TY - JOUR ID - 43374 TI - Adjoint relations for the category of local dcpos JO - Categories and General Algebraic Structures with Applications JA - CGASA LA - en SN - 2345-5853 AU - Zhao, Bin AU - Lu, Jing AU - Wang, Kaiyun AD - Shaanxi Normal University Y1 - 2017 PY - 2017 VL - 7 IS - Special Issue on the Occasion of Banaschewski's 90th Birthday (II) SP - 89 EP - 105 KW - Dcpo KW - local dcpo KW - $S$-ldcpo KW - forgetful functor DO - N2 - 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: (1) The forgetful functor $U$ : {\bf LDcpo} $\longrightarrow$ {\bf Pos} has a left adjoint, but does not have a right adjoint;(2) The inclusion functor $I$ : {\bf Dcpo} $\longrightarrow$ {\bf LDcpo} has a left adjoint, but does not have a right adjoint;(3) The forgetful functor $U$ : {\bf LDcpo}-$S$ $\longrightarrow$ {\bf LDcpo} hasboth left and right adjoints;(4) If $(S,\cdot,1)$ is a good ldcpo-monoid, then the forgetful functor $U$: {\bf LDcpo}-$S$ $\longrightarrow$ {\bf Pos}-$S$ has a left adjoint. UR - https://cgasa.sbu.ac.ir/article_43374.html L1 - https://cgasa.sbu.ac.ir/article_43374_e3ba4928af107559409d8a2f182b5716.pdf ER -