抄録
It is known that for a semigroup S and any two idempotent elements
e, f of S, the following assertions are satisfied: (1) Reg(eSf) = Reg(eS) ∩ Reg(Sf);
(2) Gr(eSf) = gr(eSf) = Gr(eS) ∩ Gr(Sf); (3) E(eSf) = E(eS) ∩ E(Sf); (4)
Gr(Se) = gr(Se) and Gr(eS) = gr(eS). Also that for a semigroup S and an idempotent
element e of S, the following conditions are equivalent: (i) Reg(eSe) = Reg(Se);
(ii) Reg(Se) ⊆ Reg(eS); (iii) E(eSe) = E(Se); (iv) E(Se) ⊆ E(eS). We extend these
results in ordered semigroups. As an application of the result of the present paper, the
above mentioned results hold for any elements a, b of a semigroup S and not only for
idempotent elements of S. Some additional information are also obtained.
