On Connections Between Inverse Semigroups and Topology Structures
by
Grigory Zhitomirsky
Bar-Ilan University, Israel

It is known that sometimes the algebraic structure of a topological algebra determines some of its topological properties (for example, in topological groups). Now we study some connections between topological and algebraic structures in a topological inverse semigroup.

A topological inverse semigroup \bold S as a structure of the kind (S, o , ^-1, O), where O is a topology on S, (S, o , ^-1) is an inverse semigroup and its operations are continuous with respect to the topology O. We assume only hat \goth O is a T\sb 0-topology. Let as usual: (a, b) in \pi₁ <===> aa^-1 = bb^-1 , (a, b) in \omega <===> aa^-1 b = a , \chi₁ = \omega o \pi₁ and \kappa be the greatest congruence relation included in \pi\sb 1. Let , (a, b) in \sigma <===> aa^-1 b = bb^-1a ba^-1a = ab^-1b.

If U is an open subset of S then so are \chi₁ (U), \omega(U), and (\omega o \kappa)(U). Let \goth O \sb \omega be the set of all U in \goth O such that \omega(U) subset U. With respect to \goth O \sb \omega, S is also a topological inverse semigroup. A topological inverse semigroup \bold S will be called \omega-separable if the topology O _\omega satisfies the T ₀-axiom.

\bold S is \omega-separable iff for every a in S the set \omega^-1 < a > is closed. The relation \omega\sb t = {(a, b)|a in [`{b}]} is a stable order relation on S, and therefore \omega\sb t subset \sigma. If \bold S is \omega-separable then \omega\sb t subset \omega and \omega\sb t o \omega^-1\sb t is a congruence relation on S. If \goth O is a T\sb 1-topology then the subsets \omega < a > , \pi\sb 1 < a > , \chi\sb 1 < a > and \kappa < a > are closed for every a in S.

Proposition Let \bold S be \omega-separable. The following conditions are equivalent: (i) \goth O is T₁-topology; (ii) all sets \omega < a > , a in S are closed; (iii) all sets \pi₁ < a > , a in S are closed; (iv) all sets \chi₁ < a > , a in S are closed; (v) all sets \kappa < a > , a in S are closed. If the topology \goth O is Hausdorff then the sets \omega^-1 < a > , \sigma < a > are closed for every a in S and therefore \bold S is \omega-separable. We call a pair of elements of a topological space a close pair if every neighborhoods of these elements have non-empty intersections. All such pairs form a stable tolerance relation \xi on this space. If a topological inverse semigroup is \omega-separable then \xi subset \sigma. This generalizes the well known fact that the topology of every topological group is Hausdorff.

Every topological inverse semigroup can be regarded as a fiber space which base is a topological antigroup and fibers are topological groups. If S is an antigroup its algebraic structure determines the least \omega-separable topology such that every topological complete representation of S is continuous. Properties of this topology are studied.

Date received: May 31, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caec-38.