# Limits and colimits in the category of pre-directed complete pre-ordered sets

Document Type : Research Paper

Author

Depatment of Mathematcs, Faculty of science, University of Jiroft, Jiroft, Iran

10.29252/as.2020.1833

Abstract

In this paper, some categorical properties of the category { Pre-Dcpo} of all pre-dcpos; pre-ordered sets which are also pre-directed complete, with pre-continuous maps between them is considered. In particular, we characterize products and coproducts in this category. Furthermore, we show that this category is neither complete nor cocomplete. Also, epimorphisms and monomorphisms in {Pre-Dcpo} are described.
Finally, some adjoint relations between the category {Pre-Dcpo} and others are considered.
More precisely, we consider the forgetful functors between this category and some well-known categories, and study the existence of their left and right adjoints.

Keywords

#### References

[1] S. Abramsky and A. Jung, Domain theory. In: Handbook of Computer Science and logic, vol. 3, Clarendon Press, Oxford, (1995).
[2] J. Adamek, H. Herrlich and G. Strecker, Abstract and Concrete Categories. The Joy of Cats, http://katmat.math.uni-bremen.de/acc/acc.pdf
[3] A. Fiech, Colimits in the category DCPO, Math. Structures Comput. Sci. Vol. 6 (1996), pp. 455-468.
[4] A. Jung, Cartesian closed categories of algebraic cpos, Theoret. Comp. Sci. Vol. 70 (1990), pp. 233-250.
[5] M. Mahmoudi, H. Moghbeli and K. Pi oro, Natural congruences and isomorphism theorem for directed complete posets, Algebra Universalis. Vol. 77 No. 1 (2017), pp. 79-99, DOI: 10.1007/s00012-017-0424-5.
[6] M. Mahmoudi, H. Moghbeli and K. Pi oro, Directed complete poset congruences, Journal of Pure and Applied Algebra. Vol. 223 No. 10 (2019), pp. 4161-4170.
[7] S. Bulman-Fleming and M. Mahmoudi, The category of S-posets, Semigroup Forum. Vol. 71 No. 3 (2005), pp. 443-461.
[8] R. L. Crole, Categories for types, Cambridge University Press, Cambridge, (1994).
[9] S. Vickers and C. Townsend, A universal characterization of the double powerlocale, Theoretical Computer Science. Vol. 316 (2004), pp. 297-321.
[10] D. S. Scott, Domains for denotational semantics, In Nielsen, M., Schmidt, E.M., eds.: Automata, Languages and Programming, Proceedings of the 9th Colloquium, July 1982, Aarhus, Denmark, Lect. Notes Comp. Sci. Vol. 140 (1982), pp. 577-613.