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On polymorphism-homogeneous relational structures and their clones

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Abstract

A relational structure is homomorphism-homogeneous if every homomorphism between finite substructures extends to an endomorphism of the structure. This notion was introduced recently by Cameron and Nešetřil. In this paper, we consider a strengthening of homomorphism-homogeneity–we call a relational structure polymorphism-homogeneous if every partial polymorphism with a finite domain extends to a global polymorphism of the structure. It turns out that this notion (under various names and in completely different contexts) has existed in algebraic literature for at least 30 years. Motivated by this observation, we dedicate this paper to the topic of polymorphism-homogeneous structures. We study polymorphism-homogeneity from a model-theoretic, an algebraic, and a combinatorial point of view. E.g., we study structures that have quantifier elimination for positive primitive formulae, and show that this notion is equivalent to polymorphism-homogeneity for weakly oligomorphic structures. We demonstrate how the Baker-Pixley theorem can be used to show that polymorphism-homogeneity is a decidable property for finite relational structures. Finally, we completely characterize the countable polymorphism-homogeneous graphs, the polymorphism-homogeneous posets of arbitrary size, and the countable polymorphism- homogeneous strict posets.

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  1. Maja Pech

    Present address: Institute of Algebra, Technische Universität Dresden, Dresden, Germany

Authors and Affiliations

  1. Institute of Algebra, Technische Universität Dresden, Dresden, Germany

    Christian Pech

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

    Maja Pech

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  1. Christian Pech
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  2. Maja Pech
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Presented by K. Kaarli.

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Pech, C., Pech, M. On polymorphism-homogeneous relational structures and their clones. Algebra Univers. 73, 53–85 (2015). https://doi.org/10.1007/s00012-014-0310-3

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  • DOI: https://doi.org/10.1007/s00012-014-0310-3

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