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The shape of congruence lattices

About this Title

Keith A. Kearnes, Department of Mathematics, University of Colorado, Boulder, Colorado 80309-0395 and Emil W. Kiss, Loránd Eötvös University, Department of Algebra and Number Theory, 1117 Budapest, Pázmány Péter sétány 1/c, Hungary

Publication: Memoirs of the American Mathematical Society
Publication Year: 2013; Volume 222, Number 1046
ISBNs: 978-0-8218-8323-5 (print); 978-0-8218-9515-3 (online)
Published electronically: September 18, 2012
Keywords:Abelian, almost congruence distributivity, commutator theory, compatible semilattice operation, congruence identity, congruence modularity, 8congruence semidistributivity, Maltsev condition, meet continuous lattice, rectangulation, residual smallness, solvable, tame congruence theory, term condition, variety, weak difference term
MSC: Primary 08B05; Secondary 08B10

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Preliminary notions
  • Chapter 3. Strong term conditions
  • Chapter 4. Meet continuous congruence identities
  • Chapter 5. Rectangulation
  • Chapter 6. A theory of solvability
  • Chapter 7. Ordinary congruence identities
  • Chapter 8. Congruence meet and join semidistributivity
  • Chapter 9. Residually small varieties
  • Problems
  • Appendix A. Varieties with special terms


We develop the theories of the strong commutator, the rectangular commutator, the strong rectangular commutator, as well as a solvability theory for the nonmodular TC commutator. These theories are used to show that each of the following sets of statements are equivalent for a variety  $\mathcal {V}$ of algebras.
  1. [(I)]
    1. $\mathcal {V}$ satisfies a nontrivial congruence identity.
    2. $\mathcal {V}$ satisfies an idempotent Maltsev condition that fails in the variety of semilattices.
    3. The rectangular commutator is trivial throughout  $\mathcal {V}$.
  2. [(II)]
    1. $\mathcal {V}$ satisfies a nontrivial meet continuous congruence identity.
    2. $\mathcal {V}$ satisfies an idempotent Maltsev condition that fails in the variety of sets.
    3. The strong commutator is trivial throughout  $\mathcal {V}$.
    4. The strong rectangular commutator is trivial throughout  $\mathcal {V}$.
  3. [(III)]
    1. $\mathcal {V}$ is congruence semidistributive.
    2. $\mathcal {V}$ satisfies an idempotent Maltsev condition that fails in the variety of semilattices and in any nontrivial variety of modules.
    3. The rectangular and TC commutators are both trivial throughout  $\mathcal {V}$.
We prove that a residually small variety that satisfies a congruence identity is congruence modular.

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