On cofree S-spaces and cofree S-flows

Behnam Khosravi

Abstract

Let S-Tych be the category of Tychonoff S-spaces for a topological monoid S. We study the cofree S-spaces and cofree S-flows over topological spaces and we prove that for any topological space X and a topological monoid S, the function space C(S,X) with the compact-open topology and the action s · f = (t → f(st)) is the cofree S-space over X if and only if the compact-open topology is admissible and Tychonoff. Finally we study injective S-spaces and we characterize injective cofree S-spaces, when the compact-open topology is admissible and Tychonoff. As a consequence of this result, we characterize the cofree S-spaces and cofree S-flows, when S is a locally compact topological monoid.

Keywords

S-space; S-flow; Cofree S-space; Cofree S-flow Compact-open topology; Injective S-space

Full Text:

PDF

References

R. N. Ball, S. Geschke and J. N. Hagler, Injective and projective T-Boolean algebras, J. Pure Appl. Algebra 209, no. 1 (2007), 1–36. http://dx.doi.org/10.1016/j.jpaa.2006.05.004

J. F. Berglund, H. D. Junghenn and P. Milnes, Analysis on semigroups. Function spaces, compactifications, representations Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. John Wiley and Sons, Inc., New York, 1989.

J. Dugundji, Topology, Wm. C. Brown Publishers, 1989.

M. M. Ebrahimi and M. Mahmoudi, The Category of M-sets, Ital. J. Pure Appl. Math. 9 (2001), 123–132.

M. H. Escardo, Function-Space Compactifications of Function Spaces, Topology Appl. 120, no. 3 (2002), 441–463. http://dx.doi.org/10.1016/S0166-8641(01)00089-X

M. H. Escardo, J. Lawson and A. Simpson, Comparing Cartesian closed categories of (core) compactly generated spaces, Topology Appl. 143, no. 1-3 (2004), 105–144. http://dx.doi.org/10.1016/j.topol.2004.02.011

E. Glasner and M. G. Megrelishvili, New algebras of functions on topological groups arising from G-spaces, Fund. Math. 201 (2008), 1–51. http://dx.doi.org/10.4064/fm201-1-1

D. N. Georgiou, Topologies on function spaces and hyperspaces, Appl. Gen. Topol. 10, no. 1 (2009), 159–170.

N. Hindman and D. Strauss, Algebra in the Stone-Cech compactification. Theory and applications, Walter de Gruyter & Co., Berlin, 1998. http://dx.doi.org/10.1515/9783110809220

B. Khosravi, On free and projective S-spaces and flows over a topological monoid, Asian European Journal of Math. 3, no. 3 (2010), 443–456. http://dx.doi.org/10.1142/S1793557110000350

B. Khosravi, Free topological acts over a topological monoid, Quasigroup and Related

M. Kilp, U. Knauer and A. Mikhalev, Monoids, Acts and Categories, Walter deGruyter, Berlin, New York, 2000. http://dx.doi.org/10.1515/9783110812909

J. Lawson and A. Lisan, Flows, congruences, and factorizations, Topology Appl. 58, no. 1 (1994), 35–46. http://dx.doi.org/10.1016/0166-8641(94)90072-8

M. G. Megrelishvili, Free topological G-groups, New Zealand J. Math. 25, no. 1 (1996), 59–72.

M. G. Megrelishvili, Topological Transformation Groups: Selected Topics, in: Elliott Pearl, Editor, Open problems in topology. II. Elsevier B. V., Amsterdam, 2007. http://dx.doi.org/10.1016/B978-044452208-5/50043-0

M. G. Megrelishvili, Reflexively representable but not Hilbert representable compact flows and semitopological semigroups, Colloq. Math. 110 (2006), 383–407. http://dx.doi.org/10.4064/cm110-2-5

J. R. Munkres, Topology, 2nd Ed., Prentice-Hall, New Jersy, 2000.

P. Normak, Topological S-acts: Preliminaries and Problems, Transformation semigroups 199 (1993), 60–69, , Univ. Essex, Colchester.systems 18, no. 1 (2010), 25–42.

Abstract Views

2841
Metrics Loading ...

Metrics powered by PLOS ALM




Esta revista se publica bajo una licencia de Creative Commons Reconocimiento-NoComercial-SinObraDerivada 4.0 Internacional.

Universitat Politècnica de València

e-ISSN: 1989-4147   https://doi.org/10.4995/agt