It is known that for a semigroup $S$ and any two idempotent
elements $e,f$ of $S$, the following assertions are satisfied: (1)
$Reg(eSf)=Reg(eS)\cap Reg(Sf)$; (2) $Gr(eSf)=gr(eSf)=Gr(eS)\cap
Gr(Sf)$; (3) $E(eSf)=E(eS)\cap E(Sf)$; (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) $Reg(Se)\subseteq Reg(eS)$; (iii)
$E(eSe)=E(Se)$; (iv) $E(Se)\subseteq E(eS)$. 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.