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.

**1.**C. Bergman, R. McKenzie,*Minimal varieties and quasivarieties*, J. Austral. Math. Soc.**48**(1990), 133-147. MR**91c:08013****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.**D. Hobby and R. McKenzie,*The Structure of Finite Algebras*, Contemporary Mathematics vol. 76, American Mathematical Society, 1988. MR**89m:08001****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.**R. McKenzie, G. McNulty and W. Taylor,*Algebras, Lattices, Varieties Volume 1*, Wadsworth and Brooks/Cole, Monterey, California, 1987. MR**88e:08001****9.**P. P. Pálfy, P. Pudlák,*Congruence lattices of finite algebras and intervals in subgroup lattices of finite groups*, Algebra Universalis**11**(1980), 22-27. MR**82g:08003****10.**D. Scott,*Equationally complete extensions of finite algebras*, Nederl. Akad. Wetensch. Proc. Ser. A**59**(1956), 35-38. MR**18:636c****11.**Á. Szendrei,*A survey on strictly simple algebras and minimal varieties*, in Universal Algebra and Quasigroup Theory, A. Romanowska and J. D. H. Smith (eds.), Heldermann Verlag Berlin, 1992. MR**93e:08001****12.**Á. Szendrei,*Term minimal algebras*, Algebra Universalis**32**(1994), 439-477. MR**95i:08002****13.**Á. Szendrei,*Maximal non-affine reducts of simple affine algebras*, Algebra Universalis**34**(1995), 144-174. MR**96i:08001****14.**Á. Szendrei,*Strongly abelian minimal varieties*, Acta Sci. Math. (Szeged)**59**(1994), 25-42. MR**95g:08002****15.**W. Taylor,*The fine spectrum of a variety*, Algebra Universalis**5**(1975), 263-303. MR**52:10547**

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