Luca Carai
Università degli Studi di Milano
Free Heyting algebras via duality
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Free Heyting algebras play a fundamental role in the study of intuitionistic logic, as they arise as Lindenbaum–Tarski algebras of the intuitionistic propositional calculus. Their fascinating structure is both rich and intricate, and has been extensively investigated, yet they remain in part mysterious. In this tutorial, we will explore how Esakia duality, an extension of Stone duality to Heyting algebras, sheds some light on the structure of free Heyting algebras. Along the way, we will also see how Stone and Priestley dualities can be used to describe free Boolean algebras and free distributive lattices. The tutorial will also serve as a gentle introduction to Priestley and Esakia dualities. Show slides

Brett McLean
Universiteit Gent
Duality for algebras of functions
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Functions constitute a fundamental and ubiquitous concept in the formal sciences. We can investigate functions algebraically in much the same way as Boole taught us to investigate propositions algebraically. And just like there are dualities that apply to algebras of propositions (e.g. Stone duality), there are dualities that apply to algebras of functions. I will present two such dualities: a duality for algebras of partial bijections [1] and a duality for algebras of partial functions [2], along with the necessary background on the algebraic study of functions.
[1] Mark V. Lawson, A noncommutative generalization of Stone duality, Journal of the Australian Mathematical Society 88 (2010), no. 3, 385–404.
[2] Brett McLean, A categorical duality for algebras of partial functions, Journal of Pure and Applied Algebra 225 (2021), no. 11, 106755.

Sam van Gool
Université Paris Cité
Duality for automata and regular languages
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The theory of regular languages and automata is a computer science topic with a rich theory having close connections to logic, and with a wide and still growing range of practical and theoretical applications. One of the main strengths of this theory is the availability of mathematical tools from algebra and topology, in particular, finite and profinite monoids. In this tutorial, which is based on Chapter 8 of the book [1], we will see the rudiments of this theory from a duality point of view.
[1] M. Gehrke and S. v. Gool, Topological duality for distributive lattices: Theory and applications, Cambridge University Press (2024), 296pp.

Schedule for Both Days (16–17 June 2025)

• 10:15–11:00 – Luca Carai
• 11:00–11:30 – Coffee Break
• 11:30–12:15 – Luca Carai
• 12:15–13:30 – Lunch
• 13:30–14:15 – Brett McLean
• 14:15–14:25 – Short Break
• 14:25–15:10 – Brett McLean
• 15:10–15:40 – Coffee Break
• 15:40–16:25 – Sam van Gool
• 16:25–16:30 – Short Break
• 16:30–17:15 – Sam van Gool

The workshop is free of charge and open to all interested participants. No registration is required to attend the workshop.

Immediately after the workshop, the conference AAA107 – Workshop on General Algebra will take place in Bern from 20–22 June 2025. In addition, two mini-courses (18–20 June 2025) — on Constraint Satisfaction Problems and Quantum Computing — as well as an application session will be offered. For more information, please consult their website.