Finite equational bases for congruence modular varieties. (English) Zbl 0648.08006
In the paper the finite axiomatizability of a congruence modular variety \({\mathcal V}\) of a finite type is studied. Two theorems are obtained as principal results. One is that \({\mathcal V}\) has a finite equational basis provided that \({\mathcal V}\) is generated by a finite algebra and is residually small. This result is a generalization of K. A. Baker’s finite basis theorem. As the other one it is proved that every finite algebra in \({\mathcal V}\) belongs to a finitely axiomatizable locally finite variety provided that a certain relation defined by using commutators of principal congruences is first order definable uniformly in all algebras of \({\mathcal V}\). This hypothesis is satisfied by finite groups and rings.
Reviewer: H.Yutani
MSC:
| 08B10 | Congruence modularity, congruence distributivity |
Keywords:
finite axiomatizability; congruence modular variety; finite equational basis; residually small; locally finite variety; commutatorsReferences:
| [1] | K. A. Baker Finite equational bases for finite algebras in a congruence-distributive equational class, Advances in Math.24 (1977), 207-243. · Zbl 0356.08006 |
| [2] | K. A.Baker, G. F.McNulty, and H.Werner Shift-automorphism methods for inherently nonfinitely based varieties of algebras [manuscript]. · Zbl 0677.08005 |
| [3] | R.Bryant The laws of finite pointed groups, Bull. London Math. Soc. [to appear]. · Zbl 0451.08006 |
| [4] | S.Burris and H. P.Sankappanavar A COURSE IN UNIVERSAL ALGEBRA, Springer-Verlag, Graduate Texts in Mathematics v. 78. |
| [5] | R. Freese andR. McKenzie 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 |
| [6] | R.Freese and R.McKenzie Commutator theory for congruence modular varieties, London Math. Soc. Lecture Note series, vol. 125, 1987. · Zbl 0636.08001 |
| [7] | H. P. Gumm An easy way to the commutator in modular varieties, Archiv der Math. (Basel)34 (1980), 220-228. · Zbl 0438.08004 |
| [8] | H. P. Gumm Congruence modularity is permutability composed with distributivity, Archiv der Math. (Basel)36 (1981), 569-576. · Zbl 0465.08005 |
| [9] | H. P.Gumm Geometrical methods in congruence modular algebras, Memoirs of the Amer. Math. Soc.286 (1983). |
| [10] | D.Hobby and R.McKenzie The Structure of Finite Algebras (Tame congruence theory), Amer. Math. Soc. Contemporary Mathematics series [to appear]. · Zbl 0721.08001 |
| [11] | R. McKenzie Nilpotent and solvable radicals in locally finite congruence modular varieties, Algebra Universal24 (1987) 251-266. · Zbl 0648.08007 · doi:10.1007/BF01195264 |
| [12] | G. F. McNulty How to construct finite algebras which are not finitely based in Universal Algebra and Lattice Theory, S. D. Comer Ed., Lecture Notes in Mathematics, vol. 1149, Springer-Verlag, New York (1985) 167-174. |
| [13] | S. Oates-MacDonald andM. Vaughan-Lee Varieties that make one Cross, J. Austral. Math. Soc. (Series A)26 (1978), 368-382. · Zbl 0393.17001 · doi:10.1017/S1446788700011897 |
| [14] | S. V. Polin Identities of finite algebras, Siberian Math. J.17 (1976), 992-999. · Zbl 0404.17001 · doi:10.1007/BF00968027 |
| [15] | W. Taylor Baker’s finite basis theorem, Algebra Universalis8 (1978), 191-196. · Zbl 0377.08004 · doi:10.1007/BF02485388 |
| [16] | M. R. Vaughan-Lee Nilpotence in permutable varieties, in Universal Algebra and Lattice Theory, R. Freese and O. Garcia, eds., Lectures Notes in Mathematics, vol. 1004, Springer-Verlag, New York (1983) 293-308. |
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.
