Quasi-Ideals and \(\mathcal{H}\)-Classes on the Direct Product of Two Semigroups

Let $S$ be a semigroup and $x \in S$. The principal quasi-ideal of $S$ containing $x$ is denoted by $Q(x)$. An $\mathcal{H}$-class of $S$ containing $x$ is denoted by $H_x$. Let $S_1, S_2$ be semigroups. The direct product $S_1 \times S_2$ is defined as the Cartesian product of $S_1$ and $S_2$ equipped with the componentwise binary operation. Let $(a,b) \in S_1 \times S_2$. The direct product of $Q(a) \times Q(b)$ need not to be $Q((a,b))$. In this paper, we provide necessary and sufficient conditions when $Q(a) \times Q(b) = Q((a,b))$ and the conditions when $H_{(a,b)} = H_{a} \times H_{b}$.