On pseudocompact topological Brandt λ0-extensions of semitopological monoids

Oleg Gutik 1  and Kateryna Pavlyk 2
  • 1 Department of Mechanics and Mathematics, National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine
  • 2 Institute of Mathematics, University of Tartu, J. Liivi 2, 50409, Tartu, Estonia

Abstract

In the paper we investigate topological properties of a topological Brandt λ0-extension B0λ(S) of a semitopological monoid S with zero. In particular we prove that for every Tychonoff pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) semitopological monoid S with zero there exists a unique semiregular pseudocompact (resp., Hausdorff countably compact, Hausdorff compact) extension B0λ(S) of S and establish their Stone-Cˇ ech and Bohr compactifications. We also describe a category whose objects are ingredients in the constructions of pseudocompact (resp., countably compact, sequentially compact, compact) topological Brandt λ0- extensions of pseudocompact (resp., countably compact, sequentially compact, compact) semitopological monoids with zeros.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] A. V. Arhangel’skij, Function spaces in the topology of pointwise convergence, and compact sets, Uspekhi Mat. Nauk 39:5 (1984), 11–50 (in Russian); English version: Russ. Math. Surv. 39:5 (1984), 9–56.

  • [2] A. V. Arkhangel’skii, Topological Function Spaces, Kluwer Publ., Dordrecht, 1992.

  • [3] A. H. Clifford, Matrix representations of completely simple semigroups, Amer. J. Math. 64 (1942), 327–342.

  • [4] I. Bucur and A. Deleanu, Introduction to the Theory of Categories and Functors, John Willey and Sons, Ltd., London, New York and Sidney, 1968.

  • [5] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vol. I. Amer. Math. Soc. Surveys 7, 1961; Vol. II. Amer. Math. Soc. Surveys 7, 1967.

  • [6] W. W. Comfort and K. A. Ross, Pseudocompactness and uniform continuity in topological groups, Pacif. J. Math. 16 (1966), 483–496.

  • [7] K. DeLeeuw, and I. Glicksberg, Almost-periodic functions on semigroups, Acta Math. 105 (1961), 99–140.

  • [8] R. Engelking, General Topology, 2nd ed., Heldermann, Berlin, 1989.

  • [9] O. V. Gutik, On Howie semigroup, Mat. Metody Phys.-Mech. Fields 42:4 (1999), 127–132 (in Ukrainian).

  • [10] O. V. Gutik and K. P. Pavlyk, H-closed topological semigroup and Brandt λ-extensions, Mat. Metody Phys.-Mech. Fields 44:3 (2001), 20–28 (in Ukrainian).

  • [11] O. V. Gutik and K. P. Pavlyk, Topological semigroups of matrix units, Algebra Discrete Math. no. 3 (2005), 1–17.

  • [12] O. V. Gutik and K. P. Pavlyk, On Brandt λ0-extensions of semigroups with zero, Mat. Metody Phis.-Mech. Polya. 49:3 (2006), 26–40.

  • [13] O. Gutik, K. Pavlyk, and A. Reiter, Topological semigroups of matrix units and countably compact Brandt λ0- extensions, Mat. Stud. 32:2 (2009), 115–131.

  • [14] O. V. Gutik, K. P. Pavlyk and A. R. Reiter, On topological Brandt semigroups, Math. Methods and Phys.-Mech. Fields 54:2 (2011), 7–16 (in Ukrainian); English Version in: J. Math. Sc. 184:1 (2012), 1–11.

  • [15] O. Gutik and D. Repovš, On countably compact 0-simple topological inverse semigroups, Semigroup Forum 75:2 (2007), 464–469.

  • [16] O. Gutik and D. Repovš, On Brandt λ0-extensions of monoids with zero, Semigroup Forum 80:1 (2010), 8–32.

  • [17] J. M. Howie, Fundamentals of Semigroup Theory, London Math. Monographs, New Ser. 12, Clarendon Press, Oxford, 1995.

  • [18] W. D. Munn, Matrix representations of semigroups, Proc. Cambridge Phil. Soc. 53 (1957), 5–12.

  • [19] M. Petrich, Inverse Semigroups, John Wiley & Sons, New York, 1984.

  • [20] E. A. Reznichenko, Extension of functions defined on products of pseudocompact spaces and continuity of the inverse in pseudocompact groups, Topology Appl. 59:3 (1994), 233–244.

  • [21] W. Ruppert, Compact Semitopological Semigroups: An Intrinsic Theory, Lecture Notes in Mathematics, Vol. 1079, Springer, Berlin, 1984.

OPEN ACCESS

Journal + Issues

Search