arXiv subjects

Luca Spada

Publications and source records attributed to Luca Spada.

11 recordsLinked to original sources

arXiv2025

The unification type of Lukasiewicz logic with a bounded number of variables

Building on the correspondence between finitely axiomatised theories in {\L}ukasieiwcz logic and rational polyhedra, we prove that the unification type of the fragment of {\L}ukasiewicz logic with $n\geq 2$ variables is nullary. This solves a problem left open in [V. Marra and L. Spada. Ann. Pure Appl. Logic 164 2013, p. 192-210]. Furthermore, we refine the study of unification with bounds on the number of variables. Our proposal distinguishes the number $m$ of variables allowed in the problem and the number $n$ in the solution. We prove that the unification type of {\L}ukasiewicz logic for all $m,n \geq 2$ is nullary.

math.LO↗

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 $\mathcal{U}^{\kappa}$ 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↗

arXiv2022

Stone-Gelfand duality for metrically complete lattice-ordered groups

We extend Yosida's 1941 version of Stone-Gelfand duality to metrically complete unital lattice-ordered groups that are no longer required to be real vector spaces. This calls for a generalised notion of compact Hausdorff space whose points carry an arithmetic character to be preserved by continuous maps. The arithmetic character of a point is (the complete isomorphism invariant of) a metrically complete additive subgroup of the real numbers containing $1$, namely, either $\frac{1}{n}\mathbb{Z}$ for an integer $n = 1, 2, \dots$, or the whole of $\mathbb{R}$. The main result needed to establish the extended duality theorem is a substantial generalisation of Urysohn's Lemma to such "arithmetic" compact Hausdorff spaces. The original duality is obtained by considering the full subcategory of spaces whose each point is assigned the entire group of real numbers. In the introduction we indicate motivations from and connections with the theory of dimension groups.

math.FA↗

arXiv2022

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↗

arXiv2021

Are locally finite MV-algebras a variety?

We answer Mundici's problem number 3 (D. Mundici. Advanced {\L}ukasiewicz calculus. Trends in Logic Vol. 35. Springer 2011, p. 235): Is the category of locally finite MV-algebras equivalent to an equational class? We prove: (i) The category of locally finite MV-algebras is not equivalent to any finitary variety. (ii) More is true: the category of locally finite MV-algebras is not equivalent to any finitely-sorted finitary quasi-variety. (iii) The category of locally finite MV-algebras is equivalent to an infinitary variety; with operations of at most countable arity. (iv) The category of locally finite MV-algebras is equivalent to a countably-sorted finitary variety. Our proofs rest upon the duality between locally finite MV-algebras and the category of multisets by R. Cignoli, E. J. Dubuc and D. Mundici, and categorical characterisations of varieties and quasi-varieties proved by J. Duskin, J. R. Isbell, F. W. Lawvere and others. In fact no knowledge on MV-algebras is needed, apart from the aforementioned duality.

math.CT↗

arXiv2017

Join-completions of ordered algebras

We present a systematic study of join-extensions and join-completions of ordered algebras, which naturally leads to a refined and simplified treatment of fundamental results and constructions in the theory of ordered structures ranging from properties of the Dedekind-MacNeille completion to the proof of the finite embeddability property for a number of varieties of ordered algebras.

math.LO↗

arXiv2016

MV-algebras, infinite dimensional polyhedra, and natural dualities

We connect the dual adjunction between MV-algebras and Tychonoff spaces with the general theory of natural dualities, and provide a number of applications. In doing so, we simplify the aforementioned construction by observing that there is no need of using presentations of MV-algebras in order to obtain the adjunction. We also provide a description of the dual maps that is intrinsically geometric, and thus avoids the syntactic notion of definable map. Finally, we apply these results to better explain the relation between semisimple tensor products and coproducts of MV-algebras, and we extend beyond the finitely generated case the characterisations of strongly semisimple and polyhedral MV-algebras.

math.RA↗

arXiv2015

Canonical formulas for k-potent commutative, integral, residuated lattices

Canonical formulas are a powerful tool for studying intuitionistic and modal logics. Actually, they provide a uniform and semantic way to axiomatise all extensions of intuitionistic logic and all modal logics above K4. Although the method originally hinged on the relational semantics of those logics, recently it has been completely recast in algebraic terms. In this new perspective canonical formulas are built from a finite subdirectly irreducible algebra by describing completely the behaviour of some operations and only partially the behaviour of some others. In this paper we export the machinery of canonical formulas to substructural logics by introducing canonical formulas for $k$-potent, commutative, integral, residuated lattices ($k$-$\mathsf{CIRL}$). We show that any subvariety of $k$-$\mathsf{CIRL}$ is axiomatised by canonical formulas. The paper ends with some applications and examples.

math.LO↗

arXiv2014

General affine adjunctions, Nullstellens\"atze, and dualities

We introduce and investigate a category-theoretic abstraction of the standard "system-solution" adjunction in affine algebraic geometry. We then look further into these geometric adjunctions at different levels of generality, from syntactic categories to (possibly infinitary) equational classes of algebras. In doing so, we discuss the relationships between the dualities induced by our framework and the well-established theory of concrete dual adjunctions. In the context of general algebra we prove an analogue of Hilbert's Nullstellensatz, thereby achieving a complete characterisation of the fixed points on the algebraic side of the adjunction.

math.CT↗

arXiv2013

Two isomorphism criteria for directed colimits

Using the general notions of finitely presentable and finitely generated object introduced by Gabriel and Ulmer in 1971, we prove that, in any (locally small) category, two sequences of finitely presentable objects and morphisms (or two sequences of finitely generated objects and monomorphisms) have isomorphic colimits (=direct limits) if, and only if, they are confluent. The latter means that the two given sequences can be connected by a back-and-forth chain of morphisms that is cofinal on each side, and commutes with the sequences at each finite stage. In several concrete situations, analogous isomorphism criteria are typically obtained by ad hoc arguments. The abstract results given here can play the useful r\^ole of discerning the general from the specific in situations of actual interest. We illustrate by applying them to varieties of algebras, on the one hand, and to dimension groups---the ordered $K_0$ of approximately finite-dimensional C*-algebras---on the other. The first application encompasses such classical examples as Kurosh's isomorphism criterion for countable torsion-free Abelian groups of finite rank. The second application yields the Bratteli-Elliott Isomorphism Criterion for dimension groups. Finally, we discuss Bratteli's original isomorphism criterion for approximately finite-dimensional C*-algebras, and show that his result does not follow from ours.

math.CT↗

arXiv2010

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.

math.LO↗