On non-topologizable semigroups
Анотація
We find anti-isomorphic submonoids $\mathscr{C}_{+}(a,b)$ and $\mathscr{C}_{-}(a,b)$ of the bicyclic monoid $\mathscr{C}(a,b)$ with the following properties: every Hausdorff left-continuous (right-continuous) topology on $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) is discrete and there exists a compact Hausdorff topological monoid $S$ which contains $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) as a submonoid. Also, we construct a non-discrete right-continuous (left-continuous) topology $\tau_p^+$ ($\tau_p^-$) on the semigroup $\mathscr{C}_{+}(a,b)$ ($\mathscr{C}_{-}(a,b)$) which is not left-continuous (right-continuous).
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.
A. Chornenka and O. Gutik, On topologization of the bicyclic monoid, Visnyk L’viv Univ., Ser. Mech.-Math. 95 (2023), 45-56. DOI: 10.30970/vmm.2023.95.046-056
A. H. Clifford and G. B. Preston, The algebraic theory of semigroups, Vol. I, Amer. Math. Soc. Surveys 7, Providence, R.I., 1961.
W. W. Comfort, Topological groups, Handbook of set-theoretic topology, Kunen K., Vaughan J. (eds.) Elsevier, 1984, pp. 11431263.
L. Descalco and N. Ruskuc, Subsemigroups of the bicyclic monoid, Int. J. Algebra Comput. 15 (2005), no. 1, 37-57. DOI: 10.1142/S0218196705002098
L. Descalco and N. Ruskuc, Properties of the subsemigroups of the bicyclic monoid, Czech. Math. J. 58 (2008), no. 2, 311-330. DOI: 10.1007/s10587-008-0018-7
D. N. Dikranjan, L. N. Stoyanov, and I. R. Prodanov, Topological groups. Characters, dualities, and minimal group topologies, Pure and Appl. Math., 130. Marcel Dekker, Inc., New York, 1990, 287p.
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), 29-34.
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
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
K. H. Hovsepyan, Inverse subsemigroups of the bicyclic semigroup, Math. Notes 108 (2020), no. 4, 550-556). DOI: 10.1134/S0001434620090278
A. A. Markov, On free topological groups, Izvestia Akad. Nauk SSSR 9 (1945), 364 (in Russian); English version in: Transl. Amer. Math. Soc. 8 (1962), no. 1, 195-272.
D. Montgomery and L. Zippin, Topological transformation groups, Interscience Publ., New York-London, 1955. 282p.
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 version in: Algebra and Logic 12 (1973), no. 1, 64-65. DOI: 10.1007/BF02218642
A. D. Taimanov, On the topologization of commutative semigroups, Mat. Zametki 17 (1975), no. 5, 745-748 (in Russian); English version in: Math. Notes 17 (1975), no. 5, 443-444. DOI: 10.1007/BF01155800
Посилання
- Поки немає зовнішніх посилань.