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Joins of subalgebras and normals in 0-regular varieties

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Abstract

In any 0-normal variety (0-regular variety in which {0} is a subalgebra), every congruence class containing 0 is a subalgebra. These “normal subalgebras” of a fixed algebra constitute a lattice, isomorphic to its congruence lattice. We are interested in those 0-normal varieties for which the join of two normal subalgebras in the lattice of normal subalgebras of an algebra equals their join in the lattice of subalgebras, as happens with groups and rings. We characterise this property in terms of a Mal’cev condition, and use examples to show it is strictly stronger than being ideal determined but strictly weaker than being 0-coherent (classically ideal determined) and does not imply congruence permutability.

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Authors and Affiliations

  1. Department of Defence, Locked Bag 5076, Kingston, ACT, 2604, Australia

    Nicholas McConnell

  2. Department of Mathematics, The University of Waikato, Hamilton, New Zealand

    Tim Stokes

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  1. Nicholas McConnell
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  2. Tim Stokes
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Correspondence to Nicholas McConnell.

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Presented by T. Kowalski.

Dedicated to Brian Davey on the occasion of his 65th birthday

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McConnell, N., Stokes, T. Joins of subalgebras and normals in 0-regular varieties. Algebra Univers. 74, 293–304 (2015). https://doi.org/10.1007/s00012-015-0344-1

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  • DOI: https://doi.org/10.1007/s00012-015-0344-1

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