On a Topological Semigroup Finite Transformations of a Hausdorff Topological Space
by
Andriy Reiter
Ivan Franko National University of Lviv
Coauthors: Oleg Gutik

We shall follow the terminology of [CHK].

A topological (inverse) semigroup is a topological space together with a continuous multiplication (and an inversion, respectively). If S is a semigroup and t is a topology on S such that (S, t) is a topological semigroup, then we shall call t a semigroup topology on S.

Let X be a Hausdorff topological space. Let I_fin(X) denote the set of all partial one-to-one finite transformations of X together with the following semigroup operation:

x(ab)=(xa)b   if    x ∈ dom(ab)={ y ∈ doma | ya ∈ domb},     for     a, b ∈ Ifin(X).

The semigroup I_fin(X) is called the finite symmetric inverse semigroup over the topological space X.

Definition ([2], [4]): Let S be a class of topological semigroups. A semigroup S is called algebraically h-closed in the class S if for any topology t on S such that (S, t) ∈ S and for any topological semigroups T ∈ S and any continuous homomorphism h: S→ T we have that h(S) is a closed subsemigroup of T.

Theorem 1 For any Hausdorff topological space the semigroup I_fin(X) is algebraically h-closed in the class of topological inverse semigroups.

Definition [2]: A Bohr compactification of a topological semigroup S is a pair (b, B(S)) such that B(S) is a compact topological semigroup, b: S→ B(S) is a continuous homomorphism, and if g: S→ T is a continuous homomorphism of S into a compact semigroup T, then there exists a unique continuous homomorphism f: B(S)→ T such that f○b = g.

Theorem 2 Let X be an infinite Hausdorff topological space and t be a Hausdorff semigroup topology on I_fin(X). Then the Bohr compactification of (I_fin(X), t) is a trivial semigroup.

References:

[1] J.H. Carruth, J.A. Hildebrant, 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] K. DeLeeuw, and I. Glicksberg, Almost-periodic functions on semigroups, Acta Math. 105 (1961), 99-140.

[3] O. V. Gutik and K. P. Pavlyk, On topological semigroups of matrix units, Semigroup Forum 71:3 (2005), 389-400.

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

Date received: May 11, 2008