Adjoint relations for the category of local dcpos

Document Type: Research Paper


Shaanxi Normal University


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.


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.