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
Representation of MV-algebras by regular ultrapowers of [0,1]
We present a uniform version of Di Nola Theorem, this enables to embed all MV-algebras of a bounded cardinality in an algebra of functions with values in a single non-standard ultrapower of the real interval [0,1]. This result also implies the existence, for any cardinal \alpha, of a single MV-algebra in which all infinite MV-algebras of cardinality at most $\alpha$ embed. Recasting the above construction with iterated ultrapowers, we show how to construct such an algebra of values in a definable way, thus providing a sort of “canonical” set of values for the functional representation.
Representation of MV-algebras by regular ultrapowers of [0,1]
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
Advances in the theory of muLPi algebras
Recently an expansion of \rm L {\rm\Pi\frac{1}{2}} logic with fixed points has been considered. In the present work we study the algebraic semantics of this logic, namely \mu L\Pi algebras, from algebraic, model theoretic and computational standpoints.
We provide a characterisation of free \mu L\Pi algebras as a family of particular functions from [0,1]^{n} to [0,1]. We show that the first-order theory of linearly ordered \mu L\Pi-algebras enjoys quantifier elimination, being, more precisely, the model completion of the theory of linearly ordered \rm L {\rm\Pi\frac{1}{2}} algebras. Furthermore, we give a functional representation of any \rm L {\rm\Pi\frac{1}{2}} algebra in the style of Di Nola Theorem for MV-algebras and finally we prove that the equational theory of \mu L\Pi algebras is in PSPACE.
Advances in the theory of muLPi 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
