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THE UNDECIDABILITY OF THE DEFINABILITY OF PRINCIPAL SUBCONGRUENCES

Published online by Cambridge University Press:  22 April 2015

MATTHEW MOORE
MATTHEW MOORE*
Affiliation:
VANDERBILT UNIVERSITY NASHVILLE, TN 37240, USAE-mail:matthew.moore@vanderbilt.edu
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Abstract

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For each Turing machine ${\cal T}$, we construct an algebra $\mathbb{A}$′$\left( {\cal T} \right)$ such that the variety generated by $\mathbb{A}$′$\left( {\cal T} \right)$ has definable principal subcongruences if and only if ${\cal T}$ halts, thus proving that the property of having definable principal subcongruences is undecidable for a finite algebra. A consequence of this is that there is no algorithm that takes as input a finite algebra and decides whether that algebra is finitely based.

Keywords

definable principal subcongruencesundecidablefinite basisfinite residual bound
Type
Articles
Information
The Journal of Symbolic Logic , Volume 80 , Issue 2 , June 2015 , pp. 384 - 432
Copyright
Copyright © The Association for Symbolic Logic 2015 

References

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