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Antonio Di Nola

Publications and source records attributed to Antonio Di Nola.

11 recordsLinked to original sources
arXiv2025

Tropicalization through the lens of Łukasiewicz logic, with a topos theoretic perspective

The main aim of this paper is to show the interconnections between Łukasiewicz logic and algebraic geometry using algebraic, geometric and logical instruments. We continue our investigation into a new algebraic geometry based on idempotent semifields, in particular those related with MV-algebras, describing categorical equivalence between different structures related to tropical geometry and many valued logics. Further, we describe such connections in terms of topoi.

math.LO↗
arXiv2023

The spectrum problem for $\ell$-groups and for MV-algebras: a categorical approach

As a main result, we characterize prime spectra of abelian lattice ordered groups. Further we introduce some categories based on spectral spaces, lattices and Priestley spaces, and we relate these categories with each other and with the category of presented MV-algebras, by means of functors. We turn to lattices and offer a simple characterization of 1) maps whose Stone dual preserves closed sets, and 2) closed epimorphisms between distributive lattices as well as their Stone duals. We have a characterization of the variety generated by the Chang MV-algebra and we study this variety. Next we generalize the results to every variety generated by a Komori chain. Finally we discuss homogeneous polynomials in MV-algebras.

math.RA↗
arXiv2019

The spectrum problem for Abelian l-groups and MV-algebras

This paper deals with the problem of characterizing those topological spaces which are homeomorphic to the prime spectra of MV-algebras or Abelian l-groups. As a first main result, we show that a topological space $X$ is the prime spectrum of an MV-algebra if and only if: (1) $X$ is spectral, and (2) the lattice of compact open subsets of $X$ is an epimorphic image of a lattice of "cylinder rational polyhedra" (a natural generalization of rational polyhedra) of some hypercube. As a second main result we extend our results to Abelian l-groups. That is, let $X$ be a spectral space and $K(X)$ the lattice of its compact open sets. The following are equivalent: (1) $X$ is the spectrum of some Abelian l-group; (2) $X$ is homeomorphic to $Spec(K(X))$ and $K(X)\cup\{\infty\}$ is isomorphic to the lattice of the compact open sets of a local MV-algebra, where $\infty>x$ for every $x\in K(X)$. Finally we axiomatize, in monadic second order logic, the lattices of cylinder rational polyhedra of dimension $1$ and $2$.

math.LO↗
arXiv2018

Infinitary logic and basically disconnected compact Hausdorff spaces

We extend Łukasiewicz logic obtaining the infinitary logic $\mathcal{IR}Ł$ whose models are algebras $C(X,[0,1])$, where $X$ is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in $σ$-complete Riesz spaces with strong unit. The Lindenbaum-Tarski algebra of $\mathcal{IR}Ł$ is, up to isomorphism, an algebra of $[0,1]$-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval $[0,1]$.

math.LO↗
arXiv2017

An analysis of the logic of Riesz Spaces with strong unit

We study Łukasiewicz 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.

math.LO↗
arXiv2017

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.

math.RA↗
arXiv2015

The semiring-theoretic approach to MV-algebras: a survey

In this paper we review some of the main achievements of the semiring-theoretic approach to MV-algebras initiated and pursued mainly by the present authors and their collaborators. The survey focuses mainly on the connections between MV-algebras and other theories that such a semiringbased approach enabled, and on an application of such a framework to Digital Image Processing. We also give some suggestions for further developments by stating several open problems and possible research lines.

math.LO↗
arXiv2013

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 Łukasiewicz $\infty$-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 Łukasiewicz 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 ${\mathbb R}{\cal L}$.

math.LO↗
arXiv2010

Stone Duality Type Theorems for MV-algebras with Internal State

Recently in \cite{FM, FlMo}, the language of MV-algebras was extended by adding a unary operation, an internal operator, called also a state-operator. In \cite{DD1}, a stronger version of state MV-algebras, called state-morphism MV-algebras was given. In this paper, we present Stone Duality Theorems for (i) the category of Boolean algebras with a fixed state-operator and the category of compact Hausdorff topological spaces with a fixed idempotent continuous function, and for (ii) the category of weakly divisible $σ$-complete state-morphism MV-algebras and the category of Bauer simplices whose set of extreme points is basically disconnected and with a fixed idempotent continuous function.

math.FA↗
arXiv2010

The Logic of CMV-Algebras

In this paper, once recalled some properties of CMV-algebras, we introduce an expansion of the one-variable fragment of Lukasiewicz propositional logic whose algebraic semantics is the variety of CMV-algebras.

math.LO↗