  In this paper we focus on Rees $I\times \Lambda$ matrix semigroups without
zero over a semigroup $S$ with $\Lambda\times I$ sandwich matrix $P$, where $I$
is a singleton, $\Lambda$ is the factor semigroup of $S$ modulo the kernel
$\theta_S$ of the right regular representation of $S$, and $P$ is a choice
function on the collection of all $\theta _S$-classes of $S$. We describe the
kernel of the right regular representation of this type of Rees matrix
semigroups, and prove embedding theorems on them. Motivated by one of embedding
theorems, we show how right commutative right cancellative semigroups can be
constructed. We define the concept of a right regular sequence of semigroups,
and show that every congruence on an arbitrary semigroup defines such a
sequence.
