The second isomorphism theorem for B-algebras

Abstract

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.

Some properties of cyclic B-algebras

“…If A ⊆ X, then we denote |A| B as the cardinality of A. Definition 2.7 [3] Let H, K be subalgebras of X. Define the subset HK of X to be the set HK = {x ∈ X : x = h * (0 * k) for some h ∈ H, k ∈ K}.…”

On cyclic B-algebras

“…In [3], if {N α : α ∈ A } is any nonempty collection of subalgebras of a Balgebra X, then α∈A N α is a subalgebra of X. Thus, we can have the following definition:…”

On JB-semigroups

“…A normal subset of X is a subalgebra of X. Moreover, in [2], the intersection of any nonempty collection of (normal) subalgebras of X is also a (normal) subalgebra of X. Definition 2.2 A JB-semigroup is a nonempty set X together with two binary operations * and • and a constant 0 satisfying the following: (i) implies that if (X; * , •, 0) is a JB-semigroup, then all properties pertaining to the binary operation * with respect to the B-algebra (X; * , 0) also hold for the JB-semigroup (X; * , •, 0).…”

“…Then I is a sub JB-semigroup of X and I is a JB-ideal for every sub JB-semigroup of X containing I. In [2], if A is a subalgebra of a B-algebra (X; * , 0), then A * A = A. Moreover, if A and B are normal subalgebras of X, then A * B = B * A is a normal subalgebra of X.…”

“…In [2], if J is a normal subalgebra of Y , then ϕ −1 (J) is a normal subalgebra of X. If I is a normal subalgebra of X and ϕ is onto, then ϕ(I) is a normal subalgebra of Y .…”