On sequences of homomorphisms into measure algebras and the Efimov problem

Damian Sobota; Piotr Borodulin-Nadzieja

doi:10.1017/jsl.2021.70

On sequences of homomorphisms into measure algebras and the Efimov problem

Damian Sobota
, Piotr Borodulin-Nadzieja

Department of Mathematics

Publications: Contribution to journal › Article › Peer Reviewed

Abstract

For given Boolean algebras A and B we endow the space H(A,B) of all Boolean homomorphisms from A to B with various topologies and study convergence properties of sequences in H(A,B). We are in particular interested in the situation when B is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on A in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on H(A,B) for a Boolean algebra B carrying a strictly positive measure and convergence properties of sequences of measures on A.

Original language	English
Pages (from-to)	191-218
Number of pages	28
Journal	Journal of Symbolic Logic
Volume	88
Issue number	1
Early online date	13 Sept 2021
DOIs	https://doi.org/10.1017/jsl.2021.70
Publication status	Published - 13 Mar 2023

Funding

We would like to thank Grzegorz Plebanek and Krzysztof Majcher for valuable discussions concerning homomorphisms, Hamming codes, and painting fences. We would also like to thank Lyubomyr Zdomskyy for reading the first draft of the paper and sharing with us many relevant comments. The first author was supported by National Science Center project no. 2018/29/B/ST1/00223. The second author was supported by the Austrian Science Fund, grant M2500-N35.

Austrian Fields of Science 2012

101013 Mathematical logic
101024 Probability theory

Keywords

algebraic convergence
Efimov problem
measure algebras
perfect Hamming codes.
random forcing
weak convergence of measures
perfect Hamming codes
TOPOLOGY
Algebraic convergence
Weak convergence of measures
Measure algebras
Random forcing

Access to Document

10.1017/jsl.2021.70

https://arxiv.org/abs/2101.00513

Cite this

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abstract = "For given Boolean algebras A and B we endow the space H(A,B) of all Boolean homomorphisms from A to B with various topologies and study convergence properties of sequences in H(A,B). We are in particular interested in the situation when B is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on A in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on H(A,B) for a Boolean algebra B carrying a strictly positive measure and convergence properties of sequences of measures on A.",

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AB - For given Boolean algebras A and B we endow the space H(A,B) of all Boolean homomorphisms from A to B with various topologies and study convergence properties of sequences in H(A,B). We are in particular interested in the situation when B is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on A in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on H(A,B) for a Boolean algebra B carrying a strictly positive measure and convergence properties of sequences of measures on A.

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KW - Efimov problem

KW - measure algebras

KW - perfect Hamming codes.

KW - random forcing

KW - weak convergence of measures

KW - perfect Hamming codes

KW - TOPOLOGY

KW - Algebraic convergence

KW - Weak convergence of measures

KW - Measure algebras

KW - Random forcing

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DO - 10.1017/jsl.2021.70

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VL - 88

SP - 191

EP - 218

JO - Journal of Symbolic Logic

JF - Journal of Symbolic Logic

IS - 1

ER -

On sequences of homomorphisms into measure algebras and the Efimov problem

Abstract

Funding

Austrian Fields of Science 2012

Keywords

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