(untitled) (European Journal of Pure and Applied Mathematics, 2025)
Let S be a semigroup and x S. The principal quasi-ideal of S containing x is denoted by Q(x). An H-class of S containing x is denoted by H_x. Let S_1, S_2 be semigroups. The direct product S_1 × S_2 is defined as the Cartesian product of S_1 and S_2 equipped with the componentwise binary operation. Let (a,b) S_1 × S_2. The direct product of Q(a) × Q(b) need not to be Q((a,b)). In this paper, we provide necessary and sufficient conditions when Q(a) × Q(b) = Q((a,b)) and the conditions when H_(a,b) = H_a × H_b.