Tarski's finite basis problem via

Author:
Ross Willard

Journal:
Trans. Amer. Math. Soc. **349** (1997), 2755-2774

MSC (1991):
Primary 03C05; Secondary 08B05

DOI:
https://doi.org/10.1090/S0002-9947-97-01807-2

MathSciNet review:
1389791

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Abstract | References | Similar Articles | Additional Information

Abstract: R. McKenzie has recently associated to each Turing machine a finite algebra having some remarkable properties. We add to the list of properties, by proving that the equational theory of is finitely axiomatizable if halts on the empty input. This completes an alternate (and simpler) proof of McKenzie's negative answer to A. Tarski's finite basis problem. It also removes the possibility, raised by McKenzie, of using to answer an old question of B. Jónsson.

**1.**Kirby A. Baker,*Finite equational bases for finite algebras in a congruence-distributive equational class*, Advances in Math.**24**(1977), no. 3, 207–243. MR**447074**, https://doi.org/10.1016/0001-8708(77)90056-1**2.**David Hobby and Ralph McKenzie,*The structure of finite algebras*, Contemporary Mathematics, vol. 76, American Mathematical Society, Providence, RI, 1988. MR**958685****3.**K. Kearnes and Á. Szendrei, Self-rectangulating varieties of type 5, manuscript, 1995.**4.**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**, https://doi.org/10.1007/BF01190604**5.**Ralph McKenzie,*Para primal varieties: A study of finite axiomatizability and definable principal congruences in locally finite varieties*, Algebra Universalis**8**(1978), no. 3, 336–348. MR**469853**, https://doi.org/10.1007/BF02485404**6.**-, The residual bounds of finite algebras,*Int. J. Algebra and Computation***6**(1996), 1-28. CMP**96:07****7.**-, The residual bound of a finite algebra is not computable,*Int. J. Algebra and Computation***6**(1996), 29-48. CMP**96:07****8.**-, Tarski's finite basis problem is undecidable,*Int. J. Algebra and Computation***6**(1996), 49-104. CMP**96:07****9.**Peter Perkins,*Unsolvable problems for equational theories*, Notre Dame J. Formal Logic**8**(1967), 175–185. MR**236012****10.**A. Tarski,*Equational logic and equational theories of algebras*, Contributions to Math. Logic (Colloquium, Hannover, 1966) North-Holland, Amsterdam, 1968, pp. 275–288. MR**0237410****11.**Walter Taylor,*Equational logic*, Houston J. Math.**Surve, Survey**(1979), iii+83. MR**546853****12.**R. Willard, On McKenzie's method,*Periodica Math. Hungarica***32**(1996), 149-165. CMP**97:01**

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Additional Information

**Ross Willard**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada

Email:
rdwillar@flynn.uwaterloo.ca

DOI:
https://doi.org/10.1090/S0002-9947-97-01807-2

Keywords:
Finite algebra,
equational theory,
finitely axiomatizable

Received by editor(s):
October 18, 1995

Additional Notes:
This research was supported by the NSERC of Canada

Article copyright:
© Copyright 1997
American Mathematical Society