Product of TBCH-algebras and TBCH-algebras involving ideals

A BCH-algebra (H, *, 0) equipped with a topology tau on H (also called a BCH-topology on H) is called a topological BCH-algebra (or TBCH-algebra) if the operation * : H x H -> H, defined by *((x, y)) = x * y for any x, y is an element of H, is continuous, where the Cartesian product topology on H x H is furnished by tau. In this paper, we show that given two BCH-algebras (H-1, *(1), 0(1)) and (H-2, *(2), 0(2)), an operation * can be defined on the product H = H-1 x H-2 so that (H, *, 0), where 0 = (0(1), 0(2)), is a BCH-algebra. Moreover, if (H-1, tau(1)) and (H-2, tau(2)) are TBCH-algebras, then (H, tau) is a TBCH-algebra, where tau is the product topology. We also consider in this paper TBCH-algebras involving ideals.