Countably Compact Primitive topological Inverse semigroups
by
Kateryna Pavlyk
Department of Algebra, Pidstrygach Institute for Applied Problems of Mechanics and Mathematics of National Academy of Sciences, Naukova 3b, Lviv, 79060, Ukraine
Coauthors: Oleg Gutik

We shall follow the terminology of [1], [2], [4]. If S is a semigroup then we denote the subset of idempotents of S by E(S).

A topological (inverse) semigroup is a topological space together with a continuous multiplication (and an inversion, respectively).

An inverse semigroup S with zero is called primitive if every idempotent of S is primitive. A completely 0-simple inverse semigroup is called Brandt semigroup.

Theorem 1 Any countably compact primitive topological inverse semigroup S is an orthogonal sum of topological inverse Brandt semigroups with finite subsemigroups of idempotents. Moreover, E(S) is homeomorphic to the one-point Alexandroff compactification of infinite discrete space with zero as a remainder if and only if E(S) is an infinite set.

Theorem 2 Any countably compact primitive topological inverse semigroup is a dense subsemigroup of a compact primitive topological inverse semigroup.

Let S be a class of topological semigroups.

A semigroup S ∈ S is called H-closed in S, if S is a closed subsemigroup of any topological semigroup T ∈ S which contains S as subsemigroup. If S coincides with the class of all topological semigroups, then the semigroup S is called H-closed. H-closed topological semigroups were introduced by J. W. Stepp in [5], and there they were called maximal semigroups.

A topological semigroup S ∈ S is called absolutely H-closed in the class S, if any continuous homomorphic image of S into T ∈ S is H-closed in S [3], [6].

Theorem 3 A primitive topological inverse semigroup S is (absolutely) H-closed in the class of topological inverse semigroups if and only if any maximal subgroup of S is an (absolutely) H-closed semigroup in the class of topological inverse semigroups.

References:

[1] 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.

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

[3] O. V. Gutik and K. P. Pavlyk, H-closed topological semigroups and topological Brandt l-extensions, Math. Methods and Phys.-Mech. Fields 44:3 (2001), 20-28. (in Ukrainian)

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

[5] J. W. Stepp, A note on maximal locally compact semigroups, Proc. Amer. Math. Soc. 20:1 (1969), 251-253.

[6] J. W. Stepp, Algebraic maximal semilattices, Pacific J. Math. 58:1 (1975), 243-248.

Date received: May 11, 2008