On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal

  • O. V. Gutik Ivan Franko National University of Lviv, Lviv
  • M. B. Khylynskyi Ivan Franko National University of Lviv, Lviv, Ukraine

Анотація

Let $[0,\infty)$ be the set of all non-negative real numbers. The set $\boldsymbol{B}_{[0,\infty)}=[0,\infty)\times [0,\infty)$ with the following binary operation $(a,b)(c,d)=(a+c-\min\{b,c\},b+d-\min\{b,c\})$ is a bisimple inverse semigroup.
In the paper we study Hausdorff locally compact shift-continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal of the following tree types.
The semigroup $\boldsymbol{B}_{[0,\infty)}$ with the induced usual topology $\tau_u$ from $\mathbb{R}^2$, with the topology $\tau_L$ which is generated by the natural partial order on the inverse semigroup $\boldsymbol{B}_{[0,\infty)}$, and the discrete topology are denoted by $\boldsymbol{B}^1_{[0,\infty)}$, $\boldsymbol{B}^2_{[0,\infty)}$, and $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$, respectively. We show that if $S_1^I$ ($S_2^I$) is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^1_{[0,\infty)}$ ($\boldsymbol{B}^2_{[0,\infty)}$) with an adjoined compact ideal $I$ then either $I$ is an open subset of $S_1^I$ ($S_2^I$) or the topological space $S_1^I$ ($S_2^I$) is compact. As a corollary we obtain that the topological space of a Hausdorff locally compact shift-continuous topology on $S^1_{\boldsymbol{0}}=\boldsymbol{B}^1_{[0,\infty)}\cup\{\boldsymbol{0}\}$ (resp. $S^2_{\boldsymbol{0}}=\boldsymbol{B}^2_{[0,\infty)}\cup\{\boldsymbol{0}\}$) with an adjoined zero $\boldsymbol{0}$ is either homeomorphic to the one-point Alexandroff compactification of the topological space $\boldsymbol{B}^1_{[0,\infty)}$ (resp. $\boldsymbol{B}^2_{[0,\infty)}$) or zero is an isolated point of $S^1_{\boldsymbol{0}}$ (resp. $S^2_{\boldsymbol{0}}$).
Also, we proved that if $S_{\mathfrak{d}}^I$ is a Hausdorff locally compact semitopological semigroup $\boldsymbol{B}^{\mathfrak{d}}_{[0,\infty)}$ with an adjoined compact ideal $I$ then $I$ is an open subset of $S_{\mathfrak{d}}^I$.

Біографії авторів

O. V. Gutik, Ivan Franko National University of Lviv, Lviv

Department of Mechanics and Mathematics
Ivan Franko National University of Lviv, Lviv

M. B. Khylynskyi, Ivan Franko National University of Lviv, Lviv, Ukraine

Ivan Franko National University of Lviv

Lviv, Ukraine

Посилання

K.R. Ahre, Locally compact bisimple inverse semigroups, Semigroup Forum 22 (1981), №3, 387–389. doi: 10.1007/BF02572817

K.R. Ahre, On the closure of $boldsymbol{B}^1_{[0,infty)}$, Istanbul Tek. Univ. Bul. 36 (1983), №4, 553–562.

K.R. Ahre, On the closure of $boldsymbol{B}^1_{[0,infty)}$, Semigroup Forum 33 (1986), 269–272. doi: 10.1007/BF02573200

K.R. Ahre, On the closure of $boldsymbol{B}^2_{[0,infty)}$, Bull. Tech. Univ. Istanbul 42 (1989), №3, 387–390.

L.W. Anderson, R.P. Hunter, 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, O. Gutik, The Rees-Suschkiewitsch Theorem for simple topological semigroups, Mat. Stud. 31 (2009), №2, 211–218.

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

S. Bardyla, Classifying locally compact semitopological polycyclic monoids, Mat. Visn. Nauk. Tov. Im. Shevchenka 13 (2016), 21–28.

S. Bardyla, On locally compact semitopological graph inverse semigroups, Mat. Stud. 49 (2018), №1, 19–28. doi: 10.15330/ms.49.1.19-28

S. Bardyla, On topological McAlister semigroups, J. Pure Appl. Algebra 227 (2023), №4, 107274. doi:10.1016/j.jpaa.2022.107274

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

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

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

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

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

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

C. Eberhart, 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.

V.A. Fortunatov, Congruences on simple extensions of semigroups, Semigroup Forum 13 (1976), 283–295. doi: 10.1007/BF02194949

G.L. Fotedar, On a semigroup associated with an ordered group, Math. Nachr. 60 (1974), 297–302. doi: 10.1002/mana.19740600128

G.L. Fotedar, On a class of bisimple inverse semigroups, Riv. Mat. Univ. Parma (4) 4 (1978), 49–53.

G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M.W. Mislove, D.S. Scott, Continuous lattices and domains. Cambridge Univ. Press, Cambridge, 2003.

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

O. Gutik, P. Khylynskyi, On a locally compact submonoid of the monoid cofinite partial isometries of $mathbb{N}$ with adjoined zero, Topol. Algebra Appl. 10 (2022), №1, 233–245. doi: 10.1515/taa-2022-0130

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

O.V. Gutik, K.M. Maksymyk, On a semitopological extended bicyclic semigroup with adjoined zero, Mat. Metody Fiz.-Mekh. Polya 62 (2019), №4, 28–38. Reprinted version: O.V. Gutik, K.M. Maksymyk, On a semitopological extended bicyclic semigroup with adjoined zero, J. Math. Sci. 265 (2022), №3, 369–381. doi: 10.1007/s10958-022-06058-6

O. Gutik, M. Mykhalenych, On a semitopological semigroup $boldsymbol{B}_{omega}^{mathscr{F}}$ when a family $mathscr{F}$ consists of inductive non-empty subsets of $omega$}, Mat. Stud. 59 (2023), №1, 20–28. doi: 10.30970/ms.59.1.20-28

O. Gutik, D. Pagon, K. Pavlyk, Congruences on bicyclic extensions of a linearly ordered group, Acta Comment. Univ. Tartu. Math. 15 (2011), №2, 61–80. doi: 10.12697/ACUTM.2011.15.10

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

J.A. Hildebrant, R.J. Koch, Swelling actions of $Gamma$-compact semigroups, Semigroup Forum 33 (1986), 65–85. doi: 10.1007/BF02573183

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

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

K. Maksymyk, On locally compact groups with zero, Visn. Lviv Univ., Ser. Mekh.-Mat. 88 (2019), 51–58. (in Ukrainian).

T. Mokrytskyi, On the dichotomy of a locally compact semitopological monoid of order isomorphisms between principal filters of $mathbb{N}^n$ with adjoined zero, Visn. Lviv Univ., Ser. Mekh.-Mat. 87 (2019), 37–45.

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

Опубліковано
2024-03-19
Як цитувати
Gutik, O. V., & Khylynskyi, M. B. (2024). On locally compact shift continuous topologies on the semigroup $\boldsymbol{B}_{[0,\infty)}$ with an adjoined compact ideal. Математичні студії, 61(1), 10-21. https://doi.org/10.30970/ms.61.1.10-21
Розділ
Статті