For a subsemigroup $T$ of a semigroup $S$, $Reg(T)$ denotes the set of regular elements of $T$, $LReg(T)$ the set of left regular elements of $T$ and $reg(T)$ the set of elements of $T$ which are regular in $S$. Characterizations of a semigroup $S$ for which $reg(Se)=Reg(Se)$ for each idempotent element $e$ of $S$ have been given in [3]. This type of semigroups is the semigroups $S$ in which each element of the subsemigroup $Se$ of $S$ which is regular in $S$ is a left regular element of $Se$ for every idempotent element $e$ of $S$. Moreover, this type of semigroups is the semigroups $S$ in which the regular elements are left regular, equivalently the sets of regular and completely regular elements coincide [3]. In the present paper we prove that the type of semigroups mentioned above is actually the semigroups in which $reg(Sa)=reg(Sa)$ for every $a\in S$.