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A characterization of idempotent strong Mal’cev conditions for congruence meet-semidistributivity in locally finite varieties

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Abstract

We prove a characterization of all idempotent, linear, strong Mal’cev conditions in two variables which hold in all locally finite congruence meet-semidistributive varieties. This is an alternative proof to the one previously given by Z. Brady, and has some advantages, some disadvantages, to his approach. Along the way we prove that such a strong Mal’cev condition holds in all locally finite congruence meet-semidistributive varieties iff it is realized in a certain four-element algebra.

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Acknowledgements

The authors thank Z. Brady for suggesting we “just make the full binary composition of p(x, y) with itself” which allowed us to prove the last sentence of Theorems 5.4 and 5.5. The anonymous, but very knowledgeable, referee also contributed a lot in an exceptionally detailed review. Along with several other improvements to our paper, the referee’s proofs of Proposition 4.2 and Corollary 6.2 replaced our inferior ones.

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Authors and Affiliations

  1. Department of Mathematics, ETH Zurich, Zurich, Switzerland

    Nemanja Draganić

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

    Petar Marković, Vlado Uljarević & Samir Zahirović

Authors
  1. Nemanja Draganić
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  2. Petar Marković
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  3. Vlado Uljarević
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  4. Samir Zahirović
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Corresponding author

Correspondence to Petar Marković.

Additional information

To Bjarni Jónsson, who invented congruence meet-semidistributivity, who made an area out of Mal’cev conditions, who taught us all how to write down a pretty proof.

Presented by J.B. Nation.

This article is part of the topical collection “In memory of Bjarni Jónsson” edited by J.B. Nation.

The second, third and fourth authors were supported by the Grant no. 174018 of the Ministry of Education and Science of Serbia.

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Draganić, N., Marković, P., Uljarević, V. et al. A characterization of idempotent strong Mal’cev conditions for congruence meet-semidistributivity in locally finite varieties. Algebra Univers. 79, 53 (2018). https://doi.org/10.1007/s00012-018-0533-9

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  • DOI: https://doi.org/10.1007/s00012-018-0533-9

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