Countable infinite existentially closed models of universally axiomatizable theories

In the present article, we obtain a new criterion for amodel of a universally axiomatizable theory to be existentially closed. The notion of a maximal existential type is used in the proof and for investigating properties of countable infinite existentially closed structures. The notions of a prime and a homogeneous model, which are classical for the general model theory, are introduced for such structures. We study universal theories with the joint embedding property admitting a single countable infinite existentially closed model. We also construct, for every natural n, an example of a complete inductive theory with a countable infinite family of countable infinite models such that n of them are existentially closed and exactly two are homogeneous.     

Sofic mean dimension

We introduce mean dimensions for continuous actions of countable sofic groups on compact metrizable spaces. These generalize the Gromov–Lindenstrauss–Weiss mean dimensions for actions of countable amenable groups, and are useful for distinguishing continuous actions of countable sofic groups with infinite entropy.     

Minimal first-order structures

We prove a dichotomy theorem for minimal structures and use it to prove that the number of non-isomorphic countable elementary extensions of an arbitrary countable, infinite first-order structure is infinite.     

Homogeneous and strictly homogeneous criteria for partial structures

The Galois closure on the set of relations invariant to all finite partial automorphisms (automorphisms) of a countable partial structure is established via quantifier-free infinite predicate languages (infinite languages with finite string of quantifiers respectively). Based on it the homogeneous and strictly homogeneous criteria for a countable partial structure as well as an ultrahomogeneous criterion for a countable relational structure are found. Next it is shown that infinite languages with a finite string of quantifiers cannot determine the corresponding Galois closure for relations invariant to all automorphisms of an uncountable partial structure.     

Convex games with an infinite number of players and sequencing situations

In this paper we study convex games with an infinite countable set of agents and provide characterizations of this class of games. To do so, and in order to overcome some shortcomings related to the difficulty of dealing with infinite orderings, we need to use a continuity property. Infinite sequencing situations where the number of jobs is infinite countable can be related to convex cooperative TU games. It is shown that some allocations turn out to be extreme points of the core of an infinite sequencing game.     

Abstract structure for Lie pseudogroups

We give the solution of Lie's third fundamental problem for the class of infinite dimensional Lie algebras corresponding to the flat Lie pseudogroups of infinite type. The associated groupoids are infinite dimensional Lie groupoids of the second kind and of countable order.     

On Retracts of the Random Graph and Their Natural Order

 We prove that the natural order on the idempotents of the endomorphism monoid of the countably infinite random graph is -universal; that is, it embeds every countable order. We therefore extend in a strong fashion a result of [2] which showed that the natural order embeds every countable linear order. We consider a refinement of the natural order which embeds every countable quasi-order. Received 26 January 2001; in revised form 3 October 2001     

Topologically free subsets of universal algebras

We prove that every infinite Abelian algebra and every countable field contain infinite topologically free subsets. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 49, No. 5, pp. 651–661, May, 1997.     

Some Generalizations of the Pinwheel Tiling

We introduce a new family of nonperiodic tilings, based on a substitution rule that generalizes the pinwheel tiling of Conway and Radin. In each tiling the tiles are similar to a single triangular prototile. In a countable number of cases, the tiles appear in a finite number of sizes and an infinite number of orientations. These tilings generally do not meet full-edge to full-edge, but can be forced through local matching rules. In a countable number of cases, the tiles appear in a finite number of orientations but an infinite number of sizes, all within a set range, while in an uncountable number of cases both the number of sizes and the number of orientations is infinite. Received April 9, 1996, and in revised form September 16, 1996.     

Sub-additive ergodic theorems for countable amenable groups

In this paper we generalize Kingman's sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.     

Theories with constants and three countable models

We prove that a countable, complete, first-order theory with infinite dcl() and precisely three non-isomorphic countable models interprets a variant of Ehrenfeucht’s or Peretyatkin’s example.     

On constants and the strict order property

Let T be a complete, countable, first-order theory with a finite number of countable models. Assuming that dcl(∅) is infinite we show that T has the strict order property. The author is supported by Ministry of Science and Technology of Serbia Thanks to the referee for the comments; thanks to Anand Pillay and the referee for a very quick procession of the paper.     

Simpler counterexamples to the edge-reconstruction conjecture for infinite graphs

A nonisomorphic, edge-hypomorphic pair of countable forests is constructed, hereby providing a counterexample to the edge-reconstruction conjecture for infinite graphs that is simpler than the counterexamples previously given by C. Thomassen. In addition, to answer the questions posed by C. Thomassen in a previous paper, it is shown that there is a countable forest that is vertex-reconstructible but not edge-reconstructible, and that there is a countable, connected graph with these properties.     

Invertibility in Fréchet Algebras

Using tensor products and a generalised spectrum, we extend to infinite dimensional domains classical results on the ideals generated by a finite or countable set of elements in a Fréchet algebra.     

The list-chromatic number of infinite graphs

We investigate the list-chromatic number of infinite graphs. It is easy to see that Chr(X) ≤ List(X) ≤ Col(X) for each graph X. It is consistent that List(X) = Col(X) holds for every graph with Col(X) infinite. It is also consistent that for graphs of cardinality ? ₁, List(X) is countable iff Chr(X) is countable.     

On extensions of countable filterbases to ultrafilters and ultrafilter compactness

We show in the Zermelo-Fraenkel set theory ZF without the axiom of choice:

1. Given an infinite set X, the Stone space S(X) is ultrafilter compact.

2. For every infinite set X, every countable filterbase of X extends to an ultra-filter i? for every infinite set X, S(X) is countably compact.

3. ω has a free ultrafilter i? every countable, ultrafilter compact space is countably compact.

   We also show the following:

4. There are a permutation model 𝒩 and a set X ∈ 𝒩 such that X has no free ultrafilters and S(X) is not compact but S(X) is countably compact and every countable filterbase of X extends to an ultrafilter.

5. It is relatively consistent with ZF that every countable filterbase of ω extends to an ultrafilter but there exists a countable filterbase of ? which does not extend to an ultrafilter. Hence, it is relatively consistent with ZF that ? has free ultrafilters but there exists a countable filterbase of ? which does not extend to an ultrafilter.

     

Infinite lexicographic products

We generalize the lexicographic product of first-order structures by presenting a framework for constructions which, in a sense, mimic iterating the lexicographic product infinitely and not necessarily countably many times. We then define dense substructures in infinite products and show that any countable product of countable transitive homogeneous structures has a unique countable dense substructure, up to isomorphism. Furthermore, this dense substructure is transitive, homogeneous and elementarily embeds into the product. This result is then utilized to construct a rigid elementarily indivisible structure.     

APPROXIMATION OF COMMON FIXED POINT OF FAMILIES OF NONLINEAR MAPPINGS WITH APPLICATIONS

It is our purpose in this paper to show that some results obtained in uniformly convex real Banach space with uniformly G?ateaux differentiable norm are extendable to more general reflexive and strictly convex real Banach space with uniformly G?ateaux differentiable norm. Demicompactness condition imposed in such results is dispensed with. Furthermore,Applications of our theorems to approximation of common fixed point of countable infinite family of continuous pseudocontractive mappings and approximation of common solution of countable infinite family of generalized mixed equilibrium problems are also discussed. Our theorems improve, generalize, unify and extend several recently announced results.     

Duality theorems on an infinite network *

We study conjugate duality for optimization problems on an infinite, but locally finite network with countable node set X and countable are set Y In contrast to earlier approaches we do not employ Hilbert or Banach space methods. Instead we work in the spaces R^X and R^Y which are siven their Droduct toDolosv, As an application we obtain generalizations of some basic inverse relations from discrete potential theory     

The hierarchical approach to modeling knowledge and common knowledge

One approach to representing knowledge or belief of agents, used by economists and computer scientists, involves an infinite hierarchy of beliefs. Such a hierarchy consists of an agent's beliefs about the state of the world, his beliefs about other agents' beliefs about the world, his beliefs about other agents' beliefs about other agents' beliefs about the world, and so on. (Economists have typically modeled belief in terms of a probability distribution on the uncertainty space. In contrast, computer scientists have modeled belief in terms of a set of worlds, intuitively, the ones the agent considers possible.) We consider the question of when a countably infinite hierarchy completely describes the uncertainty of the agents. We provide various necessary and sufficient conditions for this property. It turns out that the probability-based approach can be viewed as satisfying one of these conditions, which explains why a countable hierarchy suffices in this case. These conditions also show that whether a countable hierarchy suffices may depend on the “richness” of the states in the underlying state space. We also consider the question of whether a countable hierarchy suffices for “interesting” sets of events, and show that the answer depends on the definition of “interesting”.