A characterization of minimal locally finite varieties. (English) Zbl 0871.08005
Summary: We describe a one-variable Mal’tsev-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’tsev-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.
MSC:
| 08B15 | Lattices of varieties |
| 08B30 | Injectives, projectives |
| 08B05 | Equational logic, Mal’tsev conditions |
Keywords:
minimal variety; locally finite variety; strictly simple algebra; strictly simple generator; projective; abelian algebra; structure theorem; Mal’tsev conditionReferences:
| [1] | Clifford Bergman and Ralph McKenzie, Minimal varieties and quasivarieties, J. Austral. Math. Soc. Ser. A 48 (1990), no. 1, 133 – 147. · Zbl 0692.08011 |
| [2] | R. Freese and R. McKenzie, Commutator Theory for Congruence Modular Varieties, LMS Lecture Notes vol. 125, Cambridge University Press, 1987. · Zbl 0636.08001 |
| [3] | David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. · Zbl 0721.08001 |
| [4] | K. A. Kearnes, A Hamiltonian property for nilpotent algebras, to appear in Algebra Universalis. · Zbl 0903.08001 |
| [5] | K. A. Kearnes, E. W. Kiss and M. A. Valeriote, Minimal sets and varieties, to appear in Trans. Amer. Math. Soc. · Zbl 0889.08011 |
| [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 · Zbl 0859.08006 |
| [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. · Zbl 0611.08001 |
| [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. · Zbl 0386.06002 · doi:10.1007/BF02483080 |
| [10] | D. Scott, Equationally complete extensions of finite algebras, Nederl. Akad. Wetensch. Proc. Ser. A 59 (1956), 35-38. · Zbl 0073.24602 |
| [11] | A. Romanowska and J. D. H. Smith , 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. |
| [12] | Á. Szendrei, Term minimal algebras, Algebra Universalis 32 (1994), no. 4, 439 – 477. · Zbl 0812.08001 · doi:10.1007/BF01195723 |
| [13] | Á. Szendrei, Maximal non-affine reducts of simple affine algebras, Algebra Universalis 34 (1995), no. 1, 144 – 174. · Zbl 0836.08002 · doi:10.1007/BF01200496 |
| [14] | Ágnes Szendrei, Strongly abelian minimal varieties, Acta Sci. Math. (Szeged) 59 (1994), no. 1-2, 25 – 42. · Zbl 0817.08008 |
| [15] | Walter Taylor, The fine spectrum of a variety, Algebra Universalis 5 (1975), no. 2, 263 – 303. · Zbl 0336.08004 · doi:10.1007/BF02485261 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
