Ultraproduct Construction Research Articles

Compactness in team semantics

AbstractWe provide two proofs of the compactness theorem for extensions of first‐order logic based on team semantics. First, we build upon Lück's [16] ultraproduct construction for team semantics and prove a suitable version of Łoś' Theorem. Second, we show that by working with suitably saturated models, we can generalize the proof of Kontinen and Yang [13] to sets of formulas with arbitrarily many variables.

Free polynilpotent groups and the Magnus property

Abstract Motivated by a classic result for free groups, one says that a group G has the Magnus property if the following holds: Whenever two elements generate the same normal subgroup of G, they are conjugate or inverse-conjugate in G. It is a natural problem to find out which relatively free groups display the Magnus property. We prove that a free polynilpotent group of any given class row has the Magnus property if and only if it is nilpotent of class at most 2. For this purpose we explore the Magnus property more generally in soluble groups, and we produce new techniques, both for establishing and for disproving the property. We also prove that a free center-by-(polynilpotent of given class row) group has the Magnus property if and only if it is nilpotent of class at most 2. On the way, we display 2-generated nilpotent groups (with non-trivial torsion) of any prescribed nilpotency class with the Magnus property. Similar examples of finitely generated, torsion-free nilpotent groups are hard to come by, but we construct a 4-generated, torsion-free, class-3 nilpotent group of Hirsch length 9 with the Magnus property. Furthermore, using a weak variant of the Magnus property and an ultraproduct construction, we establish the existence of metabelian, torsion-free, nilpotent groups of any prescribed nilpotency class with the Magnus property.

Ultraproducts and Related Constructions

In this work, we survey some research directions in which the ultraproduct construction and methods based on ultrafilters play significant roles. Rather different areas of mathematics have been considered: topics we are reviewing here include some aspects of the model theory of first-order and second-order existential logics, finite Ramsey theory and general topology. Special emphasis has been made for producing a uniform treatment and for highlighting interconnections between these different subjects.

The isomorphism theorem for linear fragments of continuous logic

AbstractThe ultraproduct construction is generalized to p‐ultramean constructions () by replacing ultrafilters with finitely additive measures. These constructions correspond to the linear fragments of continuous logic and are very close to the constructions in real analysis. A powermean variant of the Keisler‐Shelah isomorphism theorem is proved for . It is then proved that ‐sentences (and their approximations) are exactly those sentences of continuous logic which are preserved by such constructions. Some other applications are also given.

A uniform stability principle for dual lattices

We prove a highly uniform stability or ``almost-near'' theorem for dual lattices of lattices $L \subseteq \Bbb R^n$. More precisely, we show that, for a vector $x$ from the linear span of a lattice $L \subseteq \Bbb R^n$, subject to $\lambda_1(L) \ge \lambda > 0$, to be $\varepsilon$-close to some vector from the dual lattice $L^\star$ of $L$, it is enough that the inner products $u\,x$ are $\delta$-close (with $\delta 0$ depends on $n$, $\lambda$, $\delta$ and $\varepsilon$, only. This generalizes an analogous result previously proved by M. Macaj and the second author for integral vector lattices. The proof is nonconstructive, using the ultraproduct construction and a slight portion of nonstandard analysis.

Accessible images revisited

We extend and improve the result of Makkai and Paré (1989) that the powerful image of any accessible functor F F is accessible, assuming there exists a sufficiently large strongly compact cardinal. We reduce the required large cardinal assumption to the existence of L μ , ω L_{\mu ,\omega } -compact cardinals for sufficiently large μ \mu , and also show that under this assumption the λ \lambda -pure powerful image of F F is accessible. From the first of these statements, we obtain that the tameness of every Abstract Elementary Class follows from a weaker large cardinal assumption than was previously known. We provide two ways of employing the large cardinal assumption to prove each result — one by a direct ultraproduct construction and one using the machinery of elementary embeddings of the set-theoretic universe.

On Ultralimits of Sparse Graph Classes

The notion of nowhere denseness is one of the central concepts of the recently developed theory of sparse graphs. We study the properties of nowhere dense graph classes by investigating appropriate limit objects defined using the ultraproduct construction. It appears that different equivalent definitions of nowhere denseness, for example via quasi-wideness or the splitter game, correspond to natural notions for the limit objects that are conceptually simpler and allow for less technically involved reasonings.

Multiple Recurrence in Quasirandom Groups

We establish a new mixing theorem for quasirandom groups (finite groups with no low-dimensional unitary representations) $G$ which, informally speaking, asserts that if $g, x$ are drawn uniformly at random from $G$, then the quadruple $(g,x,gx,xg)$ behaves like a random tuple in $G^4$, subject to the obvious constraint that $gx$ and $xg$ are conjugate to each other. The proof is non-elementary, proceeding by first using an ultraproduct construction to replace the finitary claim on quasirandom groups with an infinitary analogue concerning a limiting group object that we call an \emph{ultra quasirandom group}, and then using the machinery of idempotent ultrafilters to establish the required mixing property for such groups. Some simpler recurrence theorems (involving tuples such as $(x,gx,xg)$) are also presented, as well as some further discussion of specific examples of ultra quasirandom groups.

Approximate extension of partial ε-characters of abelian groups to characters with application to integral point lattices

Let G be an abelian group, S ⊆ G be a finite set, and T denote the multiplicative group of complex unitswith the invariant arc metric | arg( a/b)|. We will show that for a mapping ƒ : S → T to be ε-close on S to a character φ : G → T it is enough that ƒ be extendable to a mapping ¯f : ( S U {1} U S −1) n → T, where n is big enough and ¯f violates the homomorphy condition at most up to an arbitrary σ < min(ε, π/2). Moreover, n can be chosen uniformly, independently of G and both ƒ and ¯f, depending just on σ, ε and the number of elements of S. The proof is non-constructive, using the ultraproduct construction and Pontryagin duality, hence yielding no estimate on the actual size of n. As one of the applications we show that, for a vector u ∈ R q to be ε-close to some vector from the dual lattice H ★ of a full rank integral point lattice H ⊆ ℤ q , it is enough for the scalar product ux to be δ-close (with δ < 1/3) to an integer for all vectors xH satisfying Σ i| x i | < n, where n depends on δ, ε and q only.

Structure theory for L*-algebras

AbstractThe structure theory of separable complex L*-algebras was given by Schue in [10] and [11]. In [3] Balachandran makes a study of infinite-dimensional complex topologically simple L*-algebras of classical type and poses the question whether these algebras exhaust the class of all infinite-dimensional complex topologically simple L*-algebras. In this paper we give an affirmative answer by determining all the complex topologically simple infinite-dimensional L*-algebras. The case of the real L*-algebras was studied previously in [4], [9] and [12] also under the separability condition. Applying the result of Balachandran, our result yields the structure theory for real L*-algebras. The main tool used here is the ‘approximation’ of the L*-algebra by topologically simple separable L*-algebras via an ultraproduct construction.

Countable ultraproducts without CH

An important application of ultrafilters is in the ultraproduct construction in model theory. In this paper we study ultraproducts of countable structures, whose universe we assume is ω, using ultrafilters on a countable index set, which we also assume to be ω. Many of the properties of the ultraproduct are in fact inherent properties of the ultrafilter. For example, if we take a sequence of countable linear orders without maximal element, then their ultraproduct will have no maximal element, and we can ask what its cofinality is. This cardinal depends only on the ultrafilter; it does not depend on what linear orders comprise the factors.

Ultraproducts and approximation in local rings I

In this paper we give new and quite elementary proofs of the strong approximation theorems of M. Greenberg (Theorem 2.1), M. Artin (Theorem 3.2) and D. Popescu (Theorem 3.3). Our proofs use the ultraproduct construction. In w ! we give a brief outline of this construction and derive the properties we shall use. Our method of proof gives a quite general way of deriving a strong approximation theorem from the corresponding approximation theorem. (By an "approximation theorem" we mean a theorem of the type of Theorem 3.1 and by a "strong approximation theorem" we mean a theorem of the type of Theorem 3.2 or 3.3.) For example if we have an approximation theorem with some of the Y's constrained to depend only on some of the X's then we can immediately derive the strong approximation theorem with the same constraints. In w 4 we give a pair of such theorems with the constraints that some of the Y's depend only on X 1. In w 5 we present some counterexamples: Let k be the algebraic closure of the field with p elements. There is a system of polynomials f ( X t, X2, Yt . . . . . Yv) over k such that the system f (X1, X2, Yt .... ,Y v)=0; Y1, Y6ek[[_X2~; Y2, Y3, Y6ek[[XI~ has a solution mod (X1, X2) J, for every j e N but has no solution in k[[_Xz, X2~. We also give a similar counterexample over the rationals. In w 6 we discuss effectivity. In Ultraproducts and Approximation in Local Rings II we shall extend the methods of this paper to give proofs of some further strong approximation theorems, e.g. the Theorem of Pfister and Popescu [17] as well as some new results. The usefulness of the ultraproduct construction (or equivalently an appropriate form of the G~Sdel Completeness and Compactness Theorems) in analyzing algebraic questions about valued fields is due to Ax and Kochen [3] (cf. also

Transfer theorems for topological structures

The sentences transferred in the above situations arebest described as almost positive, variables appearing ina negative subfoπnula are quantified in a prescribed manner.The main tools of this investigation are the manipula-tion of classical transfer theorems in the context of com-mutative diagrams, the ultraproduct construction, and the^-limit operation of Chang and Keisler's Continuous ModelTheory.Introduction* Every subgroup of an abelian group is abelian,any extension ring of a ring with zero divisors has zero divisors,and each homomorphic image of a commutative ring is commutativeare instances of classical transfer theorems (Lemma 1). In thispaper transfer theorems are obtained for the following mathematicalsituations:

Model Theoretic Approaches to Definability

When approaching the problems of mathematical logic from a model-theoretic point of view, one has at his disposal a variety of ‘methods of attack.’ We have in mind, for example, methods based on the compactness theorem and utilized by Robinson, Svenonius and others (see [18] and [20]) or methods based on the ultraproduct construction (perhaps more precisely, the Łos theorem [16]) and its generalizations and employed by Kochen, Keisler and others (see [8], [11], [12], [13], [14]). Some of the more important of these methods are considered here with a view toward revealing the underlying uniformity. The analysis is made in the context of the problems of definability. Particularly in Section 3, an attempt is made to indicate basis similarities between the various methods considered while addressing each approach to a proof of the Beth theorem [1]. These different proofs of the Beth theorem invoke Craig’s lemma (see [6], or [7]). Although much of Section 3 is expository in nature, we feel that the material should be available in the unified presentation.

Limit ultraproducts

This paper is a sequel to our earlier paper, “Limit Ultrapowers”, [6]. In that paper we introduced the limit ultrapower construction and proved that is isomorphic to a limit ultrapower of if and only if every PCΔ class which contains also contains . In Section 1 of this paper we introduce the more general limit ultraproduct construction, and in Section 2 we prove that, for any class K of relational systems, a relational system is isomorphic to a limit ultraproduct of members of K if and only if every PCΔ class which includes K also contains . As a consequence, the property of K being an intersection of PCΔ classes is characterized purely set-theoretically by the property of K being closed under isomorphisms and limit ultraproducts.In Section 3 we apply limit ultraproducts to obtain model-theoretic conditions equivalent to the set-theoretic condition that every α-complete ultrafilter is γ+-complete. The first result, Theorem 3.7, was announced in the abstract [8], and it is also closely related to a result which was stated without proof in [10], namely Theorem 2 of that paper.In Sections 4 and 5 we apply our results in order to improve a theorem of Craig in [2]. Craig considered the logic L(Q), where Q is a set of cardinals, obtained from ordinary first order logic by adding for each α ϵ Q the quantifier “there exist at least α”.

Ultraproducts in the Theory of Models

In this paper we shall study an algebraic construction which has become a powerful new tool in the theory of models.1 This construction, called the ultraproduct operation', was first described in Los [20] under the name champ logique, where its characteristic property of yielding elementary extensions of a given relational system was stated. This paper falls into two main parts, the first part entitled Ultraproducts, and the second Ultralimits. In the first part, the ultraproduct construction is defined and its fundamental model-theoretic property (Theorem 5.1) proved. This theorem is implicit in Los [20]; we give the particularly elegant and general formulation and proof due to D. Scott. After giving those set-theoretic properties of the ultraproduct necessary for model-theoretic applications (? 6), we apply the ultraproduct to obtain many classical results in logic. Chief among these is the completeness of various elementary theories, including the theory of real closed fields. In ? 8 we show that ultraproducts over division rings have a particularly simple algebraic description. In this first part, no attempt has been made to use ultraproducts to characterize logically defined concepts. In ? 9 we combine the ultraproduct operation with the direct limit operation to produce a new construction, the ultralimit. The ultralimit proves adequate in characterizing most concepts in the theory of models, including elementary equivalence, elementary classes, and elementary functions (?? 9-12). As each logical concept is characterized, a number of applications are presented. For instance, an immediate consequence of Theorem 12.1 is Beth's theorem in the theory of definition (Theorem 12.4). In ? 10, we show that there is a connection between ultraproducts and A. Robinson's method of diagrams. We remark here that H. J. Keisler has given an independent generalization of ultraproducts which allows him to obtain a characterization of elementary equivalence closely connected with Theorem 9.3 of this paper.3