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On a stronger reconstruction notion for monoids and clones

Mike Behrisch ORCID logo and Edith Vargas-García ORCID logo
From the journal Forum Mathematicum

Abstract

Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which entails automatic homeomorphicity. We further give a characterization of automatic homeomorphicity for transformation monoids on arbitrary carriers with a dense group of invertibles having automatic homeomorphicity. We then show how to lift automatic action compatibility from groups to monoids and from monoids to clones under fairly weak assumptions. We finally employ these theorems to get automatic action compatibility results for monoids and clones over several well-known countable structures, including the strictly ordered rationals, the directed and undirected version of the random graph, the random tournament and bipartite graph, the generic strictly ordered set, and the directed and undirected versions of the universal homogeneous Henson graphs.


Communicated by Manfred Droste


Funding source: OeAD-GmbH

Award Identifier / Grant number: CZ 02/2019

Funding statement: The research of the first author was partly supported by the OeAD KONTAKT project CZ 02/2019 “Function algebras and ordered structures related to logic and data fusion”. The second author gratefully acknowledges financial support by the Asociación Mexicana de Cultura A.C.

Acknowledgements

The authors are highly grateful to Christian Pech for enlightening discussions on the subject and many valuable comments. In particular, Christian Pech observed that the argument given in Corollary 5.2 was already sufficient to cover four more examples that were initially listed as open. This in turn has led to significant simplification and streamlining of several proofs in Section 5. Moreover, Christian Pech pointed out that (,betw) can be dealt with using the results from [2]. This initiated the work on Lemmas 5.7 and 5.9 concerning (,circ) and (,sep). The authors are also indebted to John Truss for helpful remarks, some clarifications regarding the four reducts of the rationals treated in the article, and for moral support of their work. Furthermore, the first named author thanks Manuel Bodirsky for pointing him to the work of Paolini and Shelah. Last but not least, both authors wish to thank the anonymous referee for his elaborate report including many comments and remarks that led to an improvement of the paper.

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Received: 2020-08-03
Revised: 2021-07-26
Published Online: 2021-09-25

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