Adjoint relations for the category of local dcpos

In this paper, we consider the forgetful functor from the category LDcpo of local dcpos (respectively, Dcpo of dcpos) to the category Pos of posets (respectively, LDcpo of local dcpos), and study the existence of its left and right adjoints. Moreover, we give the concrete forms of free and cofree S-ldcpos over a local dcpo, where S is a local dcpo monoid. The main results are: (1) The forgetful functor U : LDcpo ⟶ Pos has a left adjoint, but does not have a right adjoint; (2) The inclusion functor I : Dcpo ⟶ LDcpo has a left adjoint, but does not have a right adjoint; (3) The forgetful functor U : LDcpo-S ⟶ LDcpo has both left and right adjoints; (4) If (S,⋅,1) is a good ldcpo-monoid, then the forgetful functor U: LDcpo-S ⟶ Pos-S has a left adjoint.