Left Ideals and $\mathcal{L}$-classes in the Finite Direct Product of Semigroups

Abstract

Let $S_i$ be a semigroup for all $i \in \{1,2,\ldots,n\}$. Then the Cartesian product of $S_1, S_2,
\ldots, S_n$ becomes a semigroup under componentwise multiplication. Let $(s_1,s_2,\ldots,s_n)
\in S_1 \times S_2 \cdots \times S_n$. In this paper, we give necessary and sufficient condition when
the Cartesian product of principal left ideals $L(s_1)\times L(s_2) \times \cdots \times L(s_n)$ is
the principal left ideal $L((s_1,s_2,\ldots,s_n))$ and {the Cartesian} product of $\mathcal{L}$-classes
$L_{s_1}\times L_{s_2} \times \cdots \times L_{s_n}$ is an $\mathcal{L}$-class $L_{(s_1,s_2,\ldots,s_n)}$
in a semigroup $S_1 \times S_2 \times \cdots \times S_n$.