On compact topologies on the semigroup of finite partial order isomorphisms of a bounded rank of an infinite linear ordered set
DOI:
https://doi.org/10.3842/umzh.v77i5.8941Keywords:
Inverse semigroup, partial order isomorphism, semitopological semigroup, compact, countably compact, pseudocompact, totally separated, scattered, Bohr compactificationAbstract
UDC 512.536
We study the topologization of the semigroup $\mathscr{O\!\!I}\!_n(L)$ of finite partial order isomorphisms of bounded rank of an infinite linear ordered set $(L,\leqslant).$ In particular, we show that every $T_1$ left-topological (right-topological) semigroup $\mathscr{O\!\!I}\!_n(L)$ is an Urysohn, functionally Hausdorff, totally disconnected, and scattered space. It is proved that, on the semigroup $\mathscr{O\!\!I}\!_n(L),$ there exists a unique Hausdorff countably compact (pseudocompact) shift-continuous topology, which is compact, and that the Bohr compactification of the Hausdorff topological semigroup $\mathscr{O\!\!I}\!_n(L)$ is the trivial semigroup.
References
1. В. В. Вагнер, К теории частнчных преобразований, Докл. АН СССР, 84, 653–656 (1952).
2. В. В. Вагнер, Обобщенные группы, Докл. АН СССР, 84, 1119–1122 (1952).
3. O. Гутік, А. Рейтер, Про напівтопологічні симетричні інверсні півгрупи обмеженого скінченного рангу, Вісн. Львів. ун-ту. Сер. мех.-мат., 72, 94–106 (2010).
4. O. В. Гутік, M. Р. Щипель, Напівгрупа скінченних часткових порядкових ізоморфізмів обмеженого рангу нескінченної лінійно впорядкованої множини, Буковин. мат. журн., 12, № 2, 60–68 (2024); DOI: 10.31861/bmj2024.02.05. DOI: https://doi.org/10.31861/bmj2024.02.05
5. J. H. Carruth, J. A. Hildebrant, R. J. Koch, The theory of topological semigroups, vol. I, Marcel Dekker, Inc., New York, Basel (1983).
6. A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, vol. I, Amer. Math. Soc. Surveys, 7, Providence, R.I. (1961). DOI: https://doi.org/10.1090/surv/007.1
7. A. H. Clifford, G. B. Preston, The algebraic theory of semigroups, vol. II, Amer. Math. Soc. Surveys, 7, Providence, R.I. (1967). DOI: https://doi.org/10.1090/surv/007.2
8. K. DeLeeuw, I. Glicksberg, Almost-periodic functions on semigroups, Acta Math., 105, 99–140 (1961).
9. R. Engelking, General topology, 2nd ed., Heldermann, Berlin (1989); DOI: 10.1007/BF02559536. DOI: https://doi.org/10.1007/BF02559536
10. G. Gierz, K. Hofmann, K. Keimel, J. Lawson, M. Mislove, D. S. Scott, Continuous lattices and domains, Encyclopedia Math. and Appl., 93, Cambridge University Press (2003); DOI: 10.1017/CBO9780511542725. DOI: https://doi.org/10.1017/CBO9780511542725
11. O. Gutik, J. Lawson, D. Repovš, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum, 78, № 2, 326–336 (2009); DOI: 10.1007/s00233-008-9112-2. DOI: https://doi.org/10.1007/s00233-008-9112-2
12. O. Gutik, K. Pavlyk, Topological Brandt $λ$-extensions of absolutely $H$-closed topological inverse semigroups, Вісн. Львів. ун-ту. Сер. мех.-мат., 61, 98–105 (2003).
13. O. Gutik, K. Pavlyk, Topological semigroups of matrix units, Algebra and Discrete Math., № 3, 1–17 (2005). DOI: https://doi.org/10.1007/s00233-005-0530-0
14. O. Gutik, K. Pavlyk, A. Reiter, Topological semigroups of matrix units and countably compact Brandt $λ^0$-extensions, Мат. студ., 32, № 2, 115–131 (2009). DOI: https://doi.org/10.30970/ms.32.2.115-131
15. O. V. Gutik, A. R. Reiter, Symmetric inverse topological semigroups of finite rank $≤ n$, Мат. методи і фіз.-мех. поля, 52, № 3, 7–14 (2009); reprinted version: J. Math. Sci., 171, № 4, 425–432 (2010); DOI: 10.1007/s10958-010-0147-z. DOI: https://doi.org/10.1007/s10958-010-0147-z
16. M. Lawson, Inverse semigroups. The theory of partial symmetries, World Sci., Singapore (1998); DOI: 10.1142/3645. DOI: https://doi.org/10.1142/9789812816689
17. M. Petrich, Inverse semigroups, John Wiley & Sons, New York (1984).
18. W. Ruppert, Compact semitopological semigroups: an intrinsic theory, Lect. Notes Math., 1079, Springer, Berlin (1984); DOI: 10.1007/BFb0073675. DOI: https://doi.org/10.1007/BFb0073675
19. L. A. Steen, J. A. Seebach (Jr.), Counterexamples in topology, Reprint of the 2nd ed., Dover Publ., Mineola, New York (1995).
20. J. W. Stepp, Algebraic maximal semilattices, Pacif. J. Math., 58, № 1, 243–248 (1975); DOI: 10.2140/pjm.1975.58.243. DOI: https://doi.org/10.2140/pjm.1975.58.243
21. P. Urysohn, Über die Mächtigkeit der zusammenhängenden Mengen, Math. Ann., 94, 262–295 (1925); DOI: 10.1007/BF01208659. DOI: https://doi.org/10.1007/BF01208659
22. J. E. Vaughan, Countably compact and sequentially compact spaces, Handbook of Set-Theoretic Topology, K. Kunen, J. E. Vaughan (eds.), North-Holland, Amsterdam (1984), p. 569–602; DOI: 10.1016/B978-0-444-86580-9.50015-X. DOI: https://doi.org/10.1016/B978-0-444-86580-9.50015-X
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