Direct Product of Infinite Family of B-Algebras

Abstract

The concept of the direct product of finite family of B-algebras is introduced by Lingcong and Endam [J. A. V. Lingcong and J. C. Endam, Direct product of B-algebras, Int. J. Algebra,10(1):33-40, 2016.]. In this paper, we introduce the concept of the direct product of infinite family of B-algebras, we call the external direct product, which is a generalization of the direct product in the sense of Lingcong and Endam. Also, we introduce the concept of the weak direct product of B-algebras. Finally, we provide several fundamental theorems of (anti-)B-homomorphisms in view of the external direct product B-algebras.

A Note on External Direct Products of BP-algebras

“…The DP of GK-algebra was defined in 2021 by Kavitha and Gowri [15]. The idea of the EDP, also known as the DP of infinite family of B-algebras, was first suggested by Chanmanee et al [4] in 2022. They also established the idea of the B-algebraic WDP.…”

“…Definition 2.14. [4] Let U ξ = (U ξ ; ⋄ ξ ) and S ξ = (S ξ ; • ξ ) be algebras and ϖ ξ : U ξ → S ξ be a function, ∀ξ ∈ △. Define the function ϖ : ξ∈△ U ξ → ξ∈△ S ξ given by…”

On Direct Product of d-Algebras

“…The concept of d-algebras was first introduced by J. Neggers and H. S. Kim ([9]). A d-algebra X = (X, * , 0) is an algebra of type (2,0), that is, a nonempty set X together with a binary operation * and a constant 0 satisfying some axioms In [1], they introduced and investigated several relations between d-algebras and BCK-algebras and showed that the class of oriented digraphs corresponds in a simple way to the class of edge d-algebras and that arbitrary d-algebras also determine unique edge d-algebras in a natural manner. In 1999, J. Neggers, Y.…”

“…Also, the notion of direct product of 0-commutative BP-algebra and BP-homomorphisms were studied. In 2022, C. Chanmanee, R. Chinram, R. Prasertpong, P. Julatha, and A. Iampan ( [2]) gave the concept an external direct produc and a weak direct product of B-algebras and they provided several fundamental theorems of (anti-)B-homomorphisms in view of the external direct product B-algebras.…”