On the equivalence between MV-algebras and ll-groups with strong unit

In "A new proof of the completeness of the Lukasiewicz axioms"} (Transactions of the American Mathematical Society, 88) C.C. Chang proved that any totally ordered $MV$-algebra $A$ was isomorphic to the segment $A \cong \Gamma(A^*, u)$ of a totally ordered $l$-group with strong unit $A^*$. This was done by the simple intuitive idea of putting denumerable copies of $A$ on top of each other (indexed by the integers). Moreover, he also show that any such group $G$ can be recovered from its segment since $G \cong \Gamma(G, u)^*$, establishing an equivalence of categories. In "Interpretation of AF $C^*$-algebras in Lukasiewicz sentential calculus" (J. Funct. Anal. Vol. 65) D. Mundici extended this result to arbitrary $MV$-algebras and $l$-groups with strong unit. He takes the representation of $A$ as a sub-direct product of chains $A_i$, and observes that $A \overset {} {\hookrightarrow} \prod_i G_i$ where $G_i = A_i^*$. Then he let $A^*$ be the $l$-subgroup generated by $A$ inside $\prod_i G_i$. He proves that this idea works, and establish an equivalence of categories in a rather elaborate way by means of his concept of good sequences and its complicated arithmetics. In this note, essentially self-contained except for Chang's result, we give a simple proof of this equivalence taking advantage directly of the arithmetics of the the product $l$-group $\prod_i G_i$, avoiding entirely the notion of good sequence.Comment: 6 page

Lattice-ordered abelian groups and perfect MV-algebras: a topos-theoretic perspective

We establish, generalizing Di Nola and Lettieri's categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-intepretability holding for particular classes of formulas: irreducible formulas, geometric sentences and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, we obtain various results on its syntax and semantics also in relation to the cartesian theory of the variety generated by Chang's MV-algebra, including a concrete representation for the finitely presentable models of the latter theory as finite products of finitely presentable perfect MV-algebras. Among the results established on the way, we mention a Morita-equivalence between the theory of lattice-ordered abelian groups and that of cancellative lattice-ordered abelian monoids with bottom element.Comment: 54 page

Notes on divisible MV-algebras

In these notes we study the class of divisible MV-algebras inside the algebraic hierarchy of MV-algebras with product. We connect divisible MV-algebras with $\mathbb Q$-vector lattices, we present the divisible hull as a categorical adjunction and we prove a duality between finitely presented algebras and rational polyhedra

Representation of Perfect and Local MV-algebras

We describe representation theorems for local and perfect MV-algebras in terms of ultraproducts involving the unit interval [0,1]. Furthermore, we give a representation of local Abelian lattice-ordered groups with strong unit as quasi-constant functions on an ultraproduct of the reals. All the above theorems are proved to have a uniform version, depending only on the cardinality of the algebra to be embedded, as well as a definable construction in ZFC. The paper contains both known and new results and provides a complete overview of representation theorems for such classes

Semiring and semimodule issues in MV-algebras

In this paper we propose a semiring-theoretic approach to MV-algebras based on the connection between such algebras and idempotent semirings - such an approach naturally imposing the introduction and study of a suitable corresponding class of semimodules, called MV-semimodules. We present several results addressed toward a semiring theory for MV-algebras. In particular we show a representation of MV-algebras as a subsemiring of the endomorphism semiring of a semilattice, the construction of the Grothendieck group of a semiring and its functorial nature, and the effect of Mundici categorical equivalence between MV-algebras and lattice-ordered Abelian groups with a distinguished strong order unit upon the relationship between MV-semimodules and semimodules over idempotent semifields.Comment: This version contains some corrections to some results at the end of Section

Lukasiewicz logic and Riesz spaces

We initiate a deep study of {\em Riesz MV-algebras} which are MV-algebras endowed with a scalar multiplication with scalars from $[0,1]$. Extending Mundici's equivalence between MV-algebras and $\ell$-groups, we prove that Riesz MV-algebras are categorically equivalent with unit intervals in Riesz spaces with strong unit. Moreover, the subclass of norm-complete Riesz MV-algebras is equivalent with the class of commutative unital C$^*$-algebras. The propositional calculus ${\mathbb R}{\cal L}$ that has Riesz MV-algebras as models is a conservative extension of \L ukasiewicz $\infty$-valued propositional calculus and it is complete with respect to evaluations in the standard model $[0,1]$. We prove a normal form theorem for this logic, extending McNaughton theorem for \L ukasiewicz logic. We define the notions of quasi-linear combination and quasi-linear span for formulas in ${\mathbb R}{\cal L}$ and we relate them with the analogue of de Finetti's coherence criterion for ${\mathbb R}{\cal L}$.Comment: To appear in Soft Computin