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Topological Representation of MV-algebras

Final Report Summary - TOPREPMVALG (Topological Representation of MV-algebras)

Many-valued logics were first considered by J. Lukasiewicz in 1920. MV-algebras were introduced by C.C. Chang in 1958 to prove the completeness theorem for infinite-valued Lukasiewicz logic. In the last 20 years the importance of MV-algebras and Lukasiewicz logic has been increasing, for three main reasons: (i) the discovery of a categorical equivalence between MV-algebras and lattice-ordered Abelian groups with an Archimedean unit; (ii) the deep relations between MV-algebras and polyhedral and toric geometry: suffice to say that the strong Oda conjecture is equivalent to the joint refinability of MV-algebraic bases; (iii) the applications of many-valued logic to the handling of uncertain information, e.g. in the Rényi-Ulam game of Twenty Questions with errors, i.e. Berlekamp's theory of feedback error-correcting coding; remarkably enough, the satisfiability problem of infinite-valued logic is NP-complete, precisely as its two-valued counterpart.

The overall aim of the present project was to achieve a better understanding of Lukasiewicz infinite valued logic and the fine structure of MV-algebras. To this aim we studied and developed topological dualities for MV-algebras, strengthening the already known interrelations between MV-algebras, unital Abelian lattice-ordered groups and rational polyhedra. The project was framed into four specific objectives.
(i) Investigation of the existing representation theories for MV-algebras and their mutual relation-ships.
(ii) Investigation of the topological structure of the lattice of subalgebras of totally ordered MV-algebras.
(iii) Construction of topological dualities for specific classes of MV-algebras.
(iv) Application of representation and duality theorems to various central topics and the following open problems in MV-algebra theory and (simultaneously) proof-theoretic Lukasiewicz logic.

During this period Dr. Cabrer studied the existing topological representations and dualities for various classes of MV-algebras and related classes: Strongly Semisimple MV-algebras, Polyhedral MV-algebras, Strongly Semisimple Riesz Spaces, Germinal MV-algebras, and Semisimple MV-algebras. These dualities were applied to the study of various open problems in the theory of MV-algebras. Among them we highlight: the geometrical characterisation of finitely generated projective MV-algebras; the classification of germinal MV-algebras. These results solve two of the open problems proposed in [D. Mundici, Advanced Lukasiewicz Calculus and MV-algebras, Trends in Logic Vol. 35. Springer (2011), Section 20.3 Problems 5 and 6].

The results obtained during the span of the project have also found applications in other areas of mathematics. The study of automorphisms of free MV-algebras is intrinsically connected with the study the general affine linear group over the integer acting on real spaces. A complete invariant to classify the orbits of affine linear group over the integers was developed using the rational simplicial geometric approach used to study automorphisms of free MV-algebras [L.M. Cabrer and D. Mundici, Classifying orbits of the affine group over the integers. Ergodic Theory and Dynamical Systems. (in press)]. Using MV-algebras as a leading example and following Jerabek's work, a new methodology to measure the complexity of unification problems that we call Exact Unification has been developed. This new perspective applies to general equational classes an not only to MV-algebras.