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)

 

 

A characterization
of minimal locally finite varieties


Authors: Keith A. Kearnes and Ágnes Szendrei
Journal: Trans. Amer. Math. Soc. 349 (1997), 1749-1768
MSC (1991): Primary 08B15; Secondary 08B30
DOI: https://doi.org/10.1090/S0002-9947-97-01883-7
MathSciNet review: 1407494
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we describe a one-variable Mal'cev-like condition satisfied by any locally finite minimal variety. We prove that a locally finite variety is minimal if and only if it satisfies this Mal'cev-like condition and it is generated by a strictly simple algebra which is nonabelian or has a trivial subalgebra. Our arguments show that the strictly simple generator of a minimal locally finite variety is unique, it is projective and it embeds into every member of the variety. We give a new proof of the structure theorem for strictly simple abelian algebras that generate minimal varieties.


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

  • 1. Clifford Bergman and Ralph McKenzie, Minimal varieties and quasivarieties, J. Austral. Math. Soc. Ser. A 48 (1990), no. 1, 133–147. MR 1026844
  • 2. R. Freese and R. McKenzie, Commutator Theory for Congruence Modular Varieties, LMS Lecture Notes vol. 125, Cambridge University Press, 1987. MR 89c:0006
  • 3. David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. MR 958685
  • 4. K. A. Kearnes, A Hamiltonian property for nilpotent algebras, to appear in Algebra Universalis.
  • 5. K. A. Kearnes, E. W. Kiss and M. A. Valeriote, Minimal sets and varieties, to appear in Trans. Amer. Math. Soc.
  • 6. K. A. Kearnes, E. W. Kiss and M. A. Valeriote, Residually small varieties generated by simple algebras, preprint.
  • 7. R. McKenzie, An algebraic version of categorical equivalence for varieties and more general algebraic categories, in Logic and Algebra (Proceedings of the Magari Memorial Conference, Siena, Italy, April 1994), Marcel Dekker, New York, 1996. CMP 96:17
  • 8. Ralph N. McKenzie, George F. McNulty, and Walter F. Taylor, Algebras, lattices, varieties. Vol. I, The Wadsworth & Brooks/Cole Mathematics Series, Wadsworth & Brooks/Cole Advanced Books & Software, Monterey, CA, 1987. MR 883644
  • 9. Péter Pál Pálfy and Pavel Pudlák, Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups, Algebra Universalis 11 (1980), no. 1, 22–27. MR 593011, https://doi.org/10.1007/BF02483080
  • 10. D. Scott, Equationally complete extensions of finite algebras, Nederl. Akad. Wetensch. Proc. Ser. A 59 (1956), 35-38. MR 18:636c
  • 11. A. Romanowska and J. D. H. Smith (eds.), Universal algebra and quasigroup theory, Research and Exposition in Mathematics, vol. 19, Heldermann Verlag, Berlin, 1992. Papers from the Conference on Universal Algebra, Quasigroups and Related Systems held in Jadwisin, May 23–28, 1989. MR 1191225
  • 12. Á. Szendrei, Term minimal algebras, Algebra Universalis 32 (1994), no. 4, 439–477. MR 1300482, https://doi.org/10.1007/BF01195723
  • 13. Á. Szendrei, Maximal non-affine reducts of simple affine algebras, Algebra Universalis 34 (1995), no. 1, 144–174. MR 1344960, https://doi.org/10.1007/BF01200496
  • 14. Ágnes Szendrei, Strongly abelian minimal varieties, Acta Sci. Math. (Szeged) 59 (1994), no. 1-2, 25–42. MR 1285426
  • 15. Walter Taylor, The fine spectrum of a variety, Algebra Universalis 5 (1975), no. 2, 263–303. MR 0389716, https://doi.org/10.1007/BF02485261

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 08B15, 08B30

Retrieve articles in all journals with MSC (1991): 08B15, 08B30


Additional Information

Keith A. Kearnes
Affiliation: Department of Mathematical Sciences, University of Arkansas, Fayetteville, Arkansas 72701
Email: kearnes@comp.uark.edu

Ágnes Szendrei
Affiliation: Bolyai Institute, Aradi vértanúk tere 1, H–6720 Szeged, Hungary
Email: a.szendrei@sol.cc.u-szeged.hu

DOI: https://doi.org/10.1090/S0002-9947-97-01883-7
Received by editor(s): August 7, 1994
Additional Notes: Research of the first author supported by a fellowship from the Alexander von Humboldt Stiftung
Research of the second author supported by a fellowship from the Alexander von Humboldt Stiftung and by the Hungarian National Foundation for Scientific Research grant no. 1903.
Article copyright: © Copyright 1997 American Mathematical Society