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Tarski's finite basis problem via $ \mathbf {A}({\mathcal T})$


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: R. McKenzie has recently associated to each Turing machine ${\mathcal T}$ a finite algebra $\mathbf {A} ({\mathcal T})$ having some remarkable properties. We add to the list of properties, by proving that the equational theory of $\mathbf {A}({\mathcal T})$ is finitely axiomatizable if ${\mathcal T}$ 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 $\mathbf {A}({\mathcal T})$ to answer an old question of B. Jónsson.


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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