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A finite basis theorem for residually finite, congruence meet-semidistributive varieties

Published online by Cambridge University Press:  12 March 2014

Ross Willard
Ross Willard*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada, E-mail: [email protected]
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Abstract

We derive a Mal'cev condition for congruence meet-semidistributivity and then use it to prove two theorems. Theorem A: if a variety in a finite language is congruence meet-semidistributive and residually less than some finite cardinal, then it is finitely based. Theorem B: there is an algorithm which, given m < ω and a finite algebra in a finite language, determines whether the variety generated by the algebra is congruence meet-semidistributive and residually less than m.

Keywords

equational theoryfinitely basedresidually finitevarietycongruence meet-semidistributive
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 65 , Issue 1 , March 2000 , pp. 187 - 200
Copyright
Copyright © Association for Symbolic Logic 2000

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References

REFERENCES

[1]Baker, K., Finite equational bases for finite algebras in a congruence-distributive equational class, Advances in Mathematics, vol. 24 (1977), pp. 207–243.CrossRefGoogle Scholar
[2]Baker, K. and Wang, Ju, Approximate distributive laws and finite equational bases for finite algebras in congruence-distributive varieties, Algebra Universalis, to appear.Google Scholar
[3]Czédli, G., A characterization of congruence semi-distributivity, Universal algebra and lattice theory (Proceedings of Conference, Puebla, 1982), Springer Lecture Notes, no. 1004, 1983, pp. 104–110.CrossRefGoogle Scholar
[4]Foster, A. and Pixley, A., Semi-categorical algebras II, Mathematische Zeitschrift, vol. 85 (1964), pp. 169–184.CrossRefGoogle Scholar
[5]Hobby, D. and Mckenzie, R., The structure of finite algebras, Contemporary Mathematics, vol. 76 (1988), American Mathematical Society (Providence, RI).Google Scholar
[6]Jónsson, B., Congruence varieties, Appendix 3 in Grätzer, G., Universal Algebra, Second Edition, Springer-Verlag (1979).Google Scholar
[7]Jónsson, B., Algebras whose congruence lattices are distributive, Mathematica Scandinavica, vol. 21 (1967), pp. 110–121.CrossRefGoogle Scholar
[8]Jónsson, B., Congruence distributive varieties, Mathematica Japonica, vol. 42 (1995), pp. 353–401.Google Scholar
[9]Kalicki, J., On comparison of finite algebras, Proceedings of the American Mathematical Society, vol. 3, 1952, pp. 36–40.CrossRefGoogle Scholar
[10]Kearnes, K. and Szendrei, Á., The relationship between two commutators, International Journal of Algebra and Computation, vol. 8 (1998), pp. 497–531.CrossRefGoogle Scholar
[11]Kearnes, K. and Willard, R., Residuallyfinite, congruence meet-semidistributive varieties of finite type have a finite residual bound, Proceedings of the American Mathematical Society, vol. 127 (1999), pp. 2841–2850.CrossRefGoogle Scholar
[12]Lipparini, P., A characterization of varieties with a difference term, II: Neutral = meet semidistributive, Canadian Mathematical Bulletin, vol. 41 (1998), pp. 318–327.CrossRefGoogle Scholar
[13]McKenzie, R., Para-primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties, Algebra Universalis, vol. 8 (1978), pp. 336–348.CrossRefGoogle Scholar
[14]McKenzie, R., Finite equational bases for congruence modular varieties, Algebra Universalis, vol. 24 (1987), pp. 224–250.CrossRefGoogle Scholar
[15]McKenzie, R., Recursive inseparability for residual bounds of finite algebras, 1995, manuscript.Google Scholar
[16]McKenzie, R., The residual bound of a finite algebra is not computable, International Journal of Algebra and Computation, vol. 6 (1996), pp. 29–48.CrossRefGoogle Scholar
[17]McKenzie, R., The residual bounds of finite algebras, International Journal of Algebra and Computation, vol. 6 (1996), pp. 1–28.CrossRefGoogle Scholar
[18]McKenzie, R., Residual smallness relativized to congruence types, 1996, manuscript.Google Scholar
[19]McKenzie, R., Tarski's finite basis problem is undecidable, International Journal of Algebra and Computation, vol. 6 (1996), pp. 49–104.CrossRefGoogle Scholar
[20]McNulty, G. and Willard, R., Three-element algebras behaving badly, in preparation.Google Scholar
[21]Park, R., Equational classes of non-associative ordered algebras, Ph.D. dissertation, UCLA, 1976.Google Scholar
[22]Taylor, W., Residually small varieties, Algebra Universalis, vol. 2 (1972), pp. 33–53.CrossRefGoogle Scholar
[23]Willard, R., Tarski's finite basis problem via A(J), Transactions of the American Mathematical Society, vol. 349 (1997), pp. 2755–2774.CrossRefGoogle Scholar