Abstract
A topological semigroup is monothetic provided it contains a dense cyclic subsemigroup. The Koch problem asks whether every locally compact monothetic monoid is compact. This problem was opened for more than sixty years, till in 2018 Zelenyuk obtained a negative answer. In this paper we obtain a positive answer for Koch’s problem for some special classes of topological monoids. Namely, we show that a locally compact monothetic topological monoid S is a compact topological group if and only if S is a submonoid of a quasitopological group if and only if S has open shifts if and only if S is non-viscous in the sense of Averbukh. The last condition means that any neighborhood U of the identity 1 of S and for any element a ∈ S there exists a neighborhood V of a such that any element x ∈ S with (xV ∪ Vx) ∩ V ≠ ∅ belongs to the neighborhood U of 1.
References
[1] J.H. Carruth, J.A. Hildebrant, R.J. Koch, The theory of topological semigroups, Pure and Applied Mathematics, Marcel Dekker, Inc., 1983, vi + 244 pp.Search in Google Scholar
[2] L. Pontrjagin, Continuous groups, M.:Nauka, 1973, (in Russian).Search in Google Scholar
[3] E. Hewitt, Compact monothetic semigroups, Duke Math. J., 23 (1956), 447–457.10.1215/S0012-7094-56-02341-9Search in Google Scholar
[4] R. Koch, On monothetic semigroups, Proc. Amer. Math. Soc., 8 (1957), 397–401.10.1090/S0002-9939-1957-0087033-7Search in Google Scholar
[5] R. Ellis, Continuty and homeomorphism groups, Proc. Amer. Math. Soc. 4 (1953), 969–973.10.1090/S0002-9939-1953-0060807-0Search in Google Scholar
[6] Y. Zelenyuk, To Pontrjagin alternative for topological semigroups, Mat. zametki 44:3 (1988), 402–403, (in Russian).Search in Google Scholar
[7] Y. Zelenyuk, A locally compact noncompact monothetic semigroup with identity, Fund. Math., 245 (2019), 101–107.10.4064/fm535-3-2018Search in Google Scholar
[8] R. Ellis, A note on the continuity of the inverse, Proc. Amer. Math. Soc. 8 (1957), 372–373.10.1090/S0002-9939-1957-0083681-9Search in Google Scholar
[9] I. Guran, M. Kisil’, Pontryagin’s alternative for locally compact cancellative monoids, Visnyk of the Lviv Univ. Series Mech. Math. 77 (2012), 84–88, (in Ukrainian).Search in Google Scholar
[10] A.H. Clifford, G.B. Preston, The Algebraic Theory of Semigroups, Vol. I Providence, R.I., 1964 (Russian translation, Moskow, Mir, 1972).Search in Google Scholar
[11] J. Lawson, Embedding semigroups into Lie groups, The Analytical and Topological Theory of Semigroups: Trends and Developments, editors K. H. Hofmann, J. D. Lawson, J. S. Pym. Walter de Gruyter, 1990.10.1515/9783110856040.51Search in Google Scholar
[12] S. McKilligan, Embedding topological semigroups in topological groups, Proc. Edinburgh Math. Soc. 17 (1970/71), 127–138.10.1017/S0013091500009391Search in Google Scholar
[13] N. Rothman, Embedding of topological semigroups, Math. Ann. 139 (1960), 197–203.10.1007/BF01352911Search in Google Scholar
[14] K. Lau, J. Lawson, W. Zeng, Embedding Locally Compact Semigroups into Groups, Semigroup Forum. 56 (1998), 151–156.10.1007/PL00005971Search in Google Scholar
[15] B. Averbukh, On unitary Cauchy filters on topological monoids, Top. Algebra Appl. 1 (2013), 46–59.Search in Google Scholar
[16] B. Averbukh, On finest unitary extensions of topological monoids, Top. Algebra Appl. 3 (2015), 1–10.Search in Google Scholar
[17] B. Averbukh, On unitary extensions and unitary completions of topological monoids, Top. Algebra Appl. 4 (2016), 18–30.Search in Google Scholar
[18] B. Averbukh, A criterion of the existence of an embedding of a monothetic monoid into a topological group, Top. Algebra Appl. (to appear).Search in Google Scholar
[19] R. Engelking, General Topology, Heldermann Verlag, Berlin, 1989.Search in Google Scholar
[20] E. K. van Douwen, G. Reed, A. Roscoe, I.Tree, Star covering properties Top. Appl. 39:1 (1991), 71–103.10.1016/0166-8641(91)90077-YSearch in Google Scholar
[21] J. Vaughan, Countably compact and sequentially compact spaces, in K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, P. 569–602.10.1016/B978-0-444-86580-9.50015-XSearch in Google Scholar
[22] R. Stephenson, Jr, Initially -compact and related compact spaces, in K. Kunen, J. E. Vaughan (eds.), Handbook of Set-Theoretic Topology, Elsevier, 1984, P. 603–632.10.1016/B978-0-444-86580-9.50016-1Search in Google Scholar
[23] A. Dorantes-Aldama, D. Shakhmatov, Selective sequential pseudocompactness, Top. Appl. 222 (2017), 53–69.10.1016/j.topol.2017.02.016Search in Google Scholar
[24] M. Matveev, A Survey on Star Covering Properties, (preprint), http://at.yorku.ca/v/a/a/a/19.htm.Search in Google Scholar
[25] O. Gutik, A. Ravsky, On old and new classes of feebly compact spaces, Visnyk Lviv Univ. Ser. Mech. Mat. 85, (2018), 48–59.10.30970/vmm.2018.85.048-059Search in Google Scholar
[26] A. Kurosh, Group theory, M.:Nauka, 1967, (in Russian).Search in Google Scholar
[27] B. Bokalo, I. Guran, Sequentially compact Hausdorff cancellative semigroup is a topological group, Matematychni Studii 6 (1996), 39–40.Search in Google Scholar
[28] A.H. Tomita, The Wallace Problem: a counterexample from MAcountable and p-compactness, Canadian Math. Bulletin, 39:4 (1996), 486–498.10.4153/CMB-1996-057-6Search in Google Scholar
[29] A. Ravsky, Paratopological groups I, Matematychni Studii. 16 (2001), 37–48.Search in Google Scholar
[30] T. Banakh, A homogeneous first-countable zero-dimensional compactum failing to be a left-topological group, Matematychni Studii, 29 (2008) 215–217.Search in Google Scholar
[31] S. Bardyla, O. Gutik, A. Ravsky, H-closed quasitopological groups Topol. Appl. 217 (2017), 51–58.10.1016/j.topol.2016.12.003Search in Google Scholar
[32] A. Arhangel’skii, M. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.10.2991/978-94-91216-35-0Search in Google Scholar
[33] L. Pontrjagin, Continuous groups, 2nd ed., M.: GITTL, (1954), (in Russian).Search in Google Scholar
[34] E. K. van Douwen, The product of two countably compact topological groups, Trans. Amer. Math. Soc. 262 (Dec 1980), 417–427.10.1090/S0002-9947-1980-0586725-8Search in Google Scholar
[35] M. Hrušák, J. van Mill, U.A. Ramos-García, S. Shelah, Countably compact groups without non-trivial convergent sequences, (presentable draft), https://matmor.unam.mx/~michael/preprints_files/Countably_compact.pdfSearch in Google Scholar
[36] A.H. Tomita, A van Douwen-like ZFC theorem for small powers of countably compact groups without non-trivial convergent sequences, Topol. Appl. 259 (2019), 347Ű-364,10.1016/j.topol.2019.02.040Search in Google Scholar
[37] A. Hajnal, I. Juhász, A separable normal topological group need not be Lindelöf, General Topology and Appl., 6 (1976), no. 2, 199Ű-205.Search in Google Scholar
[38] S. García-Ferreira, A.H. Tomita, S. Watson, Countably compact groups from a selective ultrafilter, Proc. Amer. Math. Soc. 133 (2005), no. 3, 937Ű-943.Search in Google Scholar
[39] P. Szeptycki, A.H. Tomita, HFD groups in the Solovay model, Topol. Appl. 156 (2009), 1807Ű-1810.10.1016/j.topol.2009.03.008Search in Google Scholar
[40] M. Tkachenko, Countably compact and pseudocompact topologies on free Abelian groups, Soviet Math. (Iz. VUZ) 34 (1990), no. 5, 79–86.Search in Google Scholar
[41] P. Koszmider, A. Tomita, S. Watson, Forcing countably compact group topologies on a larger free Abelian group, Topology Proc. 25 (2000), 563–574.Search in Google Scholar
[42] R. Madariaga-Garcia, A. Tomita, Countably compact topological group topologies on free Abelian groups from selective ultra-filters, Topol. Appl. 154 (2007), 1470–1480.10.1016/j.topol.2006.03.028Search in Google Scholar
[43] T. Banakh, S. Dimitrova, O. Gutik, Embedding the bicyclic semigroup into countably compact topological semigroups, Topol. Appl. 157 (2010), no. 18, 2803–2814,10.1016/j.topol.2010.08.020Search in Google Scholar
[44] A. Blass, Combinatorial cardinal characteristics of the continuum, Handbook of set theory. Vols. 1, 2, 3, 395–489, Springer, Dordrecht, 2010.10.1007/978-1-4020-5764-9_7Search in Google Scholar
[45] W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, Lecture Notes in Math. 1079, Springer-Verlag, 1984.Search in Google Scholar
[46] E. Reznichenko, Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudo-compact groups, Topol. Appl. 59 (1994), 233–244,10.1016/0166-8641(94)90021-3Search in Google Scholar
[47] H. Pfister, Continuity of the inverse, Proc. Amer. Math. Soc. 95 (1985), 312–314.10.1090/S0002-9939-1985-0801345-5Search in Google Scholar
[48] A. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis. Lviv University, (2002) (in Ukrainian).Search in Google Scholar
[49] A. Ravsky, Paratopological groups II, Matematychni Studii. 17 (2002), 93–101.Search in Google Scholar
[50] V.V. Uspenskij, On subgroups of minimal topological groups, Top. Appl. 155 (2008), no. 14, 1580–1606.Search in Google Scholar
[51] A. Bouziad, J.-P. Troallic, A precompactness test for topological groups in the manner of Grothendieck, Topology Proc. 31 (2007), no. 1, 19–30.Search in Google Scholar
[52] T. Banakh, A. Ravsky, Feebly compact paratopological groups, preprint, https://arxiv.org/abs/1003.5343v7Search in Google Scholar
[53] T. Banakh, A. Ravsky, On subgroups of saturated or totally bounded paratopological groups, Algebra and Discrete Mathematics 4 (2003), 1–20.Search in Google Scholar
[54] A. Korovin, Continuous actions of Abelian groups and topological properties in Cp-theory, Ph.D. Thesis, Moskow State University, Moskow (1990) (in Russian).Search in Google Scholar
[55] A. Korovin, Continuous actions of pseudocompact groups and the topological group axioms, Deposited in VINITI, #3734-D, Moskow (1990) (in Russian).Search in Google Scholar
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