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Local Homomorphisms of Topological Groups

Part of: Connections with other structures, applications Generalities Topological and differentiable algebraic systems Peculiar spaces

Published online by Cambridge University Press:  09 April 2009

Yevhen Zelenyuk
Yevhen Zelenyuk
Affiliation:
School of MathematicsUniversity of the WitwatersrandPrivate Bag 3 Wits 2050 South Africazelenyuk@maths.wits.ac.za
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Abstract

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A mapping f : G → s from a left topological group G into a semigroup S is a local homomorphism if for every x є G \ {e}, there is a neighborhood Ux of e such that f (xy) = f (x)f (y) for all y є Ux \ {e}. A local homomorphism f : G → S is onto if for every neighborhood U of e, f(U \ {e}) = S. We show that

(1) every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto N,

(2) it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup,

(3) it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

MSC classification

Secondary: 22A30: Other topological algebraic systems and their representations 54H11: Topological groups 54A35: Consistency and independence results 54G05: Extremally disconnected spaces, $F$-spaces, etc.
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 83 , Issue 1 , August 2007 , pp. 135 - 148
Copyright
Copyright © Australian Mathematical Society 2007

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