Abstract
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 ∈ S.
