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
Two isomorphism criteria for directed colimits
Using the general notions of finitely presentable and finitely generated object introduced by Gabriel and Ulmer in 1971, we prove that, in any (locally small) category, two sequences of finitely presentable objects and morphisms (or two sequences of finitely generated objects and monomorphisms) have isomorphic colimits (=direct limits) if, and only if, they are confluent. The latter means that the two given sequences can be connected by a back-and-forth chain of morphisms that is cofinal on each side, and commutes with the sequences at each finite stage. In several concrete situations, analogous isomorphism criteria are typically obtained by ad hoc arguments. The abstract results given here can play the useful rôle of discerning the general from the specific in situations of actual interest. We illustrate by applying them to varieties of algebras, on the one hand, and to dimension groups—the ordered $K_{0}$ of approximately finite-dimensional C*-algebras—on the other. The first application encompasses such classical examples as Kurosh’s isomorphism criterion for countable torsion-free Abelian groups of finite rank. The second application yields the Bratteli-Elliott Isomorphism Criterion for dimension groups. Finally, we discuss Bratteli’s original isomorphism criterion for approximately finite-dimensional C*-algebras, and show that his result does not follow from ours.
Two isomorphism criteria for directed colimits
Algebra|Coalgebra seminar
Starting from October 2013 I am organising the Algebra|Coalgebra seminar at the ILLC. The webpage of the seminar are here. See you there!
Course on Capita Selecta in Modal Logic
Starting from the 29th of October I will start teaching a course with Yde Venema at the University of Amsterdam on Algebra and Coalgebra. The webpage of the course can be found here.
ADAMS project will start soon (1st of August 2013 — 31th July 2015)
Starting from the 1st of August 2013, I will be on leave from the University of Salerno. For two years form this date, I will be affiliated at the Institute for Logic, Language, and Computation of the University van Amsterdam.
This opportunity is funded by a Marie Curie scholarship awarded to the research project “ADAMS” (A Dual Approach to Many-valued Semantics).
