Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

Nick Bezhanishvili; Mai Gehrke

doi:10.2168/LMCS-7(2:9)2011

Nick Bezhanishvili ; Mai Gehrke - Finitely generated free Heyting algebras via Birkhoff duality and coalgebra

lmcs:702 - Logical Methods in Computer Science, May 17, 2011, Volume 7, Issue 2 - https://doi.org/10.2168/LMCS-7(2:9)2011

Finitely generated free Heyting algebras via Birkhoff duality and coalgebraArticle

Authors: Nick Bezhanishvili ; Mai Gehrke

Algebras axiomatized entirely by rank 1 axioms are algebras for a functor and thus the free algebras can be obtained by a direct limit process. Dually, the final coalgebras can be obtained by an inverse limit process. In order to explore the limits of this method we look at Heyting algebras which have mixed rank 0-1 axiomatizations. We will see that Heyting algebras are special in that they are almost rank 1 axiomatized and can be handled by a slight variant of the rank 1 coalgebraic methods.

https://doi.org/10.2168/LMCS-7(2:9)2011

Source: arXiv.org:1102.2828

Volume: Volume 7, Issue 2

Secondary volumes: Selected Papers of the 3rd Conference on Algebra and Coalgebra in Computer Science (CALCO 2009)

Published on: May 17, 2011

Imported on: October 5, 2010

Keywords: Computer Science - Logic in Computer Science, Mathematics - Logic, F.4.1

Licence: arXiv.org - Non-exclusive license to distribute

Funding:

Source : OpenAIRE Graph

Order-topological and model-theoretic methods for modal logics; Funder: UK Research and Innovation; Code: EP/F032102/1
Relational Semantics for Substructural and Other Logics; Funder: UK Research and Innovation; Code: EP/E029329/1

Classifications

Mathematics Subject Classification 2020¹

06D20 - Heyting algebras (lattice-theoretic aspects)

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[1] zbMATH Open.

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Review available on zbMATH Open, by Florentina Chirteş (Craiova) (English)

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Bibliographic References

14 Documents citing this article

Mai Gehrke;Tomáš Jakl;Luca Reggio, 2023, A Cook’s Tour of Duality in Logic: From Quantifiers, Through Vietoris, to Measures, arXiv (Cornell University), pp. 129-158, 10.1007/978-3-031-24117-8_4, http://arxiv.org/abs/2007.15415.

José Luis Castiglioni;Víctor Fernández;Héctor Federico Mallea;Hernán Javier San Martín, 2022, On a variety of hemi-implicative semilattices, Soft Computing, 26, 7, pp. 3187-3195, 10.1007/s00500-022-06807-4.

Nick Bezhanishvili;Tommaso Moraschini, 2022, Hereditarily Structurally Complete Intermediate Logics: Citkin’s Theorem Via Duality, Studia Logica, 111, 2, pp. 147-186, 10.1007/s11225-022-10012-7, https://doi.org/10.1007/s11225-022-10012-7.

Ramon Jansana;Hernán Javier San Martín, 2018, On Principal Congruences in Distributive Lattices with a Commutative Monoidal Operation and an Implication, Studia Logica, 107, 2, pp. 351-374, 10.1007/s11225-018-9796-6, https://doi.org/10.1007/s11225-018-9796-6.

José Luis Castiglioni;Sergio Arturo Celani;Hernán Javier San Martín, 2017, Kleene algebras with implication, El Servicio de Difusión de la Creación Intelectual (National University of La Plata), 77, 4, pp. 375-393, 10.1007/s00012-017-0433-4, https://link.springer.com/article/10.1007/s00012-017-0433-4.

Hernán Javier San Martín, 2016, Principal congruences in weak Heyting algebras, Algebra Universalis, 75, 4, pp. 405-418, 10.1007/s00012-016-0381-4.

Mai Gehrke, 2016, Duality in Computer Science, Zenodo (CERN European Organization for Nuclear Research), pp. 12-26, 10.1145/2933575.2934575, https://zenodo.org/record/3494879.

Nick Bezhanishvili;Silvio Ghilardi;Frederik Möllerström Lauridsen, 2016, One-step Heyting Algebras and Hypersequent Calculi with the Bounded Proof Property, Journal of Logic and Computation, pp. exw029, 10.1093/logcom/exw029.

Nick Bezhanishvili;Dion Coumans;Samuel J. van Gool;Dick de Jongh, 2015, Duality and Universal Models for the Meet-Implication Fragment of IPC, arXiv (Cornell University), pp. 97-116, 10.1007/978-3-662-46906-4_7, http://arxiv.org/abs/1403.0710.

Sergio A. Celani;Hernán J. San Martín, 2015, Frontal operators in distributive lattices with a generalized implication, Asian-European Journal of Mathematics, 08, 03, pp. 1550039, 10.1142/s1793557115500394.

Mai Gehrke, 2014, Canonical Extensions, Esakia Spaces, and Universal Models, Outstanding contributions to logic, pp. 9-41, 10.1007/978-94-017-8860-1_2.

Nick Bezhanishvili;Silvio Ghilardi;Mamuka Jibladze, 2014, Free Modal Algebras Revisited: The Step-by-Step Method, UvA-DARE (University of Amsterdam), pp. 43-62, 10.1007/978-94-017-8860-1_3, https://handle.uba.uva.nl/personal/pure/en/publications/free-modal-algebras-revisited-the-stepbystep-method(7e4b5da8-651a-4906-bd3e-48731481ad44).html.

Sergio A. Celani;Ramon Jansana, 2014, Easkia Duality and Its Extensions, Outstanding contributions to logic, pp. 63-98, 10.1007/978-94-017-8860-1_4.

D. C. S. Coumans;S. J. van Gool, 2012, On generalizing free algebras for a functor, Journal of Logic and Computation, 23, 3, pp. 645-672, 10.1093/logcom/exs016.

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