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Cartesian closed varieties II: links to algebra and self-similarity

Part of: Varieties Selfadjoint operator algebras Finite-dimensional Hamiltonian, Lagrangian, contact, and nonholonomic systems Semigroups

Published online by Cambridge University Press:  14 May 2025

Richard Garner Open the ORCID record for Richard Garner [Opens in a new window]
Richard Garner*
Affiliation:
School of Mathematical & Physical Sciences, Macquarie University, NSW 2109, Australia (richard.garner@mq.edu.au) (corresponding author)
*
*(Corresponding author)
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Abstract

This article is the second in a series investigating cartesian closed varieties. In first of these, we showed that every non-degenerate finitary cartesian variety is a variety of sets equipped with an action by a Boolean algebra B and a monoid M which interact to form what we call a matched pair ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$. In this article, we show that such pairs ${\left[\smash{{B} \mathbin{\mid}{M} }\right]}$ are equivalent to Boolean restriction monoids and also to ample source-étale topological categories; these are generalizations of the Boolean inverse monoids and ample étale topological groupoids used to encode self-similar structures such as Cuntz and Cuntz–Krieger $C^\ast$-algebras, Leavitt path algebras, and the $C^\ast$-algebras associated with self-similar group actions. We explain and illustrate these links and begin the programme of understanding how topological and algebraic properties of such groupoids can be understood from the logical perspective of the associated varieties.

Keywords

cartesian closed varietiesBoolean inverse monoidsself-similar groupsCuntz-Krieger algebrasLeavitt path algebras

MSC classification

Primary: 37J40: Perturbations, normal forms, small divisors, KAM theory, Arnol'd diffusion
Secondary: 20M50: Connections of semigroups with homological algebra and category theory 20M18: Inverse semigroups 08B25: Products, amalgamated products, and other kinds of limits and colimits 46L89: Other “noncommutative” mathematics based on $C^*$-algebra theory

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 45
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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