In this paper, we introduce the notion of JB-semigroup. We prove that every ring determines a JB-semigroup, but the converse need not be true. We also introduce the notions of JB-field and JB-domain, and we prove that every JB-field is a JB-domain and every finite JBdomain is a JB-field. Moreover, we introduce the notion of JB-ideal of JB-semigroup, and we construct quotient JB-semigroup via JB-ideal. Furthermore, we introduce the notion of JB-homomorphism of JBsemigroups, and we provide some of its properties.
In this paper, some properties of B-homomorphism are provided and the Second Isomorphism Theorem for B-algebras is proved. Furthermore, a sufficient and necessary condition for the product HK of subalgebras to be a subalgebra is proved.
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