Ultraproducts of continuous posets

We consider local partial clones defined on an uncountable set E having the form Polp(\({\mathfrak{R}}\)), where \({\mathfrak{R}}\) is a set of relations on E. We investigate the notion of weak extendability of partial clones of the type Polp(\({\mathfrak{R}}\)) (in the case of E countable, this coincides with the notion of extendability previously introduced by the author in 1987) which allows us to expand to uncountable sets results on the characterization of Galois-closed sets of relations as well as model-theoretical properties of a relational structure \({\mathfrak{R}}\). We establish criteria for positive primitive elimination sets (sets of positive primitive formulas over \({\mathfrak{R}}\) through which any positive primitive definable relation over \({\mathfrak{R}}\) can be expressed without existential quantifiers) for finite \({\mathfrak{R}}\) as well as for \({\mathfrak{R}}\) having only finite number of positive primitive definable relations of any arity. Emphasizing the difference between countable and uncountable sets, we show that, unlike in the countable case, the characterization of Galois-closed sets InvPol(\({\mathfrak{R}}\)) (that is, all relations which are invariant under all operations from the clone Pol(\({\mathfrak{R}}\)) defined on an uncountable set) cannot be obtained via the application of finite positive primitive formulas together with infinite intersections and unions of updirected sets of relations from \({\mathfrak{R}}\).     

Topological dynamics and the complexity of strong types

We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g., we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup.We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over Ø is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let \(\mathfrak{C}\) be a monster model of a countable theory, p ∈ S(Ø), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p(\(\mathfrak{C}\)). Then, exactly one of the following holds: (1) E is relatively definable (on p(\(\mathfrak{C}\))), smooth, and has finitely many classes, (2) E is not relatively definable, but it is type-definable, smooth, and has 2^?0 classes, (3) E is not type definable and not smooth, and has 2^?0 classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups.     

Towards a Ryll‐Nardzewski‐type theorem for weakly oligomorphic structures

A structure is called weakly oligomorphic if its endomorphism monoid has only finitely many invariant relations of every arity. The goal of this paper is to show that the notions of homomorphism‐homogeneity, and weak oligomorphy are not only completely analogous to the classical notions of homogeneity and oligomorphy, but are actually closely related. We first prove a Fraïssé‐type theorem for homomorphism‐homogeneous relational structures. We then show that the countable models of the theories of countable weakly oligomorphic structures are mutually homomorphism‐equivalent (we call first order theories with this property weakly ω‐categorical). Furthermore we show that every weakly oligomorphic homomorphism‐homogeneous structure contains (up to isomorphism) a unique homogeneous, homomorphism‐homogeneous core, to which it is homomorphism‐equivalent. As a consequence we obtain that every countable weakly oligomorphic structure is homomorphism‐equivalent to a finite or ω‐categorical structure. As a corollary we obtain a characterization of positive existential theories of weakly oligomorphic structures as the positive existential parts of ω‐categorical theories.     

Invariance of selfinjective algebras of quasitilted type under stable equivalences

We prove that the class of finite dimensional selfinjective algebras over a field which admit Galois coverings by the repetitive algebras of the quasitilted algebras, with Galois groups generated by compositions of the Nakayama automorphisms with strictly positive automorphisms, is invariant under stable and derived equivalences. Dedicated to Claus Michael Ringel on the occasion of his sixtieth birthday     

Oligomorphic clones

A permutation group on a countably infinite domain is called oligomorphic if it has finitely many orbits of finitary tuples. We define a clone on a countable domain to be oligomorphic if its set of permutations forms an oligomorphic permutation group. There is a close relationship to ω-categorical structures, i.e., countably infinite structures with a first-order theory that has only one countable model, up to isomorphism. Every locally closed oligomorphic permutation group is the automorphism group of an ω-categorical structure, and conversely, the canonical structure of an oligomorphic permutation group is an ω-categorical structure that contains all first-order definable relations. There is a similar Galois connection between locally closed oligomorphic clones and ω-categorical structures containing all primitive positive definable relations. In this article we generalise some fundamental theorems of universal algebra from clones over a finite domain to oligomorphic clones. First, we define minimal oligomorphic clones, and present equivalent characterisations of minimality, and then generalise Rosenberg’s five types classification to minimal oligomorphic clones. We also present a generalisation of the theorem of Baker and Pixley to oligomorphic clones. Presented by A. Szendrei. Received July 12, 2005; accepted in final form August 29, 2006.     

Positive Galois coverings of self-injective algebras

In the article, we study the structure of Galois coverings of self-injective artin algebras with infinite cyclic Galois groups. In particular, we characterize all basic, connected, self-injective artin algebras having Galois coverings by the repetitive algebras of basic connected artin algebras and with the Galois groups generated by positive automorphisms of the repetitive algebras.     

Generic automorphisms of the universal partial order

We show that the countable universal-homogeneous partial order has a generic automorphism as defined by the second author, namely that it lies in a comeagre conjugacy class of Aut. For this purpose, we work with `determined' partial finite automorphisms that need not be automorphisms of finite substructures (as in the proofs of similar results for other countable homogeneous structures) but are nevertheless sufficient to characterize the isomorphism type of the union of their orbits.

     

Foliations of polynomial growth are hyperfinite

We define a class of equivalence relations with polynomial growth and show that such relations always support finite invariant measures and are hyperfinite. In particular, foliations of polynomial growth define hyperfinite equivalence relations with respect to any family of finite invariant measures on transversals. We also extend a result of Dye for countable groups to show that if a locally compact second countable groupG acts freely on a Lebesgue spaceX with finite invariant measure, so that the orbit relation onX is hyperfinite, thenG is amenable.     

Invariant cocycles, random tilings and the super- and strong Markov properties

We consider -cocycles with values in locally compact, second countable abelian groups on discrete, nonsingular, ergodic equivalence relations. If such a cocycle is invariant under certain automorphisms of these relations, we show that the skew product extension defined by the cocycle is ergodic. As an application we obtain an extension of many recent results of the author and K. Petersen to higher-dimensional shifts of finite type, and prove a transitivity result concerning rearrangements of certain random tilings.

     

Elliptic equations for invariant measures on Riemannian manifolds: existence and regularity of solutions

We obtain sufficient conditions in terms of Lyapunov functions for the existence of invariant measures for diffusions on finite dimensional manifolds and prove some global regularity results for such measures. These results are extended to countable products of finite dimensional manifolds. A new concept of weak elliptic equations for measures on infinite dimensional manifolds is introduced. As an application, we obtain some a priori estimates for Gibbs measures on countable products of manifolds and prove a new existence result for such measures.     

Théorie du potentiel sur les produits infinis et les groupes localement compacts

Let F. be a connected amd locally connected locally compact group having a countable basis for its topology. Does E admit a translation invariant Brelot harmonic sheaf? For which E does the elliptic Bauer theory coincides with the Brelot theory for all invariant harmonic elliptic sheaves? This note announces the following solutions: (a) Any E carries invariant Brelot harmonic sheaves; (b) Any invariant elliptic Bauer harmonic elliptic sheaf is a Brelot sheaf if and only if E is a finite dimensional Lie group. These results are obtained by studying product diffusions on infinite products of manifolds. e.g.. compacts Lie groups.     

Complex projective structures with maximal number of Möbius transformations

We consider complex projective structures on Riemann surfaces and their groups of projective automorphisms. We show that the structures achieving the maximal possible number of projective automorphisms allowed by their genus are precisely the Fuchsian uniformizations of Hurwitz surfaces by hyperbolic metrics. More generally we show that Galois Bely? curves are precisely those Riemann surfaces for which the Fuchsian uniformization is the unique complex projective structure invariant under the full group of biholomorphisms.     

Symmetrized Induced Ramsey Theory

We prove induced Ramsey theorems in which the monochromatic induced subgraph satisfies that all members of a prescribed set of its partial isomorphisms extend to automorphisms of the colored graph (without requirement of preservation of colors). We consider vertex and edge colorings, and extensions of partial isomorphisms in the set of all partial isomorphisms between singletons as considered by Babai and Sós (European J Combin 6(2):101–114, 1985), the set of all finite partial isomorphisms as considered by Hrushovski (Combinatorica 12(4):411–416, 1992), Herwig (Combinatorica 15:365–371, 1995) and Herwig-Lascar (Trans Amer Math Soc 5:1985–2021, 2000), and the set of all total isomorphisms. We observe that every finite graph embeds into a finite vertex transitive graph by a so called bi-embedding, an embedding that is compatible with a monomorphism between the corresponding automorphism groups. We also show that every countable graph bi-embeds into Rado’s universal countable graph Γ.     

Regular and quasiregular isometric flows on Riemannian manifolds

We study the nontrivial Killing vector fields of constant length and the corresponding flows on smooth Riemannian manifolds. We describe the properties of the set of all points of finite (infinite) period for general isometric flows on Riemannian manifolds. It is shown that this flow is generated by an effective almost free isometric action of the group S ¹ if there are no points of infinite or zero period. In the last case, the set of periods is at most countable and generates naturally an invariant stratification with closed totally geodesic strata; the union of all regular orbits is an open connected dense subset of full measure.     

On the norms of group-invariant transition operators on graphs

In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set. Thel ²-norm (spectral radius) of the simple random walk is equal to one if and only if the group is both amenable and unimodular, and this also holds for arbitrary random walks with bounded invariant measure. In general, the norm is bounded above by the Perron-Frobenius eigenvalue of a finite matrix, and this bound is attained if and only if the group is both amenable and unimodular.     

Finite rings in which commutativity is transitive

A ring is called commutative transitive if commutativity is a transitive relation on its nonzero elements. Likewise, it is weakly commutative transitive (wCT) if commutativity is a transitive relation on its noncentral elements. The main topic of this paper is to describe the structure of finite wCT rings. It is shown that every such ring is a direct sum of an indecomposable noncommutative wCT ring of prime power order, and a commutative ring. Furthermore, finite indecomposable wCT rings are either two-by-two matrices over fields, local rings, or basic rings with two maximal ideals. We characterize finite local rings as generalized skew polynomial rings over coefficient Galois rings; the associated automorphisms of the Galois ring give rise to a signature of the local ring. These are then used to further describe the structure of finite local and wCT basic rings.     

Limits of cubes

The celebrated Urysohn space is the completion of a countable universal homogeneous metric space which can itself be built as a direct limit of finite metric spaces. It is our purpose in this paper to give another example of a space constructed in this way, where the finite spaces are scaled cubes. The resulting countable space provides a context for a direct limit of finite symmetric groups with strictly diagonal embeddings, acting naturally on a module which additively is the “Nim field” (the quadratic closure of the field of order 2). Its completion is familiar in another guise: it is the set of Lebesgue-measurable subsets of the unit interval modulo null sets. We describe the isometry groups of these spaces and some interesting subgroups, and give some generalisations and speculations.