Right regular triples of semigroups

Let $\mathcal{M}(S;\mathrm{\Lambda };P)$ denote a Rees $I\times \mathrm{\Lambda }$ matrix semigroup without zero over a semigroup $S$, where $I$ is a singleton. If ${\theta }_S$ denotes the kernel of the right regular representation of a semigroup $S$, then a triple $A,B,C$ of semigroups is said to be right regular, if there are mappings $AB$ and $BC$ such that $\mathcal{M}(A;B;P)\mathrm{/}{\theta }_{\mathcal{M}(A;B;P)}\cong \mathcal{M}(C;B;P^{\mathrm{\prime }})$. In this paper we examine right regular triples of semigroups.