arXiv subjects

Serafina Lapenta

Publications and source records attributed to Serafina Lapenta.

10 recordsLinked to original sources

arXiv2024

Relative ideals in homological categories, with an application to MV-algebras

Let $A$ be a homological category and $U\colon B\to A$ be a faithful conservative right adjoint. We introduce the notion of relative ideal with respect to $U$, and we show that, under suitable conditions, any object of $A$ can be seen as a relative ideal of some object in $B$. We then develop a case study. We first prove that the category of hoops is semi-abelian and that the category of MV-algebras is protomodular, then we apply our results to the forgetful functor from the category of MV-algebras to the category of Wajsberg hoops.

math.CT↗

arXiv2023

Baker-Beynon duality beyond semisimplicity

Combining tools from category theory, model theory, and non-standard analysis we extend Baker-Beynon dualities to the classes of all Abelian $\ell$-groups and all Riesz spaces (also known as vector lattices). The extended dualities have a strong geometrical flavor, as they involve a non-standard version of the category of polyhedral cones and piecewise (homogeneous) linear maps between them. We further show that our dualities are induced by the functor $\mathtt{Spec}$, once it is understood how to endow it with "coordinates" in some ultrapower $\mathcal{U}$ of $\mathbb{R}$. This also allows us to characterize the topological spaces arising as spectra of Abelian $\ell$-groups and Riesz spaces as certain subspaces of $\mathcal{U}^κ$ endowed with the Zariski topology given by definable functions in their respective languages. Furthermore, we provide some applications of the extended duality by characterizing, in geometrical terms, semisimplicity, Archimedeanity, and the existence of weak and strong order-units. Finally, we show that our dualities afford a neat and simpler proof of Panti's celebrated characterization of the prime ideals in free Abelian $\ell$-groups and Riesz spaces.

math.RA↗

arXiv2018

A general view of the algebraic semantics of Łukasiewicz logic with product

This paper aims at connecting the various classes that provide an algebraic semantics for three different conservative expansions of Lukasiewicz logic, using algebraic and category-theoretical techniques. We connect such classes of algebras by adjunctions, using the tensor product of MV-algebras and defining the tensor PMV-algebra of a semisimple MV-algebra, inspired by the construction of the tensor algebra of a vector space. We further apply the main results to prove amalgamation properties and, via categorical equivalence, we transfer all results to the framework of lattice- ordered groups.

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↗

arXiv2016

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.

math.LO↗

arXiv2015

Notes on MV-modules over integral domains

An MV-module is an MV-algebra endowed with a scalar multiplication with scalars in a PMV-algebra (i.e. an MV-algebra endowed with a binary "ring-like" product). We investigate the class of semisimple MV-modules over a semisimple and totally ordered integral domain, and we prove an adjunction with a special class of linear spaces.

math.LO↗

arXiv2014

Towards understanding the Pierce-Birkhoff conjecture via MV-algebras

Our main issue was to understand the connection between Łukasiewicz 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.

math.LO↗

arXiv2014

Scalar extensions for algebraic structures of Lukasiewicz logic

In this paper we study the tensor product for MV-algebras, the algebraic structures of Łukasiewicz $\infty$-valued logic. Our main results are: the proof that the tensor product is preserved by the categorical equivalence between the MV-algebras and abelian lattice-order groups with strong unit and the proof of the scalar extension property for semisimple MV-algebras. We explore consequences of this results for various classes of MV-algebras and lattice-ordered groups enriched with a product operation.

math.LO↗

arXiv2014

Stochastic independence for probability MV-algebras

We prove that any MV-algebra has a faithful state can be embedded in an \em{f}MV-algebra of integrable functions. As consequence, we prove Hölder's inequality and Hausdorff moment problem for MV-algebras with product and we propose a solution for the stochastic independence of probability MV-algebras.

math.LO↗