Zhao, B., Lu, J., Wang, K. (2017). Adjoint relations for the category of local dcpos. Categories and General Algebraic Structures with Applications, 7(Special Issue on the Occasion of Banaschewski's 90th Birthday (II)), 89-105.

Bin Zhao; Jing Lu; Kaiyun Wang. "Adjoint relations for the category of local dcpos". Categories and General Algebraic Structures with Applications, 7, Special Issue on the Occasion of Banaschewski's 90th Birthday (II), 2017, 89-105.

Zhao, B., Lu, J., Wang, K. (2017). 'Adjoint relations for the category of local dcpos', Categories and General Algebraic Structures with Applications, 7(Special Issue on the Occasion of Banaschewski's 90th Birthday (II)), pp. 89-105.

Zhao, B., Lu, J., Wang, K. Adjoint relations for the category of local dcpos. Categories and General Algebraic Structures with Applications, 2017; 7(Special Issue on the Occasion of Banaschewski's 90th Birthday (II)): 89-105.

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} has both 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.

Highlights

Dedicated to Bernhard Banaschewski on the occasion of his $90^{th}$ birthday

[1] Adamek, J., Herrlich, H., and Strecker, G.E., "Abstract and Concrete Categories: The Joy of Cats", John Wiley & Sons, New York, 1990. [2] Bulman-Fleming, S. and Mahmoudi, M., The category of S-posets, Semigroup Forum 71 (2005), 443-461. [3] Crole, R.L., "Categories for Types", Cambridge University Press, Cambridge, 1994. [4] Erne, M., Minimal bases, ideal extensions, and basic dualities, Topology Proc. 29 (2005), 445-489. [5] Gierz, G., Hofmann, K.H., Keimel, K., Lawson, J.D., Mislove, M., and Scott, D.S., "Continuous Lattices and Domains", Encyclopedia of Mathematics and its Applications 93, Cambridge University Press, 2003. [6] Keimel, K. and Lawson, J.D., D-completions and the d-topology, Ann. Pure Appl. Logic 159(3) (2009), 292-306. [7] Mahmoudi, M. and Moghbeli, H., Free and cofree acts of dcpo-monoids on directed complete posets, Bull. Malaysian Math. Sci. Soc. 39 (2016), 589-603. [8] Mislove, M.W., Local dcpos, local cpos, and local completions, Electron. Notes Theor. Comput. Sci. 20 (1999), 399-412.

[9] Xu, L. and Mao, X., Strongly continuous posets and the local Scott topology, J. Math. Anal. Appl. 345 (2008), 816-824. [10] Zhao, D. and Fan, T., Dcpo-completion of posets, Theoret. Comput. Sci. 411 (2010), 2167-2173. [11] Zhao, D., Partial dcpo's and some applications, Arch. Math. (Brno) 48 (2012), 243-260.