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Substitution Principle and semidirect products

Published online by Cambridge University Press:  15 August 2023

Célia Borlido Open the ORCID record for Célia Borlido [Opens in a new window]  and
Mai Gehrke
Célia Borlido*
Affiliation:
Centre for Mathematics of the University of Coimbra (CMUC), Coimbra, Portugal
Mai Gehrke
Affiliation:
Laboratoire J. A. Dieudonné, CNRS, Université Côte d’Azur, Nice, France
*
Corresponding author: Célia Borlido; Email: cborlido@mat.uc.pt
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Abstract

In the classical theory of regular languages, the concept of recognition by profinite monoids is an important tool. Beyond regularity, Boolean spaces with internal monoids (BiMs) were recently proposed as a generalization. On the other hand, fragments of logic defining regular languages can be studied inductively via the so-called “Substitution Principle.” In this paper, we make the logical underpinnings of this principle explicit and extend it to arbitrary languages using Stone duality. Subsequently, we show how it can be used to obtain topo-algebraic recognizers for classes of languages defined by a wide class of first-order logic fragments. This naturally leads to a notion of semidirect product of BiMs extending the classical such construction for profinite monoids. Our main result is a generalization of Almeida and Weil’s Decomposition Theorem for semidirect products from the profinite setting to that of BiMs. This is a crucial step in a program to extend the profinite methods of regular language theory to the setting of complexity theory.

Keywords

Formal languagelogic on wordsquantifiersubstitutionBoolean space with an internal monoidsemidirect product
Type
Paper
Information
Mathematical Structures in Computer Science , Volume 33 , Issue 6 , June 2023 , pp. 486 - 535
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Copyright
© The Author(s), 2023. Published by Cambridge University Press

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Footnotes

*

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement No.670624). The first-named author was also partially supported by the Centre for Mathematics of the University of Coimbra – UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES.

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