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
Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras
We prove that the unification type of Lukasiewicz infinite-valued propositional logic and of its equivalent algebraic semantics, the variety of MV-algebras,is nullary. The proof rests upon Ghilardi’s algebraic characterisation of unification types in terms of projective objects, recent progress by Cabrer and Mundici in the investigation of projective MV-algebras, the categorical duality between finitely presented MV-algebras and rational polyhedra, and, finally, a homotopy-theoretic argument that exploits lifts of continuous maps to the universal covering space of the circle. We discuss the background to such diverse tools. In particular, we offer a detailed proof of the duality theorem for finitely presented MV-algebras and rational polyhedra – a fundamental result that, albeit known to specialists, seems to appear in print here for the first time.
Duality, projectivity, and unification in Łukasiewicz logic and MV-algebras
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
A discrete representation of free MV-algebras
We prove that the $n$-generated free MV-algebra is isomorphic to a quotient of the disjoint union of all the $n$-generated free MV$^{(n)}$-algebras. Such a quotient can be seen as the direct limit of a system consisting of all free MV${}^{(n)}$-algebras and special maps between them as morphisms.
A discrete representation of free MV-algebras
Consequence of compactness in Lukasiewicz first order logic
The Los-Tarski Theorem and the Chang-Los-Susko Theorem, two classical results in Model Theory, are extended to the infinite-valued Lukasiewicz logic. The latter is used to settle a characterisation of the class of generic structures introduced in the framework of model theoretic forcing for Lukasiewicz logic .
Consequence of compactness in Lukasiewicz first order logic
