Finite axiomatizability of congruence rich varieties. (English) Zbl 0840.08010
The notion of a congruence rich variety of algebras is introduced. A variety \(V\) is said to be congruence rich if for each positive integer \(n\) there is a positive integer \(m\) such that every finitely generated algebra in \(V\) with more than \(n\) elements has a homomorphic image with more than \(n\) elements but no more than \(m\) elements. There are examples of varieties of this kind: any finitely generated congruence modular variety and any variety of directoids (introduced in a paper of the first author and R. Quackenbush [Algebra Univers. 27, 49-69 (1990; Zbl 0699.08002)]). The question whether locally finite subvarieties of a congruence rich variety are relatively finitely based is discussed. The results obtained are applied to prove that some interesting five-element directoid is not finitely based but fails to be inherently nonfinitely based.
Reviewer: A.Ryaskin (Novosibirsk)
MSC:
| 08B99 | Varieties |
| 08B10 | Congruence modularity, congruence distributivity |
| 08C10 | Axiomatic model classes |
Keywords:
relatively finitely based variety; congruence rich variety; locally finite subvarieties; directoidCitations:
Zbl 0699.08002References:
| [1] | Baker, K.,Finite equational bases for finite algebras in congruence distributive varieties. Advances in Mathematics24 (1977), 207–243. · Zbl 0356.08006 |
| [2] | Baker, K., Mcnulty, G. andWerner, H.,Shift-automorphism methods for inherently nonfinitely based varieties of algebras. Czechoslovak Math. J.39 (1989), 53–69. · Zbl 0677.08005 |
| [3] | Burris, S. andSankappanavar, H. P.,A Course in Universal Algebra, Springer-Verlag, New York, 1981. · Zbl 0478.08001 |
| [4] | Freese R. andMcKenzie, R.,Residually small varieties with modular congruence lattices. Trans. Amer. Math. Soc.264 (1981), 419–430. · Zbl 0472.08008 · doi:10.1090/S0002-9947-1981-0603772-9 |
| [5] | Grätzer, G.,Universal Algebra, second edition, Springer-Verlag, New York, 1979. |
| [6] | Hajilarov, E.Inherently nonfinitely based commutative directoid. (To appear in Algebra Universalis.) · Zbl 0901.08009 |
| [7] | Isaev, I. M.,Essentially infinitely based varieties of algebras. Sibiriskii Mat. Zhurnal30:6 (1989), 75–77. (English translation: Siberian Math. J. 892–894.) · Zbl 0708.17001 |
| [8] | Ježek, J.,Nonfinitely based three-element groupoids. Algebra Universalis20 (1985), 292–301. · Zbl 0574.08004 · doi:10.1007/BF01195139 |
| [9] | Ježek, J. andQuackenbush, R.,Directoids: algebraic models of up-directed sets. Algebra Universalis27 (1990), 49–69. · Zbl 0699.08002 · doi:10.1007/BF01190253 |
| [10] | Kearnes, K. andWtllard, R.,Inherently nonfinitely based solvable algebras. To appear in the Canadian Math. Bull. · Zbl 0815.08002 |
| [11] | Kruse, R.,Identities satisfied by a finite ring. J. Algebra26 (1973), 298–318. · Zbl 0276.16014 · doi:10.1016/0021-8693(73)90025-2 |
| [12] | Lyndon, R.,Identities in finite algebras. Proc. Amer. Math. Soc.5 (1954), 8–9. · Zbl 0055.02705 · doi:10.1090/S0002-9939-1954-0060482-6 |
| [13] | L’vov, I. V.,Varieties of associative rings, I and II. Algebra i Logika12 (1973), 269–297, and 667–688. |
| [14] | McKenzle, R.,Equational bases for lattice theories. Math. Scand.27 (1970), 24–38. · Zbl 0307.08001 |
| [15] | McKenzle, R.,A new product of algebras and a type reduction theorem. Algebra Universalis18 (1984), 29–69. · Zbl 0543.08005 · doi:10.1007/BF01182247 |
| [16] | McKenzle, R.,Finite equational bases for congruence modular algebras. Algebra Universalis24 (1987), 224–250. · Zbl 0648.08006 · doi:10.1007/BF01195263 |
| [17] | McKenzie, R., Tarski’s finite basis problem is undecidable. To appear in the International Journal of Algebra and Computation. · Zbl 0844.08011 |
| [18] | McNulty, G. F. andShallon, C. R.,Inherently nonfinitely based finite algebras. In:Universal Algebra and Lattice Theory, Lecture Notes in Math.1004. 206–231. R. Freese and O. Garcia, eds., Springer Verlag, New York, 1983. |
| [19] | McKenzte, R., McNulty, G. andTaylor, W.,Algebras, Lattices Varieties, I, Brooks/Cole, Monterey, CA, 1987. |
| [20] | Murskii, V. L.,The existence in three-valued logic of a closed class with finite basis not having a finite complete set of identities. (Russian) Dokl. Akad. Nauk SSSR163 (1965), 815–818. |
| [21] | Oates, S. andPowell, M. B.,Identical relations infinite groups. J. Algebra1 (1965), 11–39. · Zbl 0121.27202 · doi:10.1016/0021-8693(64)90004-3 |
| [22] | Perkins, P.,Basis questions for general algebra. Algebra Universalis19 (1984), 16–23. · Zbl 0545.08008 · doi:10.1007/BF01191487 |
| [23] | Sapir, M.,Inherently nonfinitely based finite semigroups. Mat. Sbornik133 (1987). English translation: Math. USSR Sbornik61 (1988), 155–166. · Zbl 0655.20045 |
| [24] | Shallon, C.,Non-finitely based algebras derived from lattices. Ph. D. dissertation, UCLA, 1979. |
| [25] | Szendrei, Á.,Nonfinitely based finite groupoids generating minimal varieties. Preprint. · Zbl 0803.20041 |
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.
