Adjoint relations for the category of local dcpos

In this paper, we consider the forgetful functor from the category {\bf LDcpo} of local dcpos (respectively, {\bf Dcpo} of dcpos) to  the category {\bf Pos} of posets (respectively, {\bf 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: (۱) The forgetful functor U : {\bf LDcpo} \longrightarrow {\bf Pos} has a left adjoint, but does not have a right adjoint;(۲) The inclusion functor I : {\bf Dcpo} \longrightarrow {\bf LDcpo} has a left adjoint, but does not have a right adjoint;(۳) The forgetful functor U : {\bf LDcpo}-S \longrightarrow {\bf LDcpo} hasboth left and right adjoints;(۴) If (S,\cdot,۱) is a good ldcpo-monoid, then the forgetful functor U: {\bf LDcpo}-S \longrightarrow {\bf Pos}-S has a left adjoint.

Keywords:

Dcpo , local dcpo , S-ldcpo , forgetful functor