On feebly compact inverse primitive (semi)topological semigroups

O. Gutik; O. Ravsky

doi:10.15330/ms.44.1.3-26

We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions under which the maximal subgroup of an inverse primitive feebly compact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt $\lambda^0$-extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular, we show that the inversion in a quasi-regular primitive inverse feebly compact topological semigroup is continuous. An analogue of the Comfort--Ross Theorem is proved for such semigroups: the Tychonoff product of an arbitrary family of primitive inverse semiregular feebly compact semitopological semigroups with closed maximal subgroups is feebly compact. We describe the structure of the Stone-Cech compactification of a Hausdorff primitive inverse countably compact semitopological semigroup~$S$ such that every maximal subgroup of $S$ is a topological group.

1. A. Arhangelskii, M. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

2. A.V. Arkhangelskii, Topological function spaces, Kluwer Publ., Dordrecht, 1992.

3. A.V. Arkhangelskii, E.A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology Appl., 151 (2005), 107-119.

4. T. Banakh, S. Dimitrova, Openly factorizable spaces and compact extensions of topological semigroups, Comment. Math. Univ. Carol., 51 (2010), №1, 113-131.

5. T.O. Banakh, I.Yo. Guran, O.V. Ravsky, Boundedness and separability in topological groups. (in preparation)

6. T.O. Banakh, O.V. Gutik, On the continuity of inversion in countably compact inverse topological semigroups, Semigroup Forum, 68 (2004), №3, 411-418.

7. T. Berezovski, O. Gutik, K. Pavlyk, Brandt extensions and primitive topological inverse semigroups, Int. J. Math. Math. Sci., 2010 (2010), 13 p., doi:10.1155/2010/671401.

8. J.H. Carruth, J.A. Hildebrant, R.J. Koch, The theory of topological semigroups, V.I, Marcel Dekker, Inc., New York and Basel, 1983; V.II, Marcel Dekker, Inc., New York and Basel, 1986.

9. A.H. Clifford, G.B. Preston, The algebraic theory of semigroups, V. I and II, Amer. Math. Soc. Surveys, 7, Providence, R.I., 1961 and 1967.

10. W.W. Comfort, K.A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacif. J. Math., 16 (1966), №3, 483-496.

11. K. DeLeeuw, I. Glicksberg, Almost-periodic functions on semigroups, Acta Math., 105 (1961), 99-140.

12. C. Eberhart, J. Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc., 144 (1969), 115-126.

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

14. I. Glicksberg, Stone-Cech compactifications of products, Trans. Amer. Math. Soc., 90 (1959), 369-382.

15. O.V. Gutik, On Howie semigroup, Mat. Metody Phis.-Mech. Polya., 42 (1999), №4, 127-132. (in Ukrainian)

16. O.V. Gutik, K.P. Pavlyk, On Brandt $\lambda^0$-extensions of semigroups with zero, Mat. Metody Phis.-Mech. Polya, 49 (2006), №3, 26-40.

17. O.V. Gutik, K.P. Pavlyk, Pseudocompact primitive topological inverse semigroups, Mat. Metody Phis.- Mech. Polya, 56 (2013), №2, 7-19; reprinted version in: J. Math. Sc., 203 (2014), .1, 1-15.

18. O. Gutik, K. Pavlyk, On pseudocompact topological Brandt $\lambda^0$-extensions of semitopological monoids, Topol. Algebra Appl., 1 (2013), 60-79.

19. O. Gutik, K. Pavlyk, A. Reiter, Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions, Mat. Stud., 32 (2009), №2, 115-131.

20. O.V. Gutik, K.P. Pavlyk, A.R. Reiter, On topological Brandt semigroups, Math. Methods and Phys.- Mech. Fields, 54 (2011), №2, 7-16 (in Ukrainian); English version in: J. Math. Sc., 184 (2012), №1, 1-11.

21. O. Gutik, D. Repov.s, On countably compact 0-simple topological inverse semigroups, Semigroup Forum, 75 (2007), №2, 464-469.

22. O. Gutik, D. Repov.s, On Brandt $\lambda^0$-extensions of monoids with zero, Semigroup Forum, 80 (2010), №1, 8-32.

23. M. Kat.etov, On H-closed extensions of topological spaces, .Casopis P.est. Mat. Fys., 72 (1947), 17-32.

24. P. Koszmider, A. Tomita, S. Watson, Forcing countably compact group topologies on a larger free Abelian group, Topology Proc., 25, Summer (2000), 563-574.

25. M. Petrich, Inverse Semigroups, John Wiley & Sons, New York, 1984.

26. O.V. Ravsky, Paratopological groups II, Mat. Stud., 17, №1 (2002), 93-101.

27. O.V. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis. Lviv University, 2002. (in Ukrainian)

28. O. Ravsky, An example of a Hausdorff countably compact paratopological group which is not a topological group, Proc. IV Int. Algebraic Conf. in Ukraine, Lviv, 2003, 192 p.

29. A. Ravsky, Pseudocompact paratopological groups, Preprint arXiv:1003.5343v5.

30. A. Ravsky, Pseudocompact paratopological groups that are topological, Topology Appl., (submitted), Preprint arXiv:1406.2001.

31. O. Ravsky, E. Reznichenko, The continuity of the inverse in groups, Int. Conf. Func. Anal. Appl. dedicated 110th Anniversary of S. Banach, 2002, Lviv, Ukraine, P. 170–172.

32. W. Ruppert, Compact semitopological semigroups: an intrinsic theory, Lect. Notes Math., 1079, Springer, Berlin, 1984.

33. M.H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375–481.

	On feebly compact inverse primitive (semi)topological semigroups
Author	O. Gutik, O. Ravsky o_gutik@franko.lviv.ua, ovgutik@yahoo.com, oravsky@mail.ru Faculty of Mathematics, Ivan Franko National University of Lviv, Pidstrygach Institute for Applied Problems of Mechanics and Mathematics of NASU
Abstract	We study the structure of inverse primitive feebly compact semitopological and topological semigroups. We find conditions under which the maximal subgroup of an inverse primitive feebly compact semitopological semigroup $S$ is a closed subset of $S$ and describe the topological structure of such semiregular semitopological semigroups. Later we describe the structure of feebly compact topological Brandt $\lambda^0$-extensions of topological semigroups and semiregular (quasi-regular) primitive inverse topological semigroups. In particular, we show that the inversion in a quasi-regular primitive inverse feebly compact topological semigroup is continuous. An analogue of the Comfort--Ross Theorem is proved for such semigroups: the Tychonoff product of an arbitrary family of primitive inverse semiregular feebly compact semitopological semigroups with closed maximal subgroups is feebly compact. We describe the structure of the Stone-Cech compactification of a Hausdorff primitive inverse countably compact semitopological semigroup~$S$ such that every maximal subgroup of $S$ is a topological group.
Keywords	semigroup; primitive inverse semigroup; Brandt $\lambda^0$-extension; topological semigroup; topological group; paratopological group; semitopological semigroup; semitopological group; topological Brandt $\lambda^0$-extension; Brandt semigroup; primitive inverse semigroup; pseudocompact space; feebly compact space; countably compact space; countably pracompact space; Stone-Cech compactification
DOI	doi:10.15330/ms.44.1.3-26
Reference	1. A. Arhangelskii, M. Tkachenko, Topological groups and related structures, Atlantis Studies in Mathematics, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008. 2. A.V. Arkhangelskii, Topological function spaces, Kluwer Publ., Dordrecht, 1992. 3. A.V. Arkhangelskii, E.A. Reznichenko, Paratopological and semitopological groups versus topological groups, Topology Appl., 151 (2005), 107-119. 4. T. Banakh, S. Dimitrova, Openly factorizable spaces and compact extensions of topological semigroups, Comment. Math. Univ. Carol., 51 (2010), №1, 113-131. 5. T.O. Banakh, I.Yo. Guran, O.V. Ravsky, Boundedness and separability in topological groups. (in preparation) 6. T.O. Banakh, O.V. Gutik, On the continuity of inversion in countably compact inverse topological semigroups, Semigroup Forum, 68 (2004), №3, 411-418. 7. T. Berezovski, O. Gutik, K. Pavlyk, Brandt extensions and primitive topological inverse semigroups, Int. J. Math. Math. Sci., 2010 (2010), 13 p., doi:10.1155/2010/671401. 8. J.H. Carruth, J.A. Hildebrant, R.J. Koch, The theory of topological semigroups, V.I, Marcel Dekker, Inc., New York and Basel, 1983; V.II, Marcel Dekker, Inc., New York and Basel, 1986. 9. A.H. Clifford, G.B. Preston, The algebraic theory of semigroups, V. I and II, Amer. Math. Soc. Surveys, 7, Providence, R.I., 1961 and 1967. 10. W.W. Comfort, K.A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacif. J. Math., 16 (1966), №3, 483-496. 11. K. DeLeeuw, I. Glicksberg, Almost-periodic functions on semigroups, Acta Math., 105 (1961), 99-140. 12. C. Eberhart, J. Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc., 144 (1969), 115-126. 13. R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989. 14. I. Glicksberg, Stone-Cech compactifications of products, Trans. Amer. Math. Soc., 90 (1959), 369-382. 15. O.V. Gutik, On Howie semigroup, Mat. Metody Phis.-Mech. Polya., 42 (1999), №4, 127-132. (in Ukrainian) 16. O.V. Gutik, K.P. Pavlyk, On Brandt $\lambda^0$-extensions of semigroups with zero, Mat. Metody Phis.-Mech. Polya, 49 (2006), №3, 26-40. 17. O.V. Gutik, K.P. Pavlyk, Pseudocompact primitive topological inverse semigroups, Mat. Metody Phis.- Mech. Polya, 56 (2013), №2, 7-19; reprinted version in: J. Math. Sc., 203 (2014), .1, 1-15. 18. O. Gutik, K. Pavlyk, On pseudocompact topological Brandt $\lambda^0$-extensions of semitopological monoids, Topol. Algebra Appl., 1 (2013), 60-79. 19. O. Gutik, K. Pavlyk, A. Reiter, Topological semigroups of matrix units and countably compact Brandt $\lambda^0$-extensions, Mat. Stud., 32 (2009), №2, 115-131. 20. O.V. Gutik, K.P. Pavlyk, A.R. Reiter, On topological Brandt semigroups, Math. Methods and Phys.- Mech. Fields, 54 (2011), №2, 7-16 (in Ukrainian); English version in: J. Math. Sc., 184 (2012), №1, 1-11. 21. O. Gutik, D. Repov.s, On countably compact 0-simple topological inverse semigroups, Semigroup Forum, 75 (2007), №2, 464-469. 22. O. Gutik, D. Repov.s, On Brandt $\lambda^0$-extensions of monoids with zero, Semigroup Forum, 80 (2010), №1, 8-32. 23. M. Kat.etov, On H-closed extensions of topological spaces, .Casopis P.est. Mat. Fys., 72 (1947), 17-32. 24. P. Koszmider, A. Tomita, S. Watson, Forcing countably compact group topologies on a larger free Abelian group, Topology Proc., 25, Summer (2000), 563-574. 25. M. Petrich, Inverse Semigroups, John Wiley & Sons, New York, 1984. 26. O.V. Ravsky, Paratopological groups II, Mat. Stud., 17, №1 (2002), 93-101. 27. O.V. Ravsky, The topological and algebraical properties of paratopological groups, Ph.D. Thesis. Lviv University, 2002. (in Ukrainian) 28. O. Ravsky, An example of a Hausdorff countably compact paratopological group which is not a topological group, Proc. IV Int. Algebraic Conf. in Ukraine, Lviv, 2003, 192 p. 29. A. Ravsky, Pseudocompact paratopological groups, Preprint arXiv:1003.5343v5. 30. A. Ravsky, Pseudocompact paratopological groups that are topological, Topology Appl., (submitted), Preprint arXiv:1406.2001. 31. O. Ravsky, E. Reznichenko, The continuity of the inverse in groups, Int. Conf. Func. Anal. Appl. dedicated 110th Anniversary of S. Banach, 2002, Lviv, Ukraine, P. 170–172. 32. W. Ruppert, Compact semitopological semigroups: an intrinsic theory, Lect. Notes Math., 1079, Springer, Berlin, 1984. 33. M.H. Stone, Applications of the theory of Boolean rings to general topology, Trans. Amer. Math. Soc., 41 (1937), 375–481.
Pages	3-26
Volume	44
Issue	1
Year	2015
Journal	Matematychni Studii
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