The Rees-Suschkewitsch theorem for simple topological semigroups

T.O.Banakh; S.Dimitrova; O.V.Gutik

We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product $[X\times H\times Y]_\sigma$ of a topological group $H$ over topological spaces $X,Y$ with a continuous sandwich function $\sigma\colon Y\times X\to H$. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or the power $S^{2^{\mathfrak c}}$ is countably compact. The last item generalizes an old Wallace's result saying that each simple compact topological semigroup is a topological paragroup.

1. O.T.~Alas, M.~Sanchis, Countably compact paratopological groups, Semigroup Forum, 74 (2007), 423---438.

2. O.~Andersen, Ein Bericht über die Struktur abstrakter Halbgruppen, PhD Thesis, Hamburg, 1952.

3. T.~Banakh, S.D.~Dimitrova, Openly factorizable spaces and compact extensions of topological semigroups, preprint (arXiv:0811.4272).

4. T.~Banakh, O.~Gutik, On the continuity of the inversion in countably compact inverse topological semigroups, Semigroup Forum 68 (2004), 411---418.

5. R.H.~Bruck, A survey of binary systems, Ergebnisse der Math., Berlin, Springer, 1958, Heft 20.

6. J.H. Carruth, J.A. Hildebrant, R.J. Koch, The Theory of Topological Semigroups, I, II. Marcel Dekker, Inc., New York and Basel, 1983 and 1986.

7. A.H.~Clifford, G.B.~Preston, The Algebraic Theory of Semigroups, Vol. 1, Marcell Dekker, 1961.

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

9. R.~Ellis, A note on the continuity of inverse, Proc. Amer. Math. Soc. 8 (1957) №2, 372---373.

10. R.~Engelking, General Topology, Warsaw, PWN, 1977.

11. B. Gelbaum, G.K. Kalish, J.M. Olmstead, On the embedding of topological semigroups and integral domains, Proc. Amer. Math. Soc. 2 (1951), 807---821

12. J.~Ginsburg, V.~Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403---418.

13. O.~Gutik, Embedding topological semigroups into simple semigroups, Mat. Stud. 3 (1994), 10--14 (in Russian).

14. O.~Gutik, D.~Pagon, D.~Repov s, On sequential countably compact topological semigroups, preprint (arXiv: 0807.3102)

15. J.A.~Hildenbrandt, R.J.~Koch, Swelling actions of $\Gamma$-compact semigroups, Semigroup Forum 33 (1988) №1, 65---85.

16. N.~Hindman, D.~Strauss, Algebra in the Stone-Cech Compactification. Theory and Applications. de Gruyter Expositions in Mathematics, 27. Walter de Gruyter \&\ Co., Berlin, 1998.

17. K.H.~Hofmann, P.S.~Mostert, Elements of Compact Semigroups, Charles E. Merril Books, Inc., Columbus, Ohio, 1966.

18. K.~Iwasawa, Finite and compact groups, Sugaku 1 (1948), 31---32 (in Japanese).

19. R.J. Koch, A.D. Wallace, Stability in semigroups, Duke Math. J. 24 (1957), 193---195.

20. K.~Numakura, On bicompact semigroups, Math. J. Okayama Univ. 1 (1952), 99--108.

21. O.~Ravsky, E.~Reznichenko, The continuity of the inverse in groups, in: Int. Conf. Func. Anal. Appl. dedicated to 110th anniversary of S.Banach, May 28--31, 2002, Lviv, Ukraine. P.170---172.

22. D.~Rees, On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387---400.

23. E.A.~Reznichenko, Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl. 59 (1994) №3, 233---244.

24. W. Ruppert, Rechtstopologische Halbgruppen, J. Reine Angew. Math. 261 (1973), 123---133.

25. W.~Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, LNM 1079, Springer, 1984.

26. A.H.~Tomita, The Wallace problem: A counterexample from MA${}_\mathrm{countable}$ and $p$-compactness, Canad. Math. Bull 39 (1996) №4, 486---498.

27. A.D. Wallace, A note on mobs, I, Anais. Acad. Brasil. Ci. 24 (1952), 329---334.

28. A.D.~Wallace, The Suschkewitsch-Ress structure theorem for compact simple semigroups, Proc. Nat. Acad. Sci. 42 (1956), 430---432.

29. A.D.~Wallace, Retractions in semigroups, Pacific J. Math. 7 (1957), 1513---1517.

	The Rees-Suschkewitsch theorem for simple topological semigroups
Author	T.O.Banakh, S.Dimitrova, O.V.Gutik tbanakh@yahoo.com, o_gutik@franko.lviv.ua Department of Mathematics, Ivan Franko,National University of Lviv,Universytetska 1, Lviv, 79000, Ukraine, National Technical University "Kharkov Polytechnical,Institute", Frunze 21, Kharkiv, 61002, Ukraine
Abstract	We detect topological semigroups that are topological paragroups, i.e., are isomorphic to a Rees product $[X\times H\times Y]_\sigma$ of a topological group $H$ over topological spaces $X,Y$ with a continuous sandwich function $\sigma\colon Y\times X\to H$. We prove that a simple topological semigroup $S$ is a topological paragroup if one of the following conditions is satisfied: (1) $S$ is completely simple and the maximal subgroups of $S$ are topological groups, (2) $S$ contains an idempotent and the square $S\times S$ is countably compact or pseudocompact, (3) $S$ is sequentially compact or the power $S^{2^{\mathfrak c}}$ is countably compact. The last item generalizes an old Wallace's result saying that each simple compact topological semigroup is a topological paragroup.
Keywords	Rees product, simple topological semigroup, topological paragroup
DOI	doi:10.30970/ms.31.2.211-218
Reference	1. O.T.~Alas, M.~Sanchis, Countably compact paratopological groups, Semigroup Forum, 74 (2007), 423---438. 2. O.~Andersen, Ein Bericht über die Struktur abstrakter Halbgruppen, PhD Thesis, Hamburg, 1952. 3. T.~Banakh, S.D.~Dimitrova, Openly factorizable spaces and compact extensions of topological semigroups, preprint (arXiv:0811.4272). 4. T.~Banakh, O.~Gutik, On the continuity of the inversion in countably compact inverse topological semigroups, Semigroup Forum 68 (2004), 411---418. 5. R.H.~Bruck, A survey of binary systems, Ergebnisse der Math., Berlin, Springer, 1958, Heft 20. 6. J.H. Carruth, J.A. Hildebrant, R.J. Koch, The Theory of Topological Semigroups, I, II. Marcel Dekker, Inc., New York and Basel, 1983 and 1986. 7. A.H.~Clifford, G.B.~Preston, The Algebraic Theory of Semigroups, Vol. 1, Marcell Dekker, 1961. 8. C.~Eberhart, J.~Selden, On the closure of the bicyclic semigroup, Trans. Amer. Math. Soc. 144 (1969), 115--126. 9. R.~Ellis, A note on the continuity of inverse, Proc. Amer. Math. Soc. 8 (1957) №2, 372---373. 10. R.~Engelking, General Topology, Warsaw, PWN, 1977. 11. B. Gelbaum, G.K. Kalish, J.M. Olmstead, On the embedding of topological semigroups and integral domains, Proc. Amer. Math. Soc. 2 (1951), 807---821 12. J.~Ginsburg, V.~Saks, Some applications of ultrafilters in topology, Pacific J. Math. 57 (1975), 403---418. 13. O.~Gutik, Embedding topological semigroups into simple semigroups, Mat. Stud. 3 (1994), 10--14 (in Russian). 14. O.~Gutik, D.~Pagon, D.~Repov s, On sequential countably compact topological semigroups, preprint (arXiv: 0807.3102) 15. J.A.~Hildenbrandt, R.J.~Koch, Swelling actions of $\Gamma$-compact semigroups, Semigroup Forum 33 (1988) №1, 65---85. 16. N.~Hindman, D.~Strauss, Algebra in the Stone-Cech Compactification. Theory and Applications. de Gruyter Expositions in Mathematics, 27. Walter de Gruyter \&\ Co., Berlin, 1998. 17. K.H.~Hofmann, P.S.~Mostert, Elements of Compact Semigroups, Charles E. Merril Books, Inc., Columbus, Ohio, 1966. 18. K.~Iwasawa, Finite and compact groups, Sugaku 1 (1948), 31---32 (in Japanese). 19. R.J. Koch, A.D. Wallace, Stability in semigroups, Duke Math. J. 24 (1957), 193---195. 20. K.~Numakura, On bicompact semigroups, Math. J. Okayama Univ. 1 (1952), 99--108. 21. O.~Ravsky, E.~Reznichenko, The continuity of the inverse in groups, in: Int. Conf. Func. Anal. Appl. dedicated to 110th anniversary of S.Banach, May 28--31, 2002, Lviv, Ukraine. P.170---172. 22. D.~Rees, On semi-groups, Proc. Cambridge Philos. Soc. 36 (1940), 387---400. 23. E.A.~Reznichenko, Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl. 59 (1994) №3, 233---244. 24. W. Ruppert, Rechtstopologische Halbgruppen, J. Reine Angew. Math. 261 (1973), 123---133. 25. W.~Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, LNM 1079, Springer, 1984. 26. A.H.~Tomita, The Wallace problem: A counterexample from MA${}_\mathrm{countable}$ and $p$-compactness, Canad. Math. Bull 39 (1996) №4, 486---498. 27. A.D. Wallace, A note on mobs, I, Anais. Acad. Brasil. Ci. 24 (1952), 329---334. 28. A.D.~Wallace, The Suschkewitsch-Ress structure theorem for compact simple semigroups, Proc. Nat. Acad. Sci. 42 (1956), 430---432. 29. A.D.~Wallace, Retractions in semigroups, Pacific J. Math. 7 (1957), 1513---1517.
Pages	211-218
Volume	31
Issue	2
Year	2009
Journal	Matematychni Studii
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Table of content of issue	html