THE SET OF REGULAR ELEMENTS IN ORDERED SEMIGROUPS

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 ≤ 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 ∈ S.F or an ordered semigroup S it remains an open problem if condition Reg(S) ⊆ LReg(S) implies Reg(S )= Gr(S).