Right regular triples of semigroups

Let ${\cal M}(S; \Lambda; P)$ denote a Rees $I\times \Lambda$ 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 $A\stackrel{P}{\longleftarrow}B$ and $B\stackrel{P'}{\longrightarrow}C$ such that ${\cal M}(A; B; P)/\theta_{{\cal M}(A; B; P)}\cong {\cal M}(C; B; P')$. In this paper we examine right regular triples of semigroups.