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Algebras from congruences

Abstract

We present a functorial construction which, starting from a congruence \(\alpha \) of finite index in an algebra \(\mathbf {A}\), yields a new algebra \(\mathbf {C}\) with the following properties: the congruence lattice of \(\mathbf {C}\) is isomorphic to the interval of congruences between 0 and \(\alpha \) on \(\mathbf {A}\), this isomorphism preserves higher commutators and TCT types, and \(\mathbf {C}\) inherits all idempotent Maltsev conditions from \(\mathbf {A}\). As applications of this construction, we first show that supernilpotence is decidable for congruences of finite algebras in varieties that omit type \(\mathbf {1}\). Secondly, we prove that the subpower membership problem for finite algebras with a cube term can be effectively reduced to membership questions in subdirect products of subdirectly irreducible algebras with central monoliths. As a consequence, we obtain a polynomial time algorithm for the subpower membership problem for finite algebras with a cube term in which the monolith of every subdirectly irreducible section has a supernilpotent centralizer.

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Acknowledgements

We thank the referees for their constructive comments, especially for bringing to our attention a few additional results in the literature that are related to this paper.

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Correspondence to Peter Mayr.

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This material is based upon work supported by the Austrian Science Fund (FWF) Grant no. P24285, the National Science Foundation Grant no. DMS 1500254, the Hungarian National Foundation for Scientific Research (OTKA) Grants no. K104251 and K115518, and the National Research, Development and Innovation Fund of Hungary (NKFI) Grant no. K128042.

Presented by E.W. Kiss.

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Mayr, P., Szendrei, Á. Algebras from congruences. Algebra Univers. 82, 55 (2021). https://doi.org/10.1007/s00012-021-00740-7

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Keywords

  • Congruence
  • Commutator
  • Maltsev condition
  • Subpower membership
  • Supernilpotence

Mathematics Subject Classification

  • 08C05
  • 18C05
  • 08A30