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.

Dualities and geometry

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 spacesStudia 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