Minimal sets and varieties

Kearnes, Keith; Kiss, Emil; Valeriote, Matthew

doi:10.1090/S0002-9947-98-01594-3

Minimal sets and varieties
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by Keith A. Kearnes, Emil W. Kiss and Matthew A. Valeriote PDF

Trans. Amer. Math. Soc. 350 (1998), 1-41 Request permission

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

Joel Berman and Steven Seif, An approach to tame congruence theory via subtraces, Algebra Universalis 30 (1993), no. 4, 479–520. MR 1240569, DOI 10.1007/BF01195379
Stanley Burris and H. P. Sankappanavar, A course in universal algebra, Graduate Texts in Mathematics, vol. 78, Springer-Verlag, New York-Berlin, 1981. MR 648287, DOI 10.1007/978-1-4613-8130-3
Ralph Freese and Ralph McKenzie, Commutator theory for congruence modular varieties, London Mathematical Society Lecture Note Series, vol. 125, Cambridge University Press, Cambridge, 1987. MR 909290
Steven Givant, Universal Horn classes categorical or free in power, Ann. Math. Logic 15 (1978), no. 1, 1–53. MR 511942, DOI 10.1016/0003-4843(78)90025-6
Steven Givant, A representation theorem for universal Horn classes categorical in power, Ann. Math. Logic 17 (1979), no. 1-2, 91–116. MR 552417, DOI 10.1016/0003-4843(79)90022-6
David Hobby and Ralph McKenzie, The structure of finite algebras, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. MR 958685, DOI 10.1090/conm/076
Keith A. Kearnes, An order-theoretic property of the commutator, Internat. J. Algebra Comput. 3 (1993), no. 4, 491–533. MR 1250248, DOI 10.1142/S0218196793000299
K. Kearnes. Categorical quasivarieties via Morita equivalence. preprint, 1994.
K. Kearnes and Á. Szendrei. A characterization of minimal locally finite varieties. Trans. Amer. Math. Soc. 349:1749–1768, 1977.
E. Kiss. An easy way to minimal algebras. Internat. J. Algebra Comput. 7:55–75, 1977.
Emil W. Kiss and Péter Pröhle, Problems and results in tame congruence theory. A survey of the ’88 Budapest Workshop, Algebra Universalis 29 (1992), no. 2, 151–171. MR 1157431, DOI 10.1007/BF01190604
R. McKenzie. Algebraic version of the general Morita theorem for algebraic varieties. preprint.
Ralph McKenzie, Finite forbidden lattices, Universal algebra and lattice theory (Puebla, 1982) Lecture Notes in Math., vol. 1004, Springer, Berlin, 1983, pp. 176–205. MR 716183, DOI 10.1007/BFb0063438
Ralph McKenzie, Categorical quasivarieties revisited, Algebra Universalis 19 (1984), no. 3, 273–303. MR 779145, DOI 10.1007/BF01201096
E. A. Palyutin, Description of categorical quasivarieties, Algebra i Logika 14 (1975), no. 2, 145–185, 240 (Russian). MR 0398821
Á. Szendrei, Maximal non-affine reducts of simple affine algebras, Algebra Universalis 34 (1995), no. 1, 144–174. MR 1344960, DOI 10.1007/BF01200496
Ágnes Szendrei, Clones in universal algebra, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 99, Presses de l’Université de Montréal, Montreal, QC, 1986. MR 859550
Á. Szendrei, A survey on strictly simple algebras and minimal varieties, Universal algebra and quasigroup theory (Jadwisin, 1989) Res. Exp. Math., vol. 19, Heldermann, Berlin, 1992, pp. 209–239. MR 1191235
Ágnes Szendrei, Strongly abelian minimal varieties, Acta Sci. Math. (Szeged) 59 (1994), no. 1-2, 25–42. MR 1285426
Walter Taylor, The fine spectrum of a variety, Algebra Universalis 5 (1975), no. 2, 263–303. MR 389716, DOI 10.1007/BF02485261