The symmetric monoidal closed category of cpo $M$-sets

Document Type: Research Paper


Department of Mathematics, Faculty of Science, University of Jiroft, Jiroft, Iran


In this paper, we show that the category of directed complete posets with bottom elements (cpos) endowed with an action of a monoid $M$ on them forms a monoidal category. It is also proved that this category is symmetric closed.


[1] Abramsky, S. and Jung, A., "Domain Theory", Handbook of logic in computer science (Vol. 3). Oxford University Press, 1995.
[2] Borceux, F., "Handbook of Categorical Algebra 1: Basic Category Theory", Cambridge University Press, Cambridge, 1994.
[3] Borceux, F., "Handbook of Categorical Algebra 2: Categories and Structures", Cambridge University Press, Cambridge, 1994.
[4] Davey, B.A. and Priestly, H.A., "Introduction to Lattices and Order", Cambridge University Press, Cambridge, 1990.
[5] Day, B.J., On closed categories of functors, Reports of the midwest category seminar (Lane, S.Mac, editor), Lecture Notes in Math., Springer-Verlag, Berlin-New York, 137 (1970), 1–38.
[6] Ebrahimi, M.M. and Mahmoudi, M., The category of M-Sets, Ital. J. Pure Appl. Math. 9 (2001), 123-132.
[7] Fiech, A., Colimits in the category Dcpo, Math. Structures Comput. Sci., 6 (1996), 455-468.
[8] Jung, A., "Cartesian closed categories of Domain", Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam, 1989.
[9] Kilp, M., Knauer, U., and A. Mikhalev, "Monoids, Acts and Categories", Walter de Gruyter, Berlin, New York, 2000.
[10] Mahmoudi, M. and Moghbeli, H., The category of S-acts in the category Cpo, Bull. Iran. Math. Soc. 41(1) (2015), 159-175.
[11] Mahmoudi, M. and Moghbeli, H., The categories of actions of a dcpo-monoid on directed complete posets, Quaigroups Relatd Sytems, 23 (2015), 283-295.
[12] Moghbeli-Damaneh, H., Actions of a separately cpo-monoid on pointed directed complete posets, Categ. General Alg. Struct. Appl., 3(1) (2015), 21-42.
[13] Mac Lane, S., "Categories for the working mathematician". Vol.5. Springer Science and Business Media, 2013.
[14] Plotkin, G.D., A powerdomain construction. SIAM Journal on Computing, 5 (1976), 452-487.
[15] Plotkin, G.D., A powerdomain for countable non-determinism. In M. Nielsen and E. M. Schmidt, editors, Automata, Languages and programming, volume 140 of Lecture Notes in Computer Science, pages 412-428. EATCS, Springer Verlage, 1982.
[16] Smyth, M.B., Powerdomains. Journal of Computer and Systems Sciences, 16 (1978), 23-36.
[17] Streicher, T., "Domain-theoretic Foundations of Functional Programming". World Scientific, Singapore, 2006.
[18] Tix, R., Keimel. K., and G. D. Plotkin, "Semantic Domains for Combining Probability and Non-Determinism", Electronic Notes in Theoretical Computer Science, 222 (2009), 3-99.