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Quasiorder lattices in congruence modular varieties

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Abstract

Some elements of tame congruence theory can be applied to quasiorder lattices instead of congruence lattices. In particular, it is possible to consider minimal sets of an algebra with respect to one of its prime quasiorder quotients. It turns out that if a finite algebra is in a congruence modular variety, then it is minimal with respect to a quasiorder quotient iff it is minimal with respect to a congruence quotient — in which case it is either a two-element algebra, or has a Mal’tsev-polynomial. As an application of this fact, we prove that if an algebra is in a congruence modular variety, its congruence and quasiorder lattices satisfy the same identities.

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Authors and Affiliations

  1. Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, 6720, Szeged, Hungary

    Gergő Gyenizse

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  1. Gergő Gyenizse
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Correspondence to Gergő Gyenizse.

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Communicated by L. Zádori

The author’s research was partially supported by the Hungarian National Foundation for Scientific Research (OTKA) grant no. K115518.

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Gyenizse, G. Quasiorder lattices in congruence modular varieties. ActaSci.Math. 86, 3–10 (2020). https://doi.org/10.14232/actasm-018-024-4

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  • DOI: https://doi.org/10.14232/actasm-018-024-4

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