arXiv subjects

Giacomo Lenzi

Publications and source records attributed to Giacomo Lenzi.

5 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↗

arXiv2011

On P-transitive graphs and applications

We introduce a new class of graphs which we call P-transitive graphs, lying between transitive and 3-transitive graphs. First we show that the analogue of de Jongh-Sambin Theorem is false for wellfounded P-transitive graphs; then we show that the mu-calculus fixpoint hierarchy is infinite for P-transitive graphs. Both results contrast with the case of transitive graphs. We give also an undecidability result for an enriched mu-calculus on P-transitive graphs. Finally, we consider a polynomial time reduction from the model checking problem on arbitrary graphs to the model checking problem on P-transitive graphs. All these results carry over to 3-transitive graphs.

cs.LO↗

arXiv2010

On Modal μ-Calculus over Finite Graphs with Bounded Strongly Connected Components

For every positive integer k we consider the class SCCk of all finite graphs whose strongly connected components have size at most k. We show that for every k, the Modal mu-Calculus fixpoint hierarchy on SCCk collapses to the level Delta2, but not to Comp(Sigma1,Pi1) (compositions of formulas of level Sigma1 and Pi1). This contrasts with the class of all graphs, where Delta2=Comp(Sigma1,Pi1).

cs.LO↗