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Dualities via ComPactness, Openness, and their Symmetry

Lead Research Organisation: UNIVERSITY OF BIRMINGHAM
Department Name: School of Computer Science

Abstract

This project is positioned inside the study of Stone-like dualities and touches on themes in algebraic logic, domain theory and
pointfree topology.
The interplay between compact sets and open sets is a key tool in topology, and various conditions and constructions (such as local
compactness, the compact-open topology on function spaces, well-filtered spaces) hint at a symmetrical role between these two
notions. Indeed, a precise symmetry occurs in the spaces used by M.H. Stone to represent Boolean algebras (1936) and distributive
lattices (1938). However, in some important generalisations of Stone dualities (such as the duality between sober spaces and spatial
frames), the symmetry between compactness and openness is only partial.
This project aims at demonstrating that the aforementioned partial symmetry is in fact a slice of a formal, complete symmetry in a
larger framework. This framework will provide new explanations for the occurrence of certain conditions and constructions (such as
local compactness, the compact-open topology, sobriety), will suggest new results (such as new dualities), and will allow good
structural properties (such as Cartesian closure).

People

ORCID iD

Publications

10 25 50
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ABBADINI M (2024) DUALITY FOR COALGEBRAS FOR VIETORIS AND MONADICITY in The Journal of Symbolic Logic

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Abbadini M (2025) Stone-Gelfand duality for metrically complete lattice-ordered groups in Advances in Mathematics

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Abbadini M (2025) Quantifier-free formulas and quantifier alternation depth in doctrines in Journal of Pure and Applied Algebra

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Abbadini M (2025) Varieties of MV-monoids and positive MV-algebras in Journal of Algebra

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ABBADINI M (2025) THE UNIFICATION TYPE OF LUKASIEWICZ LOGIC WITH A BOUNDED NUMBER OF VARIABLES in The Journal of Symbolic Logic

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Marco Abbadini (2025) Barr-coexactness for metric compact Hausdorff spaces in Theory and Applications of Categorie

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Marco Abbadini (2024) MacNeille completions of subordination algebras in Cahiers de Topologie et Géométrie Différentielle Catégoriques

 
Description The main research outcome of the project so far is the development of a formal framework that clarifies and completes the symmetry between compactness and openness that underlies many Stone-type dualities.

Classical Stone duality exhibits a precise symmetry between algebraic and topological structures. In several important generalisations of Stone duality, however, the symmetry between compactness and openness appears only partially: certain constructions suggest a dual role for compact and open sets, but this symmetry is not fully visible within the traditional setting.

The preprint "On the symmetry behind duality" (arXiv:2507.18245) demonstrates that this apparent "partial symmetry" can be understood as a slice of a more general and formally complete symmetry within a larger categorical framework. The paper identifies the structural principles responsible for this symmetry and shows how they unify different duality results that were previously studied separately.

This result offers a unifying perspective on why conditions such as local compactness and well-filteredness naturally arise in duality theory.
Exploitation Route The research outcomes can be used by other researchers in the field.
Sectors Education

 
Description During the UKRI-funded project DCPOS (EP/Y015029/1), Marco Abbadini and Achim Jung advanced a research programme on Stone-like dualities by developing a clearer, more systematic understanding of the interaction between compactness and openness in duality theory. A central outcome is the preprint "On the symmetry behind duality" (arXiv:2507.18245), which proposes a broader formal setting in which the partial compact/open symmetry present in some Stone-like dualities is made complete and explicit, while still retaining the sober-spatial-frame duality as a special case. Beyond the main research direction, the project period supported substantial professional development and cross-disciplinary training. By working at the University of Birmingham and through local activities and courses, Marco Abbadini broadened his technical toolkit at the interface of logic and theoretical computer science, including exposure to domain theory, denotational semantics, proof theory, intuitionistic logic, type theory, and functional programming. The project also generated community-facing research training. Marco Abbadini delivered a 3-hour introductory mini-course on Stone duality (October 2024) and two seminars for PhD students and postdoctoral researchers in Computer Science (February 2025)-"The sequent calculus whose semantics admits the empty structure" and "The algebras of classical first-order logic"-helping early-career researchers engage with algebraic and duality-theoretic perspectives on logic. In addition, Achim Jung delivered a mini-course (organised by Marco Abbadini) on denotational semantics and domain theory. Results and ideas developed during the award were disseminated through a sustained programme of talks across logic, topology, and category theory venues, including invited presentations as well as contributions to workshops and seminar series. This dissemination supported exchanges with several research communities and helped consolidate collaborations relevant to the project's themes. Finally, the professional network and visibility developed during the award period supported Marco Abbadini's subsequent career step to UCLouvain, Belgium, where he took up an FSR Incoming Postdoctoral Fellowship.
Sector Education
Impact Types Cultural

 
Description Collaboration with Dr. Adam Prenosil 
Organisation University of Barcelona
Country Spain 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact Preprint: Duality for finitely valued algebras. M. Abbadini, A. Prenosil. https://arxiv.org/abs/2505.11490
Start Year 2022
 
Description Collaboration with Dr. Francesca Guffanti 
Organisation University of Luxembourg
Country Luxembourg 
Sector Academic/University 
PI Contribution Francesca Guffanti and I worked together to incorporate the notion of quantifier-free formulas modulo a first-order theory and the stratification of formulas by quantifier alternation depth modulo a first-order theory into the algebraic treatment of classical first-order logic. This has resulted in a paper and a preprint.
Collaborator Contribution Francesca Guffanti and I worked together to incorporate the notion of quantifier-free formulas modulo a first-order theory and the stratification of formulas by quantifier alternation depth modulo a first-order theory into the algebraic treatment of classical first-order logic. This has resulted in a paper and a preprint.
Impact 1) Journal paper. M. Abbadini, F. Guffanti. Quantifier-free formulas and quantifier alternation depth in doctrines. Journal of Pure and Applied Algebra, 229(8):108004 (2025). 2) Preprint. https://arxiv.org/abs/2410.16328 - Marco Abbadini, Francesca Guffanti. Freely adding one layer of quantifiers to a Boolean doctrine. preprint
Start Year 2021
 
Description Collaboration with Guram Bezhanishvili and Luca Carai 
Organisation New Mexico State University
Country United States 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact M. Abbadini, G. Bezhanishvili, L. Carai. MacNeille completions of subordination algebras. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 65(2):151-199 (2024).
Start Year 2021
 
Description Collaboration with Guram Bezhanishvili and Luca Carai 
Organisation University of Milan
Country Italy 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact M. Abbadini, G. Bezhanishvili, L. Carai. MacNeille completions of subordination algebras. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 65(2):151-199 (2024).
Start Year 2021
 
Description Collaboration with Paolo Agliano' and Stevano Fioravanti 
Organisation Charles University
Country Czech Republic 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact Journal paper: M. Abbadini, S. Fioravanti, P. Aglianò. Varieties of MV-monoids and positive MV-algebras. (Article). Journal of Algebra, 677:690-744 (2025). https://doi.org/10.1016/j.jalgebra.2025.04.027
Start Year 2023
 
Description Collaboration with Paolo Agliano' and Stevano Fioravanti 
Organisation University of Siena
Country Italy 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact Journal paper: M. Abbadini, S. Fioravanti, P. Aglianò. Varieties of MV-monoids and positive MV-algebras. (Article). Journal of Algebra, 677:690-744 (2025). https://doi.org/10.1016/j.jalgebra.2025.04.027
Start Year 2023
 
Description Collaboration with Prof. Daniele Mundici 
Organisation University of Florence
Country Italy 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact M. Abbadini, D. Mundici, Unital Specker l-groups and Boolean multispaces, https://arxiv.org/pdf/2508.21500, 2025. (submitted)
Start Year 2025
 
Description Collaboration with Prof. Dirk Hofmann 
Organisation University of Aveiro
Country Portugal 
Sector Academic/University 
PI Contribution Dirk Hofmann and I have obtained results in the study of compact metric spaces. We published this result: M. Abbadini, D. Hofmann. Barr-coexactness for metric compact Hausdorff spaces. Theory and Applications of Categories, 44(6):196-226 (2025).
Collaborator Contribution Dirk Hofmann and I have obtained a new result in the study of compact metric spaces. We published this result: M. Abbadini, D. Hofmann. Barr-coexactness for metric compact Hausdorff spaces. Theory and Applications of Categories, 44(6):196-226 (2025).
Impact M. Abbadini, D. Hofmann. Barr-coexactness for metric compact Hausdorff spaces. Theory and Applications of Categories, 44(6):196-226 (2025).
Start Year 2023
 
Description Collaboration with Prof. Luca Spada 
Organisation University of Salerno
Country Italy 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact M. Abbadini, L. Spada. The unification type of Lukasiewicz logic with a bounded number of variables. The Journal of Symbolic Logic, accepted. DOI: 10.1017/jsl.2025.10149
Start Year 2020
 
Description Collaboration with Vincenzo Marra and Luca Spada 
Organisation University of Milan
Country Italy 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact M. Abbadini, V. Marra, L. Spada. Stone-Gelfand duality for metrically complete lattice-ordered groups. Advances in Mathematics, 461:110067 (2025).
Start Year 2018
 
Description Collaboration with Vincenzo Marra and Luca Spada 
Organisation University of Salerno
Country Italy 
Sector Academic/University 
PI Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Collaborator Contribution In this collaboration, all authors contributed equally to the development of the main ideas and results. Through joint discussions and careful analysis, we refined the key arguments, formulated proofs, and structured the exposition. Our collective effort was essential in shaping the mathematical framework and ensuring the coherence of the final presentation.
Impact M. Abbadini, V. Marra, L. Spada. Stone-Gelfand duality for metrically complete lattice-ordered groups. Advances in Mathematics, 461:110067 (2025).
Start Year 2018
 
Description 3 hours minicourse on Stone duality to PhD students and postdocs at the University of Birmingham. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Postgraduate students
Results and Impact This was a 2-lecture 3-hour course offered to PhD students and Postdocs at the School of Computer Science at the University of Birmingham. The topic was Stone duality, which is a tool in mathematical logic.
Year(s) Of Engagement Activity 2024
 
Description Contributed talk at conference "Logic, Algebra and Truth Degrees 2023". 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I have delivered the talk "Free extensions" to present my research.
Year(s) Of Engagement Activity 2023
URL https://sites.google.com/view/latd-2023/
 
Description Invited Talk to the Special Session on Algebraic Logic of the 2024 Association for Symbolic Logic North American Annual Meeting 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I have given the talk titled "A duality for metrically complete lattice-ordered groups".
Year(s) Of Engagement Activity 2024
 
Description Invited Talks at National Workshop "4th Southern and Midlands Logic Seminar" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact I was invited to deliver the talk "Natural dualities" at the "4th Southern and Midlands Logic Seminar"
Year(s) Of Engagement Activity 2023
URL https://www.anupamdas.com/smls4/
 
Description Lightning talk at the autumn school ``Proof and Computation''. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Postgraduate students
Results and Impact I gave a 5-minute talk on some of my research, to present it, titled "Algebras of logic, step-by-step". At the Autumn school ``Proof and Computation''. Fischbachau, Germany. (19 Sept 2024)
Year(s) Of Engagement Activity 2024
 
Description Seminar at Catholic University of Louvain 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I have delivered the talk "Coalgebraic flavour of metric compact Hausdorff spaces", in which I presented a part of my research. I received many questions and I discussed possible developments of the research topic in the subsequent days of my visit to Universite Catholique de Louvain.
Year(s) Of Engagement Activity 2025
 
Description Seminar at the University of Birmingham 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I have presented part of my research with a seminar titled "An abstraction of the unit interval with Euclidean topology and denominators".
Year(s) Of Engagement Activity 2023
 
Description Seminar at the University of Milan, Italy 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I have presented part of my research with a talk titled "Quantifier-free fragments and quantifier alternation depth in doctrines".
Year(s) Of Engagement Activity 2024
 
Description Seminar at the University of Padua 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I have presented part of my research in a talk titled "Quantifier alternation depth in universal Boolean doctrines."
Year(s) Of Engagement Activity 2024
 
Description Seminar at the Université catholique de Louvain, Louvain-la-Neuve, Belgium 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I have presented part of my research with a talk titled "Soft sheaf representations in Barr-exact categories."
Year(s) Of Engagement Activity 2023
 
Description Seminar for Researchers at the University of Birmingham 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I have given a 1-hour seminar over an important topic I am passionate about in the algebraic treatment of classical first-order logic to PostDocs and Staff at the University of Birmingham. This resulted in a lively discussion in the meeting and afterwards, a request to meet again, meeting again, and one of the attendees is writing a document about an open problem mentioned in the seminar.
Year(s) Of Engagement Activity 2025
 
Description Seminar the the University of Manchester 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Professional Practitioners
Results and Impact I have presented part of my research with a talk titled "Quantifier-free fragments and quantifier alternation depth in doctrines".
Year(s) Of Engagement Activity 2024
 
Description Seminars for students at the University of Birmingham 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Postgraduate students
Results and Impact I have given a seminar titled "The sequent calculus whose semantics admits the empty structure". This resulted in a lively discussion during the seminar, which continued in subsequent meetings.
Year(s) Of Engagement Activity 2025
 
Description Talk at "104th Workshop on General Algebra "Arbeitstagung Allgemeine Algebra". 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I delivered the talk "Natural dualities with an infinite dualizing object", whose purpose was to present part of my research.
Year(s) Of Engagement Activity 2024
 
Description Talk at "108th Peripatetic Seminar on Sheaves and Logic" 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I have delivered the talk "Soft sheaf representations in Barr-exact categories", in which I presented part of my research. As an outcome, I have received many questions, and a request from Prof. Marino Gran (Catholic University of Louvain, Belgium) to present the same topic at his Institute. In turn, this led to plans for further research, and in particular we wrote a research proposal together, which then succeeded, resulting in an offer for a 3-year postdoc at the Catholic University of Louvain.
Year(s) Of Engagement Activity 2023
 
Description Talk at "Modalities in Substructural Logics: Theory Methods and Applications" Workshop 2023. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I delivered the talk "Stone duality for finitely valued algebras with a near-unanimity term", in which I presented my research.
Year(s) Of Engagement Activity 2023
URL https://sites.google.com/view/mosaic2023/
 
Description Talk at "Topology, Algebra, and Categories in Logic 2024." 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I delivered the talk "Vietoris endofunctor for closed relations and its de Vries dual", whose purpose was to present a part of my research.
Year(s) Of Engagement Activity 2024
URL https://iiia.csic.es/tacl2024/
 
Description Talk at 38th summer conference on topology and its applications. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Professional Practitioners
Results and Impact I delivered the talk "Vietoris endofunctor for closed relations and its de Vries dual", whose purpose was to report my research.
Year(s) Of Engagement Activity 2024
URL https://www.mat.uc.pt/~sumtopo/
 
Description Talk at XXVIII Logic Meeting of the Italian Association of Logic and its Applications. 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach National
Primary Audience Professional Practitioners
Results and Impact I delivered the talk "Vietoris endofunctor for closed relations and its de Vries dual", whose purpose was to report my research
Year(s) Of Engagement Activity 2024
URL https://ailameeting24.uniud.it/