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Subalgebras of FA-presentable algebras

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Abstract

Automatic presentations, also called FA-presentations, were introduced to extend finite model theory to infinite structures whilst retaining the solubility of fundamental decision problems. This paper studies FA-presentable algebras. First, an example is given to show that the class of finitely generated FA-presentable algebras is not closed under forming finitely generated subalgebras, even within the class of algebras with only unary operations. In contrast, a finitely generated subalgebra of an FA-presentable algebra with a single unary operation is itself FA-presentable. Furthermore, it is proven that the class of unary FA-presentable algebras is closed under forming finitely generated subalgebras and that the membership problem for such subalgebras is decidable.

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Authors and Affiliations

  1. Centro de Matemática, Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre 687, 4169–007, Porto, Portugal

    Alan J. Cain

  2. School of Mathematics and Statistics, University of St Andrews, North Haugh, St Andrews, Fife, KY16 9SS, United Kingdom

    Nik Ruškuc

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  1. Alan J. Cain
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  2. Nik Ruškuc
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Correspondence to Alan J. Cain.

Additional information

Presented by K. Kearnes.

The first author’s research was funded by the European Regional Development Fund through the programme COMPETE and by the Portuguese Government through the FCT (Fundação para a Ciência e a Tecnologia) under the project PEst-C/MAT/UI0144/2011 and through an FCT Ciência 2008 fellowship. Two visits by the first author to the University of St Andrews, where much of the research described in this paper was carried out, were supported by the EPSRC-funded project EP/H011978/1 ‘Automata, Languages, Decidability in Algebra’. The authors thank Richard M. Thomas for helpful discussions.

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Cain, A.J., Ruškuc, N. Subalgebras of FA-presentable algebras. Algebra Univers. 72, 101–123 (2014). https://doi.org/10.1007/s00012-014-0293-0

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  • DOI: https://doi.org/10.1007/s00012-014-0293-0

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