76 research outputs found
Epimorphisms between linear orders
We study the relation on linear orders induced by order preserving
surjections. In particular we show that its restriction to countable orders is
a bqo.Comment: 15 pages; in version 2 we corrected some typos and rewrote the
paragraphs introducing the results of subsection 3.3 (statements and proofs
are unchanged
A complexity dichotomy for poset constraint satisfaction
In this paper we determine the complexity of a broad class of problems that
extends the temporal constraint satisfaction problems. To be more precise we
study the problems Poset-SAT(), where is a given set of
quantifier-free -formulas. An instance of Poset-SAT() consists of
finitely many variables and formulas
with ; the question is
whether this input is satisfied by any partial order on or
not. We show that every such problem is NP-complete or can be solved in
polynomial time, depending on . All Poset-SAT problems can be formalized
as constraint satisfaction problems on reducts of the random partial order. We
use model-theoretic concepts and techniques from universal algebra to study
these reducts. In the course of this analysis we establish a dichotomy that we
believe is of independent interest in universal algebra and model theory.Comment: 29 page
Amenability, connected components, and definable actions
We study amenability of definable groups and topological groups, and prove
various results, briefly described below.
Among our main technical tools, of interest in its own right, is an
elaboration on and strengthening of the Massicot-Wagner version of the
stabilizer theorem, and also some results about measures and measure-like
functions (which we call means and pre-means).
As an application we show that if is an amenable topological group, then
the Bohr compactification of coincides with a certain ``weak Bohr
compactification'' introduced in [24]. In other words, the conclusion says that
certain connected components of coincide: .
We also prove wide generalizations of this result, implying in particular its
extension to a ``definable-topological'' context, confirming the main
conjectures from [24].
We also introduce -definable group topologies on a given
-definable group (including group topologies induced by
type-definable subgroups as well as uniformly definable group topologies), and
prove that the existence of a mean on the lattice of closed, type-definable
subsets of implies (under some assumption) that for any model .
Thirdly, we give an example of a -definable approximate subgroup
in a saturated extension of the group in a
suitable language (where is the free group in 2-generators) for
which the -definable group contains no
type-definable subgroup of bounded index. This refutes a conjecture by Wagner
and shows that the Massicot-Wagner approach to prove that a locally compact
(and in consequence also Lie) ``model'' exists for each approximate subgroup
does not work in general (they proved in [29] that it works for definably
amenable approximate subgroups).Comment: Version 3 contains the material in Sections 2, 3, and 5 of version 1.
Following the advice of editors and referees we have divided version 1 into
two papers, version 3 being the first. The second paper is entitled "On first
order amenability
Structures métriques et leurs groupes d’automorphismes : reconstruction, homogénéité, moyennabilité et continuité automatique
This thesis focuses on the study of Polish groups seen as automorphism groups of metric structures. The observation that every non-archimedean Polish group is the automorphism group of an ultrahomogeneous countable structure has indeed led to fruitful interactions between group theory and model theory. In the framework of metric model theory, introduced by Ben Yaacov, Henson and Usvyastov, this correspondence has been extended to all Polish groups by Melleray. In this thesis, we study various facets of this correspondence. The relationship between a structure and its automorphism group is particularly close in the setting of ℵ0-categorical structures. Indeed, the Ahlbrandt-Ziegler reconstruction theorem allows one to recover an ℵ0-categorical structure, up to bi-interpretability, from its automorphism group. In a joint work with Itai Ben Yaacov, we generalize this result to separably categorical metric structures. Besides, ultrahomogeneous countable structures have the advantage of being completely determined by their finitely generated substructures. In particular, this enabled Moore to give a combinatorial characterization of amenability for nonarchimedean Polish groups. We extend this characterization to all Polish groups and we deduce that amenability is a Gδ condition. Still in a reconstruction perspective, we are interested in the automatic continuity property for Polish groups. Sabok and Malicki introduced conditions of a combinatorial nature on an ultrahomogeneous metric structure that imply the automatic continuity property for its automorphism group. We show that these conditions carry to countable powers, which leads to the groups Aut(μ)N, U(l2)N and Iso(U)N satisfying the automatic continuity property. Those conditions are a weakening of the property of having ample generics. In a joint work with Francois Le Maitre, we exhibit the first examples of connected groups with ample generics, which answers a question of Kechris and Rosendal. Finally, in a joint work with Isabel Muller and Aristotelis Panagiotopoulos, we study the relative homogeneity of substructures in an ultrahomogeneous countable structure. We characterize it completely by a property of the types over the substructures: being determined by a finite setCette thèse porte sur l'étude des groupes polonais vus comme groupes d'automorphismes de structures métriques. L'observation que tout groupe polonais non archimédien est le groupe d'automorphismes d'une structure dénombrable ultra homogène a en effet mené à des interactions fructueuses entre la théorie des groupes et la théorie des modèles. Dans le cadre de la théorie des modèles métriques, introduite par Ben Yaacov, Henson et Usvyatsov, cette correspondance a été étendue par Melleray à tous les groupes polonais. Dans cette thèse, nous étudions diverses facettes de cette correspondance. Le lien entre une structure et son groupe d automorphismes est particulièrement étroit dans le cadre des structures ℵ0-categoriques. En effet, le théorème de reconstruction d'Ahlbrandt-Ziegler permet de retrouver une structure ℵ0-categorique, à bi-interprètabilité près, à partir de son groupe d'automorphismes. Dans un travail en commun avec Itai Ben Yaacov, nous généralisons ce résultat aux structures métriques separablement catégoriques. Les structures dénombrables ultra homogènes ont de plus l avantage d'être complètement déterminées par leurs sous-structures finiment engendrées. Cela a notamment permis a Moore de donner une caractérisation combinatoire de la moyennabilité des groupes polonais non archimédiens. Nous étendons cette caractérisation à tous les groupes polonais et nous en déduisons que la moyennabilite est une condition Gδ. Toujours dans une optique de reconstruction, nous nous intéressons à la propriété de continuité automatique pour les groupes polonais. Sabok et Malicki ont introduit des conditions de nature combinatoire sur une structure métrique ultra homogène qui impliquent la propriété de continuité automatique pour son groupe d'automorphismes. Nous montrons que ces conditions passent à la puissance dénombrable, ce qui a pour conséquence que les groupes Aut(μ)N, U(l2)N et Iso(U)N satisfont la propriété de continuité automatique. Ces conditions sont un affaiblissement du fait d'avoir des amples génériques. Dans un travail en commun avec Francois Le Maitre, nous exhibons les premiers exemples de groupes connexes qui ont des amples génériques, ce qui répond à une question de Kechris et Rosenda
Model theory of monadic predicate logic with the infinity quantifier
This paper establishes model-theoretic properties of ME∞, a variation of monadic first-order logic that features the generalised quantifier ∃ ∞ (‘there are infinitely many’). We will also prove analogous versions of these results in the simpler setting of monadic first-order logic with and without equality (ME and M, respectively). For each logic L∈ { M, ME, ME∞} we will show the following. We provide syntactically defined fragments of L characterising four different semantic properties of L-sentences: (1) being monotone and (2) (Scott) continuous in a given set of monadic predicates; (3) having truth preserved under taking submodels or (4) being truth invariant under taking quotients. In each case, we produce an effectively defined map that translates an arbitrary sentence φ to a sentence φp belonging to the corresponding syntactic fragment, with the property that φ is equivalent to φp precisely when it has the associated semantic property. As a corollary of our developments, we obtain that the four semantic properties above are decidable for L-sentences
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