respectively to the combinators B, C and I and the rules of substitution of equality and: $$x*y=y*x=0$$ implies $$x=y$$. A BCH algebra has all the above except the axiom corresponding to $$B$$ and should therefore probably be called a CI algebra.
In the present paper the authors give two examples of proper BCH algebras, and show that several standard BCI properties still hold. They also define the direct product of two algebras and show that the direct product of a BCH algebra and a proper BCH algebra is a proper BCH algebra.
Reviewer: M.W.Bunder

##### MSC:
 03G25 Other algebras related to logic
##### Keywords:
BCK algebra; BCI algebra; BCH algebra; direct product