Tutorial on Dualities
These are the slides of my tutorial on Dualities at the $16^{th}$ Latin American Symposium on Mathematical Logic. 28th July – 1st August 2014. Buenos Aires, Argentina. A shorter version can be found here.
Slides on Duality (SLALM 2014)
Dualities and geometry
Finally I wrote some slides about the long-waiting article I am writing together with Olivia Caramello and Vincenzo Marra on adjunctions, dualities, and Nullstellensätze . These slides where presented at the AILA meeting in Pisa and at the Apllied Logic seminar in Delft.
Geometrical dualities for Łukasiewicz logic
This is the transcript of a featured talk given on the 15th of September 2011 at the XIX Congeresso dell’Unione Matematica Italiana held in Bologna, Italy. It is based on a joint work with Vincenzo Marra of the University of Milan that was published in Vincenzo Marra and Luca Spada. The dual adjunction between MV-algebras and Tychonoff spaces, Studia Logica 100(1-2):253-278, 2012. Special issue of Studia Logica in memoriam Leo Esakia (L. Beklemishev, G. Bezhanishvili, D. Mundici and Y. Venema Editors).
The article develops a general dual adjunction between MV-algebras (the algebraic equivalents of Łukasiewicz logic) and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. Such a dual adjunction restricts to a duality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. Further the duality theorem for finitely presented objects is obtained from the general adjunction by a further specialisation. The treatment is aimed at emphasising the generality of the framework considered here in the prototypical case of MV-algebras.
Geometrical dualities for Łukasiewicz logic
The dual adjunction between MV-algebras and Tychonoff spaces
We offer a proof of the duality theorem for finitely presented MV-algebras and rational polyhedra, a folklore and yet fundamental result. Our approach develops first a general dual adjunction between MV-algebras and subspaces of Tychonoff cubes, endowed with the transformations that are definable in the language of MV-algebras. We then show that this dual adjunction restricts to aduality between semisimple MV-algebras and closed subspaces of Tychonoff cubes. The duality theorem for finitely presented objects is obtained by a further specialisation. Our treatment is aimed at showing exactly which parts of the basic theory of MV-algebras are needed in order to establish these results, with an eye towards future generalisations.
The dual adjunction between MV-algebras and Tychonoff spaces
