On Kreb Algebras

In this paper, kreb algebras are introduced. It is shown that that the class of kreb algebras is a wider class than the class of BCI algebras. Properties of kreb algebras are presented. A new binary relation different from the one in classical BCI  algebras is defined in kreb algebras, and it is shown that with this relation, kreb algebras are partially ordered sets. It is  established that this relation is not a partial ordering in classical BCI algebras. The partial ordering is then used to  investigate other properties of kreb algebras. New kinds of ideals and homomorphisms are also introduced and studied  in kreb algebras independent of these concepts in classical BCI algebras.