# Lógica matemática Organizadores: Manuela Busaniche (mbusaniche@santafe-conicet.gov.ar), Marcelo Esteban Coniglio (coniglio@cle.unicamp.br)

• #### Miércoles 15

15:00 - 15:45

#### Filter pairs and natural extensions of logics

Hugo Luiz Mariano (Universidade de São Paulo, Brasil), joint with P. Arndt (University of Düsseldorf, Germany) and D. C. Pinto (Federal University of Bahia, Brazil)

We adjust the notion of finitary filter pair ( Finitary Filter Pairs and Propositional Logics, South American Journal of Logic 4(2)), which was coined for creating and analyzing finitary logics, in such a way that we can treat logics of cardinality $$\kappa$$, {where $$\kappa$$ is a regular cardinal}. The corresponding new notion is called $$\kappa$$-filter pair. We show that any $$\kappa$$-filter pair gives rise to a logic of cardinality $$\kappa$$ and that every logic of cardinality $$\kappa$$ comes from a $$\kappa$$-filter pair. We use filter pairs to construct natural extensions for a given logic and work out the relationships between this construction and several others proposed in the literature ( A note on natural extensions in abstract algebraic logic, Studia Logica, 103(4); Constructing natural extensions of propositional logics, Studia Logica 104(6)). Conversely, we describe the class of filter pairs giving rise to a fixed logic in terms of the natural extensions of that logic.

15:45 - 16:30

#### Distributivity, Modularity, and Natural Deduction

Rodolfo C. Ertola-Biraben (Universidad de Campinas, Brasil)

On the one hand, from a logical point of view, we will pay attention to discussions around the notion of distributivity in the nineteenth century world of the algebra of logic. In particular, we will consider some passages from the writings of Ernst Schršoder and Charles Peirce. On the other hand, from an algebraic point of view, we will trace the origins of the notion of modularity in lattice theory. This we will do examining some writings of Richard Dedekind. Moreover, we will includesome remarks on the different ways to consider the notions of modularity and distributivity in the context of a join semilattice. In the end, paying special attention to the disjunction elimination rule in natural deduction, we will include considerations on those notions in the context of the conjunction-disjunction-negation fragment of propositional intuitionistic logic.

16:45 - 17:30

#### A topological duality for monotone expansions of semilattices

María Paula Menchón (Universidad Nacional del Centro de la Provincia de Buenos Aires, Argentina)

The aim of this work is to present a topological duality for the variety of monotone semilattices. To do this, combining the duality developed by Celani and González for semilattices together with the approach given by Celani and the author for the study of monotone distributive semilattices, we introduce a category of multirelational spaces called $$m$$S-spaces and we prove that this is dually equivalent to the category of monotone semilattices with homomorphisms. Moreover, as an application of this duality we provide a characterization of congruences of (monotone) semilattices by means of lower Vietoris type topologies. It is worth mentioning that a key tool for developing this duality is the topological description of canonical extension of semilattices that we provide. This is a joint work with Ismael Calomino (CIC and Universidad Nacional del Centro) and William J. Zuluaga Botero (Universidad Nacional del Centro).

17:30 - 18:15

#### Interpolation and Beth definability for conic idempotent Full Lambek calculus

Nick Galatos (University of Denver, Estados Unidos), joint with Wesley Fussner

The Full Lambek calculus forms the basic system for substructural logics, examples of which include relevance, linear, many-valued, intuitionistic and classical logic. We focus on substructural logics that satisfy contraction, mingle and the rule of conicity.

The algebraic semantics of substructural logics are residuated lattices and they have an independent history in the context of order algebra; they were first introduced by Ward and Dilworth as tools in the study of ideal lattices of rings. Residuated lattices have a monoid and a lattice reduct, as well as division-like operations; examples include Boolean algebras, lattice-ordered groups and relation algebras. They are also connected to mathematical linguistics and computer science (for example pointer management and memory allocation). We focus on idempotent conic residuated lattices, which are related to algebraic models of relevance logic, and which extend the class of idempotent linear ones. After establishing a decomposition result for this class, we show that it has the strong amalgamation property, and extend the result to the variety generated by this class; this implies that the corresponding logic has the interpolation property and the Beth definability property.

• #### Jueves 16

15:00 - 15:45

#### Lindström theorems in graded model theory

Carles Noguera i Clofent (Czech Academy of Sciences, República Checa)

Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this talk I will present some first steps towards an abstract formulation of this model theory. I will give a general notion of abstract logic based on many-valued models and discuss six Lindstr\"{o}m-style characterizations of maximality of first-order logics in terms of metalogical properties such as compactness, abstract completeness, the L\"{o}wenheim-Skolem property, the Tarski union property, and the Robinson property, among others. The results have been obtained in a joint work with Guillermo Badia.

15:45 - 16:30

#### Completitud estándar fuerte (fuerte finita) para lógicas de S5-modales de Łukasiewicz

José Patricio Díaz Varela (Universidad Nacional del Sur, Argentina)

En esta charla mostraremos una versión algebraica de teoremas de completitud estándar fuerte finita para lógicas de S5-modales de Łukasiewicz y algunas de sus extensiones axiomáticas de interés. Para esto mostraremos que la variedad de la MV-álgebras monádicas están generadas como cuasivariedad por cierta álgebra funcional estándar. Lo mismo se hará para ciertas extensiones axiomáticas de interés. Por último agregaremos una regla infinitaria y mostraremos la completitud estándar fuerte para esa lógica usando nuevamente técnicas algebraicas.

16:45 - 17:30

#### Locally pseudocomplemented abelian $$\ell$$-groups

Xavier Caicedo (Universidad de los Andes, Colombia)

It is a known result of Glass and Pierce that the variety of abelian lattice ordered groups ($$\ell$$-groups) does not to have a model companion. Call and abelian $$\ell$$-group\ \emph{locally pseudocomplemented} if any bounded lattice interval in the group is pseudocomplemented. These groups form a discriminator variety which has a model completion, namely the divisible locally pseudocomplemented abelian $$\ell$$-groups with dense boolean part. We have similar results forpseudocomplemented MV-algebras, the intervals of locally pseudocomplemented abelian $$\ell$$-groups, actually Heyting (Gödel) algebras and thus the algebraic semantics for the union of Łukasiewicz and Gödel-Dummett logic. Some applications follow on definability in these varieties and the corresponding logics (an expansion of abelian logic in the former case). These results are related to recent work by Metcalfe and Reggio.

• #### Viernes 17

15:00 - 15:45

#### Hilbert's tenth problem in rings of meromorphic functions

Xavier Vidaux (Universidad de Concepción, Chile)

We will give a short survey on definability of the integers in rings of meromorphic functions, together with some ideas on the ingredients in the proofs and the reasons why the case of complex entire functions, as a ring, remain an open problem. We will then present two new evidences for the latter to be undecidable, one obtained with T. Pheidas, and the other one with D. Chompitaki, N. García-Fritz, H. Pasten and T. Pheidas.

15:45 - 16:30

#### Embeddability between orderings and GCH

Rodrigo de Alvarenga Freire (Universidade de Brasília, Brasil)

We provide some statements equivalent in ZFC to GCH, and also to GCH above a given cardinal. These statements express the validity of the notions of replete and well-replete cardinals, which are introduced and proved to be specially relevant to the study of cardinal exponentiation. As a byproduct, a structure hypothesis for linear orderings is proved to be equivalent to GCH: for every linear ordering L, at least one of L and its converse is universal for the smaller well-orderings.

16:45 - 17:30

#### Existentially closed measure preserving actions of the free group.

Alexander Berenstein (Universidad de los Andes, Colombia)

In joint work with Ward Henson and Tomás Ibarlucía, we prove that the theory of atomless probability algebras expanded with a measure preserving action of the free group $$F_k$$ has a model companion for any $$k \geq 1$$ and the corresponding theory is stable. We build explicitly examples of existentially closed models. Our approach also applies to a reasonably large class of $$\aleph_0$$-categorical superstable theories.