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PROFINITENESS IN FINITELY GENERATED VARIETIES IS UNDECIDABLE

Published online by Cambridge University Press:  21 December 2018

ANVAR M. NURAKUNOV  and
MICHAŁ M. STRONKOWSKI
ANVAR M. NURAKUNOV
Affiliation:
INSTITUTE OF MATHEMATICS NATIONAL ACADEMY OF SCIENCES OF THE KYRGYZ REPUBLIC PR. CHU 265A, BISHKEK720071, KYRGYZSTANE-mail: [email protected]
MICHAŁ M. STRONKOWSKI
Affiliation:
FACULTY OF MATHEMATICS AND INFORMATION SCIENCES WARSAW UNIVERSITY OF TECHNOLOGY UL. KOSZYKOWA 75, 00-662WARSAW, POLAND E-mail: [email protected]
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Abstract

Profinite algebras are exactly those that are isomorphic to inverse limits of finite algebras. Such algebras are naturally equipped with Boolean topologies. A variety ${\cal V}$ is standard if every Boolean topological algebra with the algebraic reduct in ${\cal V}$ is profinite.

We show that there is no algorithm which takes as input a finite algebra A of a finite type and decide whether the variety $V\left( {\bf{A}} \right)$ generated by A is standard. We also show the undecidability of some related properties. In particular, we solve a problem posed by Clark, Davey, Freese, and Jackson.

We accomplish this by combining two results. The first one is Moore’s theorem saying that there is no algorithm which takes as input a finite algebra A of a finite type and decides whether $V\left( {\bf{A}} \right)$ has definable principal subcongruences. The second is our result saying that possessing definable principal subcongruences yields possessing finitely determined syntactic congruences for varieties. The latter property is known to yield standardness.

Keywords

22A3003D3503C0506E1508B25profinite algebrasstandard varietiesundecidability
Type
Articles
Information
The Journal of Symbolic Logic , Volume 83 , Issue 4 , December 2018 , pp. 1566 - 1578
Copyright
Copyright © The Association for Symbolic Logic 2018 

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