This is an expository note on the Lascar group. We also study the Lascar group over hyperimaginaries and make some new observations on the strong types over those. In particular, we show that in a simple theory $\operatorname{Ltp}\equiv\operatorname{stp}$ in real context implies that for hyperimaginary context.

We study the groups Gal L and Gal KP, and the associated equivalence relations EL and EKP, attached to a first order theory T. An example is given where EL≠ EKP. It is proved that EKP is the composition of EL and the closure of EL. Other examples are given showing this is best possible.

In this paper we study the relativized Lascar Galois group of a strong type. The group is a quasi-compact connected topological group, and if in addition the underlying theory T is G-compact, then the group is compact. We apply compact group theory to obtain model theoretic results in this note. -/- For example, we use the divisibility of the Lascar group of a strong type to show that, in a simple theory, such (...) types have a certain model theoretic property that we call divisible amalgamation. -/- The main result of this paper is that if c is a finite tuple algebraic over a tuple a, the Lascar group of stp(ac) is abelian, and the underlying theory is G-compact, then the Lascar groups of stp(ac) and of stp(a) are isomorphic. To show this, we prove a purely compact group-theoretic result that any compact connected abelian group is isomorphic to its quotient by every finite subgroup. -/- Several (counter)examples arising in connection with the theoretical development of this note are presented as well. For example, we show that, in the main result above, neither the assumption that the Lascar group of stp(ac) is abelian, nor the assumption of c being finite can be removed. (shrink)

We show that the Lascar group $\operatorname {Gal}_L$ of a first-order theory T is naturally isomorphic to the fundamental group $\pi _1|)$ of the classifying space of the category of models of T and elementary embeddings. We use this identification to compute the Lascar groups of several example theories via homotopy-theoretic methods, and in fact completely characterize the homotopy type of $|\mathrm {Mod}|$ for these theories T. It turns out that in each of these cases, $|\operatorname (...) {Mod}|$ is aspherical, i.e., its higher homotopy groups vanish. This raises the question of which homotopy types are of the form $|\mathrm {Mod}|$ in general. As a preliminary step towards answering this question, we show that every homotopy type is of the form $|\mathcal {C}|$ where $\mathcal {C}$ is an Abstract Elementary Class with amalgamation for $\kappa $ -small objects, where $\kappa $ may be taken arbitrarily large. This result is improved in another paper. (shrink)

In this paper we show that in every |T |+-resplendent model N , for every A ⊆ N such that |A | ≤ |T |, the group Autf of strong automorphisms is the least very normal subgroup of the group Aut and the quotient Aut/Autf is the Lascar group over A . Then we generalize this result to every |T |+-saturated and strongly |T |+-homogeneous model.

Morley rank and Lascar rank are equal on generic types of definable groups in differentially closed fields with finitely many commuting derivations.

The "space" of Lascar strong types, on some sort and relative to a given complete theory T, is in general not a compact Hausdorff topological space. We have at least three aims in this paper. The first is to show that spaces of Lascar strong types, as well as other related spaces and objects such as the Lascar group Gal L of T, have well-defined Borel cardinalities. The second is to compute the Borel cardinalities of the (...) known examples as well as of some new examples that we give. The third is to explore notions of definable map, embedding, and isomorphism, between these and related quotient objects. We also make some conjectures, the main one being roughly "smooth if and only if trivial". The possibility of a descriptive set-theoretic account of the complexity of spaces of Lascar strong types was touched on in the paper [E. Casanovas, D. Lascar, A. Pillay and M. Ziegler, Galois groups of first order theories, J. Math. Logic1 305–319], where the first example of a "non-G-compact theory" was given. The motivation for writing this paper is partly the discovery of new examples via definable groups, in [A. Conversano and A. Pillay, Connected components of definable groups and o-minimality I, Adv. Math.231 605–623; Connected components of definable groups and o-minimality II, to appear in Ann. Pure Appl. Logic] and the generalizations in [J. Gismatullin and K. Krupiński, On model-theoretic connected components in some group extensions, preprint, arXiv:1201.5221v1]. (shrink)

Lascar described E KP as a composition of E L and the topological closure of E L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X (...) is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/G L . We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories. (shrink)

We study relativized Lascar groups, which are formed by relativizing Lascar groups to the solution set of a partial type $$\Sigma $$. We introduce the notion of a Lascar tuple for $$\Sigma $$ and by considering the space of types over a Lascar tuple for $$\Sigma $$, the topology for a relativized Lascar group is (re-)defined and some fundamental facts about the Galois groups of first-order theories are generalized to the relativized context. In particular, (...) we prove that any closed subgroup of a relativized Lascar group corresponds to a stabilizer of a bounded hyperimaginary having at least one representative in the solution set of the given partial type $$\Sigma $$. Using this, we find the correspondence between subgroups of the relativized Lascar group and the relativized strong types. (shrink)

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an F_σ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over ∅. As an easy conclusion of our main theorem, we get the main result of [K. Krupiński, A. Pillay and T. (...) Rzepecki, Topological dynamics and the complexity of strong types, Israel J. Math.228 (2018) 863–932] which says that for any strong type defined on a single complete type over ∅, smoothness is equivalent to type-definability. We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from the paper mentioned above about bounded quotients of type-definable subgroups of definable groups. (shrink)

Let T be a complete, superstable theory with fewer than ${2^{\aleph_{0}}}$ countable models. Assuming that generic types of infinite, simple groups definable in T eq are sufficiently non-isolated we prove that ω ω is the strict upper bound for the Lascar rank of T.

We lay down the groundwork for the treatment of almost hyperdefinable groups: notions from [5] are put into a natural hierarchy, and new notions, essential to the study to such groups, fit elegantly into this hierarchy. We show that "classical" properties of definable and hyperdefinable groups in simple theories can be generalised to this context. In particular, we prove the existence of stabilisers of Lascar strong types and of the connected and locally connected components of subgroups, and that in (...) a simple one-based theory an almost hyperdefinable group is bounded-by-abelian-by-bounded. (shrink)

RésuméPour tout entier n, on construit des sous-groupes, infiniment définissables de rang de Lascar ωn, du groupe additif d’un corps séparablement clos.

We give strengthened versions of the Herwig–Lascar and Hodkinson–Otto extension theorems for partial automorphisms of finite structures. Such strengthenings yield several combinatorial and group-theoretic consequences for homogeneous structures. For instance, we establish a coherent form of the extension property for partial automorphisms for certain Fraïssé classes. We deduce from these results that the isometry group of the rational Urysohn space, the automorphism group of the Fraïssé limit of any Fraïssé class that is the class of all (...) ${\cal F}$-free structures, and the automorphism group of any free homogeneous structure over a finite relational language all contain a dense locally finite subgroup. We also show that any free homogeneous structure admits ample generics. (shrink)

We prove that has no proper superstable expansions of finite Lascar rank. Nevertheless, this structure equipped with a predicate defining powers of a given natural number is superstable of Lascar rank ω. Additionally, our methods yield other superstable expansions such as equipped with the set of factorial elements.

In this paper we deal with the model theory of differentially closed fields of characteristic zero with finitely many commuting derivations. First we observe that the only known lower bound for the Lascar rank of types in differentially closed fields, announced in a paper of McGrail, is false. This gives us a new class of regular types which are orthogonal to fields. Then we classify the subgroups of the additive group of Lascar rank omega with differential-type 1 (...) which are nonorthogonal to fields. The last parts consist of an analaysis of the quotients of the heat variety. We show that the generic type of such a quotient is locally modular. Finally, we answer a question of Phylliss Cassidy about the existence of certain Jordan-Holder type series in the negative. (shrink)

Let $M$ be a saturated model of a superstable theory and let $G = \operatorname{Aut}(M)$. We study subgroups $H$ of $G$ which contain $G_{(A)}, A$ the algebraic closure of a finite set, generalizing results of Lascar [L] as well as giving an alternative characterization of the simple superstable theories of [P]. We also make some observations about good, locally modular regular types $p$ in the context of $p$-simple types.

A study of smooth classes whose generic structures have simple theory is carried out in a spirit similar to Hrushovski 147; Simplicity and the Lascar group, preprint, 1997) and Baldwin–Shi 1). We attach to a smooth class K0, of finite -structures a canonical inductive theory TNat, in an extension-by-definition of the language . Here TNat and the class of existentially closed models of =T+,EX, play an important role in description of the theory of the K0,-generic. We show that (...) if M is the K0,-generic then MEX. Furthermore, if this class is an elementary class then Th=Th). The investigations by Hrushovski and Pillay , provide a general theory for forking and simplicity for the nonelementary classes, and using these ideas, we show that if K0,, where {,*}, has the joint embedding property and is closed under the Independence Theorem Diagram then EX is simple. Moreover, we study cases where EX is an elementary class. We introduce the notion of semigenericity and show that if a K0,-semigeneric structure exists then EX is an elementary class and therefore the -theory of K0,-generic is near model complete. By this result we are able to give a new proof for a theorem of Baldwin and Shelah 1359). We conclude this paper by giving an example of a generic structure whose first-order theory is simple. (shrink)

We prove a couple of results on NTP2 theories. First, we prove an amalgamation statement and deduce from it that the Lascar distance over extension bases is bounded by 2. This improves previous work of Ben Yaacov and Chernikov. We propose a line of investigation of NTP2 theories based on S1 ideals with amalgamation and ask some questions. We then define and study a class of groups with generically simple generics, generalizing NIP groups with generically stable generics.

We introduce Lascar strong types in excellent classes and prove that they coincide with the orbits of the group generated by automorphisms fixing a model. We define a new independence relation using Lascar strong types and show that it is well-behaved over models, as well as over finite sets. We then develop simplicity and show that, under simplicity, the independence relation satisfies all the properties of nonforking in a stable first order theory. Further, simplicity for an excellent (...) class, as well as the independence relation itself, is uniquely determined. Finally, we show that an excellent class is simple if and only if it has extensible U-rank . We deduce that any excellent class of finite U-rank is simple, and that any uncountably categorical excellent class has an expansion with countably many constants which is simple. (shrink)

In this paper we study a notion of HL-extension for a structure in a finite relational language $\mathcal {L}$. We give a description of all finite minimal HL-extensions of a given finite $\mathcal {L}$ -structure. In addition, we study a group-theoretic property considered by Herwig–Lascar and show that it is closed under taking free products. We also introduce notions of coherent extensions and ultraextensive $\mathcal {L}$ -structures and show that every countable $\mathcal {L}$ -structure can be extended to (...) a countable ultraextensive structure. Finally, it follows from our results that the automorphism group of any countable ultraextensive $\mathcal {L}$ -structure has a dense locally finite subgroup. (shrink)

A structure M is pregeometric if the algebraic closure is a pregeometry in all structures elementarily equivalent to M. We define a generalisation: structures with an existential matroid. The main examples are superstable groups of Lascar U-rank a power of ω and d-minimal expansion of fields. Ultraproducts of pregeometric structures expanding an integral domain, while not pregeometric in general, do have a unique existential matroid. Generalising previous results by van den Dries, we define dense elementary pairs of structures expanding (...) an integral domain and with an existential matroid, and we show that the corresponding theories have natural completions, whose models also have a unique existential matroid. We also extend the above result to dense tuples of structures. (shrink)

Hrushovski proved that the theory of difference-differential fields of characteristic zero has a model-companion, which we shall denote DCFA. Previously, the author proved that this theory is supersimple. In supersimple theories there is a notion of rank defined in analogy with Lascar U-rank for superstable theories. It is also possible to define a notion of dimension for types in DCFA based on transcendence degree of realization of the types. In this paper we compute the rank of a model of (...) DCFA, give some properties regarding rank and dimension, and give an example of a definable set with finite rank but infinite dimension. Finally we prove that for the case of definable subgroup of the additive group being finite-dimensional and having finite rank are equivalent. (shrink)

Every transformation monoid comes equipped with a canonical topology, the topology of pointwise convergence. For some structures, the topology of the endomorphism monoid can be reconstructed from its underlying abstract monoid. This phenomenon is called automatic homeomorphicity. In this paper we show that whenever the automorphism group of a countable saturated structure has automatic homeomorphicity and a trivial center, then the monoid of elementary self-embeddings has automatic homeomorphicity, too. As a second result we strengthen a result by Lascar (...) by showing that whenever \ is a countable \-categorical G-finite structure whose automorphism group has a trivial center and if \ is any other countable structure, then every isomorphism between the monoids of elementary self-embeddings is a homeomorphism. (shrink)

1. We show that if p is a real type which is internal in a set $\sigma$ of partial types in a simple theory, then there is a type p' interbounded with p, which is finitely generated over $\sigma$ , and possesses a fundamental system of solutions relative to $\sigma$ . 2. If p is a possibly hyperimaginary Lascar strong type, almost \sigma-internal$ , but almost orthogonal to $\sigma^{\omega}$ , then there is a canonical non-trivial almost hyperdefinable polygroup which (...) multi-acts on p while fixing $\sigma$ generically. In case p is $\sigma-internal$ and T is stable, this is the binding group of p over \sigma$. (shrink)

Borovik proposed an axiomatic treatment of Morley rank in groups, later modified by Poizat, who showed that in the context of groups the resulting notion of rank provides a characterization of groups of finite Morley rank [2]. (This result makes use of ideas of Lascar, which it encapsulates in a neat way.) These axioms form the basis of the algebraic treatment of groups of finite Morley rank undertaken in [1].There are, however, ranked structures, i.e., structures on which a Borovik-Poizat (...) rank function is defined, which are not ℵ0-stable [1, p. 376]. In [2, p. 9] Poizat raised the issue of the relationship between this notion of rank and stability theory in the following terms: “… ungroupede Borovik est une structure stable, alors qu'un univers rangé n'a aucune raison de l'être …” (emphasis added). Nonetheless, we will prove the following:Theorem 1.1.A ranked structure is superstable.An example of a non-ℵ0-stable structure with Borovik-Poizat rank 2 is given in [1, p. 376]. Furthermore, it appears that this example can be modified in a straightforward way to give ℵ0-stable structures of Borovik-Poizat rank 2 in which the Morley rank is any countable ordinal (which would refute a claim of [1, p. 373, proof of C.4]). We have not checked the details. This does not leave much room for strenghthenings of our theorem. On the other hand, the proof of Theorem 1.1 does give a finite bound for the heights of certain trees of definable sets related to unsuperstability, as we will see in Section 5. (shrink)

We introduce the notion of normal hyperimaginary and we develop its basic theory. We present a new proof of the Lascar-Pillay theorem on bounded hyperimaginaries based on properties of normal hyperimaginaries. However, the use of the Peter–Weyl theorem on the structure of compact Hausdorff groups is not completely eliminated from the proof. In the second part, we show that all closed sets in Kim-Pillay spaces are equivalent to hyperimaginaries and we use this to introduce an approximation of φ-types for (...) bounded hyperimaginaries. (shrink)

Assume that the class of partial automorphisms of the monster model of a complete theory has the amalgamation property. The beautiful automorphisms are the automorphisms of models ofT which: 1. are strong, i.e. leave the algebraic closure (inT eq) of the empty set pointwise fixed, 2. are obtained by the Fraïsse construction using the amalgamation property that we have just mentioned. We show that all the beautiful automorphisms have the same theory (in the language ofT plus one unary function symbol (...) for the automorphism), and we study this theory. In particular, we examine the question of the saturation of the beautiful automorphisms. We also prove that in some cases (in particular if the theory is ω-stable andG-trivial), almost all (in the sense of Baire categoricity) automorphisms of the saturated countable model are beautiful and conjugate. (shrink)

El artículo intenta establecer un paralelo entre el concepto de la máscara carnavalesca concebida por Mijaíl Bajtín en Rabelais y su mundo, y también en otros escritos, y la máscara del teatro Grotesco criollo en Buenos Aires, en las primeras décadas del siglo XX. El artículo comienza por definir semejanzas y diferencias en el uso de estas dos máscaras. En el caso medieval, la máscara es utilizada como herramienta de ecualización y de catarsis social y en el caso del grotesco (...) criollo, la máscara es articulada como una herramienta de conformación y adaptación social. Es así como las funciones prácticas y estéticas de ambas máscaras, aunque son complementarias en términos de sus ingredientes, son opuestas en cuanto a su implementación y valencia comunitaria. La máscara del carnaval es una máscara que intenta romper de forma momentánea el orden establecido para poder mantenerlo en el largo plazo. En cambio, la máscara grotesca es una máscara que opera desde adentro del individuo y la familia, obligándolo a estabilizar costumbres, valores o expectativas nacionales, en la comunidad inmigrante -extranjera y campesina avecindada en la periferia de Buenos Aires-, con el objeto de autoasegurar su subsistencia física, a contramano de la sanidad emocional y social creada por esta acción en estos sujetos marginales desplazados. This article attempts to establish a parallel between Mikhail Bakhtin’s concept of carnival’s mask in Rabelais and His World and other writings, and the mask in Grotesco Criollo Theater in Buenos Aires, during the first decades of the twentieth century. The article begins by defining similarities and differences in the use of these two masks. In the medieval case, the mask is used as a tool of equalization and social catharsis, and in the Grotesco Criollo, the mask is articulated as a shaping tool for conformity and social adaptation. Thus, the practical and aesthetic functions of both masks, though complementary in terms of their ingredients, are opposites in their implementation and community values. The carnival mask is a mask trying to break the established order momentarily, simply to keep it in the end. Instead, the grotesque mask is a mask that operates from within the individual and the family forcing him/ her to stabilize national customs, values, expectations for the suburban immigrant community displaced to Buenos Aires’ periphery to self-ensure their physical survival. However, such insurance often comes at the cost of emotional and social wellbeing of these displaced marginal subjects. (shrink)

We give a characterisation of forking in the context of simple theories in terms of the fundamental order.

Let M be a countable saturated structure, and assume that D(ν) is a strongly minimal formula (without parameter) such that M is the algebraic closure of D(M). We will prove the two following theorems: Theorem 1. If G is a subgroup of $\operatorname{Aut}(\mathfrak{M})$ of countable index, there exists a finite set A in M such that every A-strong automorphism is in G. Theorem 2. Assume that G is a normal subgroup of $\operatorname{Aut}(\mathfrak{M})$ containing an element g such that for all (...) n there exists $X \subseteq D(\mathfrak{M})$ such that $\operatorname{Dim}(g(X)/X) > n$. Then every strong automorphism is in G. (shrink)

We give here alternative definitions for the notions that S. Shelah has introduced in recent papers: the dimensional order property and the depth of a theory. We will also give a proof that the depth of a countable theory, when defined, is an ordinal recursive in T.

The 1996 European Summer Meeting of the Association of Symbolic Logic was held held the University of the Basque Country, at Donostia (San Se bastian) Spain, on July 9-15, 1996. It was organised by the Institute for Logic, Cognition, Language and Information (ILCLI) and the Department of Logic and Philosophy of Sciences of the University of the Basque Coun try. It was supported by: the University of Pais Vasco/Euskal Herriko Unib ertsitatea, the Ministerio de Education y Ciencia (DGCYT), Hezkuntza Saila (...) (Eusko Jaurlaritza), Gipuzkoako Foru Aldundia, and Kuxta Fun dazioa. The main topics of the meeting were Model Theory, Proof Theory, Re cursion and Complexity Theory, Models of Arithmetic, Logic for Artifi cial Intelligence, Formal Semantics of Natural Language and Philosophy of Contemporary Logic. The Program Committee consisted of K. Ambos Spies (Heidelberg), J.L. Balcazar (Barcelona), J.E. Fenstad (Oslo), D. Israel (Stanford), H. Kamp (Stuttgart), R. Kaye (Birmingham), J.M. Larrazabal (San Sebastian), D. Lascar (Paris, chairman), A. Marcja (Firenze), G. Mints (Stanford), M. Otero (Madrid), S. Ronchi della Rocca (Torino), K. Segerberg (Uppsala) and L. Vega (Madrid). The organizing Committee consisted of X. Arrazola (San Sebastian), A. Arrieta (San Sebastian), R. Beneyeto (Valencia), B. Carrascal (San Se bastian), K. Korta (San Sebastian), J.M. Larrazabal (San Sebastian, chair man), J.C. Martinez (Barcelona), J.M. Mendez (Salamanca), F. Migura (Victoria) and J. Perez (Victoria). (shrink)

We associate with every first order structure [Formula: see text] a family of invariant, locally Noetherian topologies. The structure is almost determined by the topologies, and properties of the structure are reflected by topological properties. We study these topologies in particular for stable structures. In nice cases, we get a behaviour similar to the Zariski topology in algebraically closed fields.