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

An analysis of the logic of Riesz Spaces with strong unit

We study \L ukasiewicz logic enriched with a scalar multiplication with scalars taken in $[0,1]$. Its algebraic models, called {\em Riesz MV-algebras}, are, up to isomorphism, unit intervals of Riesz spaces with a strong unit endowed with an appropriate structure. When only rational scalars are considered, one gets the class of {\em DMV-algebras} and a corresponding logical system. Our research follows two objectives. The first one is to deepen the connections between functional analysis and the logic of Riesz MV-algebras. The second one is to study the finitely presented MV-algebras, DMV-algebras and Riesz MV-algebras, connecting them from logical, algebraic and geometric perspective

Towards understanding the Pierce-Birkhoff conjecture via MV-algebras

Our main issue was to understand the connection between \L ukasiewicz logic with product and the Pierce-Birkhoff conjecture, and to express it in a mathematical way. To do this we define the class of \textit{f}MV-algebras, which are MV-algebras endowed with both an internal binary product and a scalar product with scalars from $[0,1]$. The proper quasi-variety generated by $[0,1]$, with both products interpreted as the real product, provides the desired framework: the normal form theorem of its corresponding logical system can be seen as a local version of the Pierce-Birkhoff conjecture

de Finetti's coherence and exchangeability in infinitary logic

We continue the investigation towards a logic-based approach to statistics within the infinitary conservative extension of Łukasiewicz logic IRL and prove versions of de Finetti's theorems on coherence and exchangeability. In particular we will prove a coherence criterion for a subclass of the variety of σ-complete Riesz MV-algebras in the conditional and unconditional case, and discuss de Finetti's exchangeability in a special case

A General View on Normal Form Theorems for Łukasiewicz Logic with Product

In this survey paper we explore the connection between the Pierce- Birkhoff conjecture and Lukasiewicz logic with product. Conservative extensions of Lukasiewicz logic can be defined by adding an internal product or a multiplication with scalars from [0; 1]. The corresponding models reflect an algebraic hierarchy of lattice-ordered structures, from groups to algebras. We prove a general version of the normal form theorem and we state a local version of the Pierce-Birkhoff conjecture

Stochastic independence for probability MV-algebras

We propose a notion of stochastic independence for probability MV-algebras, addressing an open problem posed by RieÄ\u8dan and Mundici. Furthermore, we prove a representation theorem for probability MV-algebras and we get MV-algebraic versions of the Hölder inequality and the Hausdorff moment problem