For a semigroup or an ordered semigroup $S$, we denote by
$Reg(S)$, $LReg(S)$, $Gr(S)$ the set of regular, left regular,
completely regular elements of $S$ respectively, and for a
subsemigroup $T$ of $S$, we denote by $reg(T)$ the set of elements
of $T$ which are regular in $S$. For a subset $H$ of an ordered
semigroup $S$, $(H]$ denotes the set of elements $t\in S$ such
that $t\le h$ for some $h\in H$. We characterize the ordered
semigroups $S$ in which the set of regular elements is a subset of
the set of left regular elements as the ordered semigroups such
that $reg(Sa)=Reg(Sa]$ for every $a\in S$. We prove that this type
of ordered semigroups is actually the class of semigroups for
which $reg(Se)=Reg(Se]$ for every $e\in S$ such that $e\le e^2$.
As a consequence, for a semigroup $S$ (without order), condition
$reg(Se)=Reg(Se)$ for every idempotent element of $S$ is
equivalent to the condition $reg(Sa)=Reg(Sa)$ for every $a\in S$.
For an ordered semigroup $S$ it remains an open problem if
condition $Reg(S)\subseteq LReg(S)$ implies $Reg(S)=Gr(S)$.