Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Minimal sets and varieties

Authors: Keith A. Kearnes, Emil W. Kiss and Matthew A. Valeriote
Journal: Trans. Amer. Math. Soc. 350 (1998), 1-41
MSC (1991): Primary 08A05; Secondary 08A40, 08B15
MathSciNet review: 1348152
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The aim of this paper is twofold. First some machinery is established to reveal the structure of abelian congruences. Then we describe all minimal, locally finite, locally solvable varieties. For locally solvable varieties, this solves problems 9 and 10 of Hobby and McKenzie. We generalize part of this result by proving that all locally finite varieties generated by nilpotent algebras that have a trivial locally strongly solvable subvariety are congruence permutable.

References [Enhancements On Off] (What's this?)

  • 1. J. Berman and S. Seif. An approach to tame congruence theory via subtraces. Algebra Universalis, 30:479-520, 1993. MR 94f:08003
  • 2. S. Burris and H.P. Sankappanavar. A Course in Universal Algebra. Springer-Verlag, 1981. MR 83k:08001
  • 3. R. Freese and R. McKenzie. Commutator Theory for Congruence Modular Varieties, volume 125 of London Mathematical Society Lecture Note Series. Cambridge University Press, 1987. MR 89c:08006
  • 4. S. Givant. Universal Horn classes categorical or free in power. Ann. Math. Logic, 15:1-53, 1978. MR 80c:03032
  • 5. S. Givant. A representation theorem for universal Horn classes categorical in power. Ann. Math. Logic, 17:91-116, 1979. MR 81b:03038
  • 6. D. Hobby and R. McKenzie. The Structure of Finite Algebras, volume 76 of Contemporary Mathematics. American Mathematical Society, 1988. MR 89m:08001
  • 7. K. Kearnes. An order-theoretic property of the commutator. International Journal of Algebra and Computation, 3:491-534, 1993. MR 95c:08002
  • 8. K. Kearnes. Categorical quasivarieties via Morita equivalence. preprint, 1994.
  • 9. K. Kearnes and Á. Szendrei. A characterization of minimal locally finite varieties. Trans. Amer. Math. Soc. 349:1749-1768, 1977. CMP 97:09
  • 10. E. Kiss. An easy way to minimal algebras. Internat. J. Algebra Comput. 7:55-75, 1977. CMP 97:06
  • 11. E. Kiss and P. Pröhle. Problems and results in tame congruence theory. Algebra Universalis, 29:151-171, 1992. MR 93g:08004
  • 12. R. McKenzie. Algebraic version of the general Morita theorem for algebraic varieties. preprint.
  • 13. R. McKenzie. Finite forbidden lattices. In Universal Algebra and Lattice Theory, volume 1004 of Springer Lecture Notes. Springer-Verlag, 1983, pp. 176-205. MR 85b:06006
  • 14. R. McKenzie. Categorical quasivarieties revisited. Algebra Universalis, 19:273-303, 1984. MR 87g:08022
  • 15. E. Palyutin. The description of categorical quasivarieties. Algebra and Logic, 14:86-111, 1975. MR 53:2672
  • 16. Á. Szendrei. Maximal non-affine reducts of simple affine algebras. Algebra Universalis. 34:144-174, 1995. MR 96i:08001
  • 17. Á. Szendrei. Clones in Universal Algebra, volume 99 of Séminaire de Mathématiques Superieures. Les Presses de l'Université de Montréal, 1986. MR 87m:08005
  • 18. Á. Szendrei. A survey on strictly simple algebras and minimal varieties. In A. Romanowska and J. D. H. Smith, editors, Universal Algebra and Quasigroup Theory. Heldermann Verlag, Berlin, 1992, pp. 209-239. MR 93h:08001
  • 19. Á. Szendrei. Strongly abelian minimal varieties. Acta Sci. Math. (Szeged), 59:25-42, 1994. MR 95g:08002
  • 20. W. Taylor. The fine spectrum of a variety. Algebra Universalis, 5:262-303, 1975. MR 52:10547

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 08A05, 08A40, 08B15

Retrieve articles in all journals with MSC (1991): 08A05, 08A40, 08B15

Additional Information

Keith A. Kearnes
Affiliation: Department of Mathematics, University of Louisville, Louisville, Kentucky 40292

Emil W. Kiss
Affiliation: Department of Algebra and Number Theory, Eötvös Lóránd University, 1088 Budapest, Múzeum krt. 6–8, Hungary

Matthew A. Valeriote
Affiliation: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, Canada, L8S 4K1

Received by editor(s): October 14, 1994
Received by editor(s) in revised form: August 18, 1995
Additional Notes: This research was partially supported by a fellowship from the Alexander von Humboldt Stiftung (to the first author), by the Hungarian National Foundation for Scientific Research, grant no. 1903 (to the second author), and by the NSERC of Canada (third author)
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society