Product of TBCH-algebras and TBCH-algebras involving ideals

A BCH-algebra (H,*,0) equipped with a topology τ on H (also called a BCH-topology on H) is called a topological BCH-algebra (or TBCH-algebra) if the operation * : H × H → H, defined by *((x, y)) = x * y for any x, y ∈ H, is continuous, where the Cartesian product topology on H × H is furnished by τ. In this paper, we show that given two BCH-algebras (H1, *1, 01) and (H2, *2, 02), an operation * can be defined on the product H = H1 × H2 so that (H, *, 0), where 0 = (01, 02), is a BCH-algebra. Moreover, if (H1, τ1) and (H2, τ2) are TBCH-algebras, then (H, τ) is a TBCH-algebra, where τ is the product topology. We also consider in this paper TBCH-algebras involving ideals. © 2022 Forum-Editrice Universitaria Udinese SRL. All rights reserved.