Session on Logic and Universal Algebra:
Monday, August 5
14.30-15.10
Isaac Goldbring (UC Irvine, USA)
On supra-SIM sets of natural numbers
A property $\mathcal P$ of the natural numbers is said to be partition regular if: whenever $A\subseteq \mathbb N$ has property $\mathcal P$ and $A=A_1\cup\cdots \cup A_n$, then some $A_i$ also has property $\mathcal P$. Examples of partition regular properties include being infinite (Pigeonhole principle), being an FS-set (Hindman's theorem), being piecewise syndetic, and having positive Banach density.
In this talk, we introduce a new partition regular property of sets of natural numbers called being supra-SIM. This notion originates in nonstandard analysis and generalizes the notion of SIM sets introduced by Leth. The basic idea behind SIM sets is that these are the sets for which is there a link between the internal notion of having small gaps and the external notion of having large measure.
While the class of SIM sets is poorly behaved from the combinatorial point of view, the fact that supra-SIM is partition regular indicates that it is a much more natural combinatorial property. Another indication of this is that it is also preserved under finite embeddability. We discuss these results as well as mention a few other new results about SIM sets and sumsets.
No knowledge of nonstandard analysis will be assumed and we discuss briefly the amount that we need for the purposes of our results.
The new results mentioned in this talk are joint with Steve Leth.
15.15-15.55
Edith Vargas (ITAM, México)
Reconstructing the topology on monoids and polymorphism clones
Transformation Monoids and Clones on a set $A$ carry a natural topology, induced by the topology of point-wise convergence. The endomorphism monoids $\mbox{End} \left(\mathcal{A}\right)$ and polymorphism clones $\mbox{Pol} \left({\mathcal{A}}\right)$ of a relational structure $\mathcal{A}$ are viewed abstractly as topological monoids and topological clones, respectively. Their topology is the natural one. In this talk we show how to reconstruct the topology on the monoid of endomorphisms and the polymorphism clone of some relational structures, among others are: Reducts of the rationals ${\mathbb Q}$.
This is joint work with Mike Behrisch and John K. Truss.
16.00-16.40
Xavier Caicedo (Universidad de los Andes, Colombia)
The model companion of a variety of abelian lattice ordered groups
The class of abelian lattice ordered groups ($\ell $-groups) does not to posses a model companion (Glass and Pierce, 1980). In particular, the divisible members of the class are not always existentially closed, in contrast with the case of abelian $o$-groups. Call an abelian $\ell $-group pseudo-complemented if for any $u>0\ $and $x\in \lbrack 0,u]$ there is maximum $x_{u}^{\ast }\leq u$ such that $x\wedge x_{u}^{\ast }=0$ (equivalently, the lattice ($[0,u],\leq )$ is a Heyting algebra). This is the case, for example, of any product of abelian $o$-groups.
We show that the class of pseudo-complemented abelian $\ell $-groups form a variety of algebras having a model companion, in fact a model completion, consisting on the divisible members of the class for which the image of $($ $ )^{\ast }$ is atomless. The proof is based in representation of these groups by boolean products of $o$-groups, a general form of Macintyre theorem on the model completeness of structures of global sections, and some facts of algebraic logic pertaining MV algebras. It follows that any operation defined implicitly by universal Horn formulas in divisible pseudo-complemented abelian $\ell $-groups is a term on $% 0,+,\wedge ,\vee ,($ $)^{\ast },$ and division by $n\in \omega $.
17.10-17.50
Ricardo Bello (UJED, México)
Generalised stability and algebraic properties of pseudofinite rings
By a pseudofinite ring (field, group) we mean an infinite model of the common theory of all finite rings (fields, groups). The systematic study of the algebraic properties of pseudofinite structures started with Ax ('68) with the algebraic classification of pseudofinite fields and related results. These results allowed to locate pseudofinite fields nicely into the generalised stability picture as simple (supersimple) structures. In this talk we will present some definitions from generalised stability and give some context on results about pseudofinite fields and groups, before presenting results on algebraic properties of pseudofinite rings that arise from model theoretic properties of these structures.
Tuesday, August 6
14.30-15.10
John Goodrick (Universidad de los Andes, Colombia)
Dp-minimal and dp-finite ordered Abelian groups
A classic result is that if the theory of an ordered Abelian group is o-minimal (definable sets in one variable are finite unions of points and intervals), then any definable unary function is piecewise continuous and piecewise monotonic, and a nice decomposition of higher-dimensional definable sets into "cells" can be obtained. There are many expansions of ordered Abelian groups in which similar theorems can be proved, even though they are not o-minimal, such as tame pairs or dense pairs of o-minimal structures.
In this talk we will focus on the case of dp-minimal and dp-finite ordered Abelian groups. Dp-minimality generalizes weak o-minimality and also includes many other structures such as $(\mathbb{Z}, <, +)$, and dp-finite structures are an even broader class which can be thought of as "NIP structures with finite weight." We will present some new results around how "tame topology" theorems from o-minimality can or cannot be generalized to the dp-minimal and dp-finite ordered context.
Some of the work presented here is joint with Alfred Dolich and Viktor Verbovskiy.
15.15-15.55
Angel Zadivar (Universidad de Guadalajara, México)
The frame of nuclei of an Alexandroff space
Frames (locales, complete Heyting algebras) are complete lattices $(L,\leq \bigvee, \wedge,\top,\bot)$ such that the following distributivity holds: \[a\wedge(\bigvee X)=\bigvee\{a\wedge x\mid x\in X\}\] for each $a\in L$ and $X\subseteq L$.
Frames can be understood as an algebraic manifestation of a topological space. Indeed, for every topological space $S$, the open sets $\mathcal{O}S$ constitute a frame but not every frame is a topology. A decent analysis of the category of topological spaces can be done in the language of frames.
As any algebraic object, to study a frame we need to look at to its quotients. There is an elegant treatment to this, a nucleus on a frame $L$, is a monotone function $j\colon L\to L$ which satisfies the following:
(1) $a\leq j(a)$ for all $a\in L$.
(2) $j^{2}=j$.
(3) $j(a\wedge b)=j(a)\wedge j(b)$.
The former of all nuclei, denoted by $N(L)$, is called the assembly of $L$.
This constitute a frame and many properties of $L$ can be capture via its assembly. An important class of frames are complete boolean algebras (frames in which every element has a complement); not every frame is a complete boolean algebra and not every frame is a topology (a spatial frame).
The study of these two properties via the frame of nuclei have been explored by many authors [1-3]. This characterization lead to a description of the complicated structure of $N(L)$.
In this talk we will see a new technique based on [4]. As we will see this technique is useful not only to obtain various theorems about the structure of the frame $N(L)$ but also to recover the topological intuition in the setting of Esakia spaces. In particular we will see a complete characterization of the frame $N(\mathcal{O}S)$ for any Alexandroff space $S$.
This is joint work with F.Avila, G. Bezhanishvili and P. Morandi.
Bibliography
[1] Isbell, John R, Atomless parts of spaces, Mathematica Scandinavica, 31 (1973) 5-32.
[2] Simmons, H, Spaces with Boolean assemblies, Colloquium Mathematicum, 43 (1980) 23-39.
[3] Niefield, SB and Rosenthal, KI, Spatial sublocales and essential primes, Topology and its Applications, 26 (1987) 263-269.
[4] Bezhanishvili, Guram and Ghilardi, Silvio, An algebraic approach to subframe logics. Intuitionistic case, Annals of Pure and Applied Logic, 147 (2007) 84-100.
16.00-16.40
Darío García (Universidad de los Andes, Colombia)
Pseudofinite structures, forking and unimodularity
The fundamental theorem of ultraproducts (Łoś's Theorem) provides a transference principle between the finite structures and their limits. It states that a formula is true in the ultraproduct $M$ of an infinite class of structures if and only if it is true for "almost every" structure in the class, which presents an interesting duality between finite structures and their infinite ultraproducts.
This kind of finite/infinite connection can sometimes be used to prove qualitative properties of large finite structures using the powerful known methods and results coming from infinite model theory, and in the other direction, quantitative properties in the finite structures often induce desirable model-theoretic properties in their ultraproducts. These ideas were used by Hrushovski (cf. [Hru1] to apply ideas from geometric model theory to additive combinatorics, locally compact groups and linear approximate subgroups.
Macpherson and Steinhorn define in [MS] the concept of one-dimensional asymptotic classes, which are classes of finite structures with strong conditions on the sizes of definable sets that imply nice model-theoretic behaviour of their ultraproducts. These classes include, among many others, the class of finite fields, the class of Paley graphs and the class of cyclic groups.
In this talk I will review the main concepts of pseudofinite structures, and present joint work with D. Macpherson and C. Steinhorn (cf. [GMS]) where we explored conditions on the (fine) pseudofinite dimension that guarantee good model-theoretic properties (simplicity or supersimplicity) of the underlying theory of an ultraproduct of finite structures, as well as a characterization of forking in terms of decrease of the pseudofinite dimension. I will also present the concept of unimodularity (for definable sets) - which is satisfied by both pseudofinite structures and omega-categorical structures - and joint with F. Wagner (cf. [GW]) about the equivalence between difference notions of unimodularity.
Bibliography
[GMS] Darío García, Dugald Macpherson, Charles Steinhorn, Pseudofinite structures and simplicity, Journal of Mathematical Logic, vol.15 (2015), no.01, 1550002.
[GW] Darío García and Frank Wagner, Unimodularity unified, Journal of Symbolic Logic, vol.82 (2017), no. 3, pp.1051-1065.
[Hru1] Ehud Hrushovski, On pseudo-finite dimensions, Notre Dame Journal of Formal Logic, vol. 53 (2013), no.3-4, pp.463-495.
[HW] Ehud Hrushovski and Frank Wagner, Counting and dimensions, London Mathematical Society Lecture Notes Series,vol. 350 (2008), no. 161,
[MS] Dugald Macpherson and Charles Steinhorn, One-dimensional asymptotic classes of finite structures, Transactions of the American Mathematical Society,vol.360 (2008), no. 1, pp.411-448.
17.10-17.50
Xavier Vidaux (Universidad de Concepción, Chile)
Indecidibilidad en anillos de enteros de extensiones infinitas totalmente reales del campo de números racionales
A una extensión algebraica $K$ de $\mathbb{Q}$ que es totalmente real, Julia Robinson asoció un número real (o $+\infty$) y mostró que una cierta propiedad topológica de este número implica la indecidibilidad del anillo de enteros de $K$ en el lenguaje de anillos. Haremos un mini-survey de resultados obtenidos gracias a una generalización de este método. Es un trabajo en conjunto con Marianela Castillo y Carlos R. Videla.
18.00-18.40
Carlos Videla (Mount Royal University, Canada)
La indecidibilidad de $\mathbb Q(2)$
$\mathbb Q(2)$ es por definición el compósito de todas las extensiones de grado 2 sobre el campo de números racionales Q. Demostraré la indecidibilidad de su teoría de primer orden en el lenguaje usual de anillos y hablaré de algunos problemas abiertos. Esto es trabajo en colaboración con Carlos Martinez-Ranero y Javier Utreras profesores de la Universidad de Concepción, Chile