489 research outputs found
Ramsey expansions of metrically homogeneous graphs
We discuss the Ramsey property, the existence of a stationary independence
relation and the coherent extension property for partial isometries (coherent
EPPA) for all classes of metrically homogeneous graphs from Cherlin's
catalogue, which is conjectured to include all such structures. We show that,
with the exception of tree-like graphs, all metric spaces in the catalogue have
precompact Ramsey expansions (or lifts) with the expansion property. With two
exceptions we can also characterise the existence of a stationary independence
relation and the coherent EPPA.
Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's
classification programme of Ramsey classes and as empirical evidence of the
recent convergence in techniques employed to establish the Ramsey property, the
expansion (or lift or ordering) property, EPPA and the existence of a
stationary independence relation. At the heart of our proof is a canonical way
of completing edge-labelled graphs to metric spaces in Cherlin's classes. The
existence of such a "completion algorithm" then allows us to apply several
strong results in the areas that imply EPPA and respectively the Ramsey
property.
The main results have numerous corollaries on the automorphism groups of the
Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity,
existence of universal minimal flows, ample generics, small index property,
21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor
revisio
Automorphism groups of randomized structures
We study automorphism groups of randomizations of separable structures, with
focus on the -categorical case. We give a description of the
automorphism group of the Borel randomization in terms of the group of the
original structure. In the -categorical context, this provides a new
source of Roelcke precompact Polish groups, and we describe the associated
Roelcke compactifications. This allows us also to recover and generalize
preservation results of stable and NIP formulas previously established in the
literature, via a Banach-theoretic translation. Finally, we study and classify
the separable models of the theory of beautiful pairs of randomizations,
showing in particular that this theory is never -categorical (except
in basic cases).Comment: 28 page
Recommended from our members
Homogeneous Structures: Model Theory meets Universal Algebra (online meeting)
The workshop "Homogeneous Structures: Model Theory meets Universal
Algebra'' was centred around transferring recently obtained advances
in universal algebra from the finite to the infinite. As it turns out,
the notion of homogeneity together with other model-theoretic concepts
like -categoricity and the Ramsey property play an
indispensable role in this endeavour
An Algebraic View on p-Admissible Concrete Domains for Lightweight Description Logics: Extended Version
Concrete domains have been introduced in Description Logics (DLs) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. To retain decidability when integrating a concrete domain into a decidable DL, the domain must satisfy quite strong restrictions. In previous work, we have analyzed the most prominent such condition, called w-admissibility, from an algebraic point of view. This provided us with useful algebraic tools for proving w-admissibility, which allowed us to find new examples for concrete domains whose integration leaves the prototypical expressive DL ALC decidable. When integrating concrete domains into lightweight DLs of the EL family, achieving decidability is not enough. One wants reasoning in the resulting DL to be tractable. This can be achieved by using so-called p-admissible concrete domains and restricting the interaction between the DL and the concrete domain. In the present paper, we investigate p-admissibility from an algebraic point of view. Again, this yields strong algebraic tools for demonstrating p-admissibility. In particular, we obtain an expressive numerical padmissible concrete domain based on the rational numbers. Although w-admissibility and p-admissibility are orthogonal conditions that are almost exclusive, our algebraic characterizations of these two properties allow us to locate an infinite class of p-admissible concrete domains whose integration into ALC yields decidable DLs
The automorphism group of the Fra\"iss\'e limit of finite Heyting algebras
Roelcke non-precompactness, non-simplicity, and non-amenability of the
automorphism group of the Fra\"iss\'e limit of finite Heyting algebras are
examined among others.Comment: Submitted to Journal of Symbolic Logic; 15 page
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