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On difunctionality of class relations

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Abstract

For a given variety \({\mathcal {V}}\) of algebras, we define a class relation to be a binary relation \(R\subseteq S^2\) which is of the form \(R=S^2\cap K\) for some congruence class K on \(A^2\), where A is an algebra in \( {\mathcal {V}}\) such that \(S\subseteq A\). In this paper we study the following property of \({\mathcal {V}}\): every reflexive class relation is an equivalence relation. In particular, we obtain equivalent characterizations of this property analogous to well-known equivalent characterizations of congruence-permutable varieties. This property determines a Mal’tsev condition on the variety and in a suitable sense, it is a join of Chajda’s egg-box property as well as Duda’s direct decomposability of congruence classes.

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Acknowledgements

We would like to thank the anonymous referees as well as Heinz-Peter Gumm, for their useful remarks on earlier versions of this paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, Stellenbosch University, Private Bag X1, Matieland, 7602, South Africa

    Michael Hoefnagel & Zurab Janelidze

  2. Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade do Algarve, Campus de Gambelas, 8005-139, Faro, Portugal

    Diana Rodelo

  3. CMUC, Department of Mathematics, University of Coimbra, 3001-501, Coimbra, Portugal

    Diana Rodelo

Authors
  1. Michael Hoefnagel
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  2. Zurab Janelidze
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  3. Diana Rodelo
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Correspondence to Michael Hoefnagel.

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Presented by Emil W. Kiss.

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Second author’s research is partially supported by the South African National Research Foundation. The third author acknowledges partial financial assistance by the Centre for Mathematics of the University of Coimbra—UID/MAT/00324/2019, funded by the Portuguese Government through FCT/MEC and co-funded by the European Regional Development Fund through the Partnership Agreement PT2020.

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Hoefnagel, M., Janelidze, Z. & Rodelo, D. On difunctionality of class relations. Algebra Univers. 81, 19 (2020). https://doi.org/10.1007/s00012-020-00651-z

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  • DOI: https://doi.org/10.1007/s00012-020-00651-z

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