Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions of topological semigroups

O.Gutik; K.Pavlyk; A.Reiter

We show that a topological semigroup of finite partial bijections $\mathscr{I}_\lambda^n$ of an infinite set with a compact subsemigroup of idempotents is absolutely $H$-closed and any countably compact topological semigroup does not contain $\mathscr{I}_\lambda^n$ as a subsemigroup. We give sufficient conditions onto a topological semigroup $\mathscr{I}_\lambda^1$ to be non-$H$-closed. Also we describe the structure of countably compact Brandt $\lambda^0$-extensions of topological monoids and study the category of countably compact Brandt $\lambda^0$-extensions of topological monoids with zero.

1. A.~Abd-Allah and R.~Brown, A compact-open topology on partial maps with open domains, J. London Math. Soc. 21 (2) (1980), 480-486.

2. B.~B.~Baird, Inverse semigroups of homeomorphisms between open subsets, J. Austral. Math. Soc. Ser. A 24 (1977), №1, 92--102.

3. B.~B.~Baird, Embedding inverse semigroups of homeomorphisms on closed subsets, Glasgow Math. J. 18 (1977), №2, 199--207.

4. B.~B.~Baird, Epimorphisms of inverse semigroups of homeomorphisms between closed subsets, Semigroup Forum 14 (1977), №2, 161--166.

5. B.~B.~Baird, Inverse semigroups of homeomorphisms are Hopfian, Canad. J. Math. 31 (1979), №4, 800--807.

6. T.~Banakh and S.~Dimitrova, Openly factorizable spaces and compact extensions of topological semigroup, Preprint (arXiv: 0811.4272).

7. A.~A.~Beda, Continuous inverse semigroups of open partial homeomorphisms, Izv. Vyssh. Uchebn. Zaved. Mat. no. 1 (1980), 64--65 (in Russian).

8. P.~I.~Booth and R.~Brown, Spaces of partial maps, fibred mapping spaces and the compact-open topology, General Topology Appl. 8 (1978), 181-195.

9. 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; Vol. II, Marcel Dekker, Inc., New York and Basel, 1986.

10. A.~H.~Clifford and G.~B.~Preston, The Algebraic Theory of Semigroups, Vol. I., Amer. Math. Soc. Surveys 7, Providence, R.I., 1961; Vol. II., Amer. Math. Soc. Surveys 7, Providence, R.I., 1967.

11. A.~Di~Concilio nad S.~Naimpally, Function space topologies on (partial) maps, Recent Progress in Function Spaces, D.~Di~Maio and L.~Holá (eds.), Quaderni di Mathematica, Vol.~3, Arace, 1998, 1-34.

12. R.~Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

13. V.~V.~Filippov, Basic topological structures of the theory of ordinary differential equations, Topology in Nonlinear Analysis, Banach Center Publ. 35 (1996), 171-192.

14. G. Gierz, K. H.~Hofmann, K.~Keimel, J.~D.~Lawson, M.~W.~Mislove, and D.~S.~Scott, Continuous Lattices and Domains. Cambridge Univ. Press, Cambridge, 2003.

15. L. M. Gluskin. Simple semigroups with zero. Dokl. AN SSSR 103 (1955), №1, 5-8 (in Russian).

16. L.~M.~Gluskin, Semigroups of homeomorphisms, Dokl. Akad. Nauk UkrSSR. Ser. A (1977), №12, 1059-1061 (in Russian).

17. L.~M.~Gluskin, B.~M.~Schein, L.~B.~Sneperman, and I.~S.~Yyaroker, Addendum to a survey of semigroups of contionuous selfmaps, Semigroup Forum 14 (1977), 95-125.

18. O.~Gutik, J.~Lawson, and D.~Repovs, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum 78 (2009), №2, 326-336.

19. O.~V.~Gutik, K.~P.~Pavlyk, H-closed topological semigroups and Brandt $\lambda-$extensions, Math. Metods Phys.-Mech. Fields 44 (2001), №3, 20-28 (in Ukrainian).

20. O.~V.~Gutik, K.~P. Pavlyk, Topological Brandt $\lambda$-extensions of absolutely $H$-closed topological inverse semigroups, Visnyk Lviv Univ., Ser. Mech.-Math. 61 (2003), 98-105.

21. O.~V.~Gutik and K.~P.~Pavlyk, On topological semigroups of matrix units, Semigroup Forum 71 (2005), №3, 389-400.

22. O. V. Gutik and K.~P. Pavlyk, On Brandt $\lambda^0$-extensions of semigroups with zero, Math. Metods Phys.-Mech. Fields 49 (2006), №3, 26-40.

23. O. V. Gutik and A.~R. Reiter, Symmetric inverse topological semigroups of finite rank $\leqslant n$, Math. Metods Phys.-Mech. Fields 52 (2009), №3 (to appear) (http://arxiv.org/abs/0912.0198v1).

24. O.~Gutik and D.~Repovs, On Brandt $\lambda^0$-extensions of monoids with zero, Semigroup Forum (to appear) (http://arxiv.org/abs/0910.0535v1).

25. K.~H.~Hofmann, M.~Mislove and A.~Stralka, The Pontryagin Duality of Compact 0-dimensional Semilattices and its Applications, Lecture Notes in Math. Vol.~396, Springer, 1974.

26. L.~Holá, Topologies on the space of partial maps, Recent Progress in Function Spaces, D.~Di~Maio and L.~Holá (eds.), Quaderni di Mathematica, Vol.~3, Arace, 1998, 157-186.

27. L.~Holá, Complete metrizability of generalized compact-open topology, Topology Appl. 91 (1999), №2, 159-167.

28. H.~P.~Künzi and L.~B.~Shapiro, On simulataneous extension of contionuous partial fuctions, Proc. Amer. Math. Soc. 125 (1997), 1853-1859.

29. K.~Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. 40 (1955), 61-67.

30. K.~D.~Magill, Jr., A survey of semigroups of contionuous selfmaps, Semigroup Forum 11 (1975/1976), 189-282.

31. S.~Mendes-Goncalves and R.~P.~Sullivan, Maximal inverse subsemigroups of the symmetric inverse semigroup on a finite-dimensional vector space, Comm. Algebra 34 (2006), №3, 1055-1069.

32. S.~D.~Orlov, Topologization of the generalized group of open partial homeomorphisms of a locally compact Hausdorff space, Izv. Vyssh. Uchebn. Zaved. Mat. (1974), №11, 61-68 (in Russian).

33. S.~D.~Orlov, On the theory of generalized topological groups, Theory of Semigroups and its Applications, Sratov Univ. Press. (1974), №3, 80-85 (in Russian).

34. B. M. Schein, Homomorphisms and subdirect decompositions of semigroups, Pacific J. Math. 24 (1966), №3, 529-547.

35. L.~B.~Sneperman, Semigroups of contionuous transformations and homeomorphisms of a simple arc, Dokl. Akad. Nauk SSSR 146 (1962), 1301-1304 (in Russian).

36. J.~W.~Stepp, A note on maximal locally compact semigroups, Proc. Amer. Math. Soc. 20 (1969), №1, 251-253.

37. J.~W.~Stepp, Algebraic maximal semilattices, Pacific J. Math. 58 (1975), №1, 243-248.

38. S.~Subbiah, The compact-open topology for semigroups of continuous self-maps, Semigroup Forum 35 (1987), №1, 29-33.

39. V.~V.~Wagner, Generalized groups, Dokl. Akad. Nauk SSSR 84 (1952), 1119-1122 (in Russian).

40. I.~S.~Yaroker, Semigroups of homeomorphisms of certain topological spaces, Dokl. Akad. Nauk UkrSSR. Ser. A. (1972), №11, 1008-1010 (in Russian).

	Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions of topological semigroups
Author	O.Gutik, K.Pavlyk, A.Reiter gutik@franko.lviv.ua, ovgutik@yahoo.com, andriy@yahoo.com, reiter@i.ua, kpavlyk@yahoo.co.uk Department of Mathematics, Ivan Franko Lviv National,University of L'viv, Universytetska 1, Lviv, 79000, Ukraine, , Pidstryhach Institute for Applied,Problems of Mechanics,and Mathematics of National Academy of,Sciences, Naukova 3b, Lviv, 79060,
Abstract	We show that a topological semigroup of finite partial bijections $\mathscr{I}_\lambda^n$ of an infinite set with a compact subsemigroup of idempotents is absolutely $H$-closed and any countably compact topological semigroup does not contain $\mathscr{I}_\lambda^n$ as a subsemigroup. We give sufficient conditions onto a topological semigroup $\mathscr{I}_\lambda^1$ to be non-$H$-closed. Also we describe the structure of countably compact Brandt $\lambda^0$-extensions of topological monoids and study the category of countably compact Brandt $\lambda^0$-extensions of topological monoids with zero.
Keywords	topological semigroup, matrix unit, Brandt extension
DOI	doi:10.30970/ms.32.2.115-131
Reference	1. A.~Abd-Allah and R.~Brown, A compact-open topology on partial maps with open domains, J. London Math. Soc. 21 (2) (1980), 480-486. 2. B.~B.~Baird, Inverse semigroups of homeomorphisms between open subsets, J. Austral. Math. Soc. Ser. A 24 (1977), №1, 92--102. 3. B.~B.~Baird, Embedding inverse semigroups of homeomorphisms on closed subsets, Glasgow Math. J. 18 (1977), №2, 199--207. 4. B.~B.~Baird, Epimorphisms of inverse semigroups of homeomorphisms between closed subsets, Semigroup Forum 14 (1977), №2, 161--166. 5. B.~B.~Baird, Inverse semigroups of homeomorphisms are Hopfian, Canad. J. Math. 31 (1979), №4, 800--807. 6. T.~Banakh and S.~Dimitrova, Openly factorizable spaces and compact extensions of topological semigroup, Preprint (arXiv: 0811.4272). 7. A.~A.~Beda, Continuous inverse semigroups of open partial homeomorphisms, Izv. Vyssh. Uchebn. Zaved. Mat. no. 1 (1980), 64--65 (in Russian). 8. P.~I.~Booth and R.~Brown, Spaces of partial maps, fibred mapping spaces and the compact-open topology, General Topology Appl. 8 (1978), 181-195. 9. 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; Vol. II, Marcel Dekker, Inc., New York and Basel, 1986. 10. A.~H.~Clifford and G.~B.~Preston, The Algebraic Theory of Semigroups, Vol. I., Amer. Math. Soc. Surveys 7, Providence, R.I., 1961; Vol. II., Amer. Math. Soc. Surveys 7, Providence, R.I., 1967. 11. A.~Di~Concilio nad S.~Naimpally, Function space topologies on (partial) maps, Recent Progress in Function Spaces, D.~Di~Maio and L.~Holá (eds.), Quaderni di Mathematica, Vol.~3, Arace, 1998, 1-34. 12. R.~Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989. 13. V.~V.~Filippov, Basic topological structures of the theory of ordinary differential equations, Topology in Nonlinear Analysis, Banach Center Publ. 35 (1996), 171-192. 14. G. Gierz, K. H.~Hofmann, K.~Keimel, J.~D.~Lawson, M.~W.~Mislove, and D.~S.~Scott, Continuous Lattices and Domains. Cambridge Univ. Press, Cambridge, 2003. 15. L. M. Gluskin. Simple semigroups with zero. Dokl. AN SSSR 103 (1955), №1, 5-8 (in Russian). 16. L.~M.~Gluskin, Semigroups of homeomorphisms, Dokl. Akad. Nauk UkrSSR. Ser. A (1977), №12, 1059-1061 (in Russian). 17. L.~M.~Gluskin, B.~M.~Schein, L.~B.~Sneperman, and I.~S.~Yyaroker, Addendum to a survey of semigroups of contionuous selfmaps, Semigroup Forum 14 (1977), 95-125. 18. O.~Gutik, J.~Lawson, and D.~Repovs, Semigroup closures of finite rank symmetric inverse semigroups, Semigroup Forum 78 (2009), №2, 326-336. 19. O.~V.~Gutik, K.~P.~Pavlyk, H-closed topological semigroups and Brandt $\lambda-$extensions, Math. Metods Phys.-Mech. Fields 44 (2001), №3, 20-28 (in Ukrainian). 20. O.~V.~Gutik, K.~P. Pavlyk, Topological Brandt $\lambda$-extensions of absolutely $H$-closed topological inverse semigroups, Visnyk Lviv Univ., Ser. Mech.-Math. 61 (2003), 98-105. 21. O.~V.~Gutik and K.~P.~Pavlyk, On topological semigroups of matrix units, Semigroup Forum 71 (2005), №3, 389-400. 22. O. V. Gutik and K.~P. Pavlyk, On Brandt $\lambda^0$-extensions of semigroups with zero, Math. Metods Phys.-Mech. Fields 49 (2006), №3, 26-40. 23. O. V. Gutik and A.~R. Reiter, Symmetric inverse topological semigroups of finite rank $\leqslant n$, Math. Metods Phys.-Mech. Fields 52 (2009), №3 (to appear) (http://arxiv.org/abs/0912.0198v1). 24. O.~Gutik and D.~Repovs, On Brandt $\lambda^0$-extensions of monoids with zero, Semigroup Forum (to appear) (http://arxiv.org/abs/0910.0535v1). 25. K.~H.~Hofmann, M.~Mislove and A.~Stralka, The Pontryagin Duality of Compact 0-dimensional Semilattices and its Applications, Lecture Notes in Math. Vol.~396, Springer, 1974. 26. L.~Holá, Topologies on the space of partial maps, Recent Progress in Function Spaces, D.~Di~Maio and L.~Holá (eds.), Quaderni di Mathematica, Vol.~3, Arace, 1998, 157-186. 27. L.~Holá, Complete metrizability of generalized compact-open topology, Topology Appl. 91 (1999), №2, 159-167. 28. H.~P.~Künzi and L.~B.~Shapiro, On simulataneous extension of contionuous partial fuctions, Proc. Amer. Math. Soc. 125 (1997), 1853-1859. 29. K.~Kuratowski, Sur l'espace des fonctions partielles, Ann. Mat. Pura Appl. 40 (1955), 61-67. 30. K.~D.~Magill, Jr., A survey of semigroups of contionuous selfmaps, Semigroup Forum 11 (1975/1976), 189-282. 31. S.~Mendes-Goncalves and R.~P.~Sullivan, Maximal inverse subsemigroups of the symmetric inverse semigroup on a finite-dimensional vector space, Comm. Algebra 34 (2006), №3, 1055-1069. 32. S.~D.~Orlov, Topologization of the generalized group of open partial homeomorphisms of a locally compact Hausdorff space, Izv. Vyssh. Uchebn. Zaved. Mat. (1974), №11, 61-68 (in Russian). 33. S.~D.~Orlov, On the theory of generalized topological groups, Theory of Semigroups and its Applications, Sratov Univ. Press. (1974), №3, 80-85 (in Russian). 34. B. M. Schein, Homomorphisms and subdirect decompositions of semigroups, Pacific J. Math. 24 (1966), №3, 529-547. 35. L.~B.~Sneperman, Semigroups of contionuous transformations and homeomorphisms of a simple arc, Dokl. Akad. Nauk SSSR 146 (1962), 1301-1304 (in Russian). 36. J.~W.~Stepp, A note on maximal locally compact semigroups, Proc. Amer. Math. Soc. 20 (1969), №1, 251-253. 37. J.~W.~Stepp, Algebraic maximal semilattices, Pacific J. Math. 58 (1975), №1, 243-248. 38. S.~Subbiah, The compact-open topology for semigroups of continuous self-maps, Semigroup Forum 35 (1987), №1, 29-33. 39. V.~V.~Wagner, Generalized groups, Dokl. Akad. Nauk SSSR 84 (1952), 1119-1122 (in Russian). 40. I.~S.~Yaroker, Semigroups of homeomorphisms of certain topological spaces, Dokl. Akad. Nauk UkrSSR. Ser. A. (1972), №11, 1008-1010 (in Russian).
Pages	115-131
Volume	32
Issue	2
Year	2009
Journal	Matematychni Studii
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