On topologization of the bicyclic monoid

Adriana Chornenka, Oleg Gutik

Анотація


We construct two non-discrete inverse semigroup T1-topologies and a compact inverse shift-continuous T1-topology on the bicyclic monoid. Also we give conditions on a T1-topology τ on C(p, q) to be discrete. In particular, we show that if τ is an inverse semigroup T1-topology on C(p, q) which satisfies one of the following conditions: τ is Baire, τ is quasi-regular or τ is semiregular, then τ is discrete.


Повний текст:

PDF (English)

Посилання


L. W. Anderson, R. P. Hunter, and R. J. Koch, Some results on stability in semigroups, Trans. Amer. Math. Soc. 117 (1965), 521-529. DOI: 10.2307/1994222

T. Banakh, S. Dimitrova, and O. Gutik, The Rees-Suschkiewitsch Theorem for simple topological semigroups, Mat. Stud. 31 (2009), no. 2, 211-218.

T. Banakh, S. Dimitrova, and O. Gutik, Embedding the bicyclic semigroup into countably compact topological semigroups, Topology Appl. 157 (2010), no. 18, 2803-2814. DOI: 10.1016/j.topol.2010.08.020

S. Bardyla and A. Ravsky, Closed subsets of compact-like topological spaces, Appl. Gen. Topol. 21 (2020), no. 2, 201-214. DOI: 10.4995/agt.2020.12258.

M. O. Bertman and T. T. West, Conditionally compact bicyclic semitopological semigroups, Proc. Roy. Irish Acad. A76 (1976), no. 21-23, 219-226.

J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The theory of topological semigroups, Vol. I, Marcel Dekker, Inc., New York and Basel, 1983.

J. H. Carruth, J. A. Hildebrant, and R. J. Koch, The theory of topological semigroups, Vol. II, Marcel Dekker, Inc., New York and Basel, 1986.

A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. I, Amer. Math. Soc. Surveys 7, Providence, R.I., 1961.

A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. II, Amer. Math. Soc. Surveys 7, Providence, R.I., 1967.

W. W. Comfort, Topological groups, Handbook of set-theoretic topology, Kunen K., Vaughan J. (eds.) Elsevier, 1984, pp. 1143-1263.

C. Eberhart and J. Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc. 144 (1969), 115-126. DOI: 10.1090/S0002-9947-1969-0252547-6

R. Engelking, General topology, 2nd ed., Heldermann, Berlin, 1989.

O. Gutik, On the dichotomy of a locally compact semitopological bicyclic monoid with adjoined zero, Visnyk L'viv Univ., Ser. Mech.-Math. 80 (2015), 33-41.

O. Gutik, Topological property of the Taimanov semigroup, Math. Bull. T. Shevchenko Sci. Soc. 13 (2016), 1-5.

O. V. Gutik and K. M. Maksymyk, On semitopological bicyclic extensions of linearly ordered groups, Mat. Metody Fiz.-Mekh. Polya 59 (2016), no. 4, 31-43; Reprinted version: O. V. Gutik and K. M. Maksymyk, On semitopological bicyclic extensions of linearly ordered groups, J. Math. Sci. 238 (2019), no. 1, 32-45. DOI: 10.1007/s10958-019-04216-x

O. Gutik and D. Repovs, On countably compact 0-simple topological inverse semigroups, Semigroup Forum 75 (2007), no. 2, 464-469. DOI: 10.1007/s00233-007-0706-x

R. C. Haworth and R. A. McCoy, Baire spaces, Dissertationes Math., Warszawa, PWN, 1977. Vol. 141.

E. Hewitt and K. A. Roos, Abstract harmonic analysis, Vol. 1, Springer, Berlin, 1963.

J. A. Hildebrant and R. J. Koch, Swelling actions of Γ-compact semigroups, Semigroup Forum 33 (1986), 65-85. DOI: 10.1007/BF02573183

R. J. Koch and A. D. Wallace, Stability in semigroups, Duke Math. J. 24 (1957), no. 2, 193-195. DOI: 10.1215/S0012-7094-57-02425-0

M. Lawson, Inverse semigroups. The theory of partial symmetries, Singapore: World Scientific, 1998.

A. A. Markov, On free topological groups, Izvestia Akad. Nauk SSSR 9 (1945), 3-64 (in Russian); English version in: Transl. Amer. Math. Soc. 8 (1962), no. 1, 195-272.

L. S. Pontryagin, Topological groups, Gordon & Breach, New York ets, 1966.

A. Yu. Ol'shanskiy, Remark on counatable non-topologized groups, Vestnik Moscow Univ. Ser. Mech. Math. 39 (1980), 1034 (in Russian).

W. Ruppert, Compact semitopological semigroups: an intrinsic theory, Lect. Notes Math., 1079, Springer, Berlin, 1984. DOI: 10.1007/BFb0073675

A. D. Taimanov, An example of a semigroup which admits only the discrete topology, Algebra i Logika 12 (1973), no. 1, 114-116 (in Russian); English transl. in: Algebra Logic 12 (1973), no. 1, 64-65. DOI: 10.1007/BF02218642

A. D. Taimanov, The topologization of commutative semigroups, Mat. Zametki 17 (1975), no. 5, 745-748 (in Russian); English transl. in: Math. Notes 17 (1975), no. 5, 443-444. DOI: 10.1007/BF01155800




DOI: http://dx.doi.org/10.30970/vmm.2023.95.046-056

Посилання

  • Поки немає зовнішніх посилань.