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Polymorphism clones of homogeneous structures: gate coverings and automatic homeomorphicity

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Abstract

Every clone of functions comes naturally equipped with a topology, the topology of pointwise convergence. A clone \(\mathfrak {C}\) is said to have automatic homeomorphicity with respect to a class \(\mathcal {K}\) of clones, if every clone isomorphism of \(\mathfrak {C}\) to a member of \(\mathcal {K}\) is already a homeomorphism (with respect to the topology of pointwise convergence). In this paper we study automatic homeomorphicity properties for polymorphism clones of countable homogeneous relational structures. Besides two generic criteria for the automatic homeomorphicity of the polymorphism clones of homogeneous structures we show that the polymorphism clone of the generic poset with strict ordering has automatic homeomorphicity with respect to the class of polymorphism clones of countable \(\omega \)-categorical structures. Our results extend and generalize previous results by Bodirsky, Pinsker, and Pongrácz.

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Authors and Affiliations

  1. Radebeul, Germany

    Christian Pech

  2. Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia

    Maja Pech

Authors
  1. Christian Pech
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  2. Maja Pech
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Correspondence to Maja Pech.

Additional information

Presented by Á. Szendrei.

C. Pech received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement no. 257039).

M. Pech was supported by the Ministry of Education and Science of the Republic of Serbia through Grant no. 174018.

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Pech, C., Pech, M. Polymorphism clones of homogeneous structures: gate coverings and automatic homeomorphicity. Algebra Univers. 79, 35 (2018). https://doi.org/10.1007/s00012-018-0504-1

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