Abstract
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 ∈ S such that t ≤ h for some h ∈ 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 ∈ S.
We prove that this type of ordered semigroups is actually the class of semigroups for
which reg(Se) = Reg(Se] for every e ∈ S such that e ≤ e2. 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 ∈ S. For
an ordered semigroup S it remains an open problem if condition Reg(S) ⊆ LReg(S)
implies Reg(S) = Gr(S).
