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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 [email protected]
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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

References

[1]Arhangel'skil, A., ‘Every extremally disconnected bicompactum of weight c is inhomogeneous’, Soviet Math. Dokl. 8 (1967), 897–900.Google Scholar
[2]Comfort, W. and van Mill, J., ‘Groups with only resolvable group topologies’, Proc. Amer. Math. Soc. 120(1994), 687–696.CrossRefGoogle Scholar
[3]Fremlin, D., Consequences of Martin's axiom (Cambridge University Press, 1984).CrossRefGoogle Scholar
[4]Hindman, N. and Strauss, D., Algebra in the Stone-Čech compactification (De Gruyter, Berlin, 1998).Google Scholar
[5]Malykhin, V., ‘Extremally disconnected and similar groups’, Soviet Math. Dokl. 16 (1975), 21–25.Google Scholar
[6]Protasov, I., ‘Indecomposable topologies on groups’, Ukranian Math. J. 50 (1998), 1879–1887.Google Scholar
[7]Shelah, S., Properforcing (Springer-Verlag, Berlin, 1982).Google Scholar
[8]Sirota, S., ‘The product of topological groups and extremal disconnectedness’, Math. USSR Sbornik 8 (1969), 169–180.CrossRefGoogle Scholar
[9]Zelenyuk, Y., ‘Finite groups in βN are trivial’, Semigroup Forum 55 (1997), 131–132.CrossRefGoogle Scholar
[10]Zelenyuk, Y., ‘On partitions of groups into dense subsets’, Topology Appl. 126 (2002), 327–339.Google Scholar