A note on co-Hopfian groups and rings

We introduce the notion of F-parametrizable model and prove some general results on elementary submodels of F-parametrizable models. Using this notion, we can uniformly characterize all elementary submodels for the field of real numbers and for the group of all permutations on natural numbers in the first order language as well as in the language of hereditarily finite superstructures. Assuming the constructibility axiom, we obtain a simpler characterization of elementary submodels of F-parametrizable models and prove some additional properties of the structure of their elementary submodels.

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic. The [Formula: see text] fragment suffices to distinguish the elementary theories of the groups in question. As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of finite type. We also show that mapping class groups of closed surfaces — a geometric analogue of braid groups — are elementarily equivalent if and only if they are isomorphic.

Given a topological space ∈ M, an elementary submodel of set theory, we define XM to be X ∩ M with topology generated by {U ∩ M : U ∈ T ∩ M}. We prove that if XM is homeomorphic to R, then X = XM . The same holds for arbitrary locally compact uncountable separable metric spaces, but is independent of ZFC if “local compactness” is omitted. Given a model of set theory, i.e. a collection W of sets which satisfies the usual set-theoretic axioms (ZFC), a set M ⊆ W is an elementary submodel of W if for every natural number n and for every formula φ with n free variables in the predicate calculus with = and a 2-place relation symbol ∈, and every x1, · · · , xn ∈ M (we will systematically confuse the membership relation and the symbol ’∈’), φ(x1, · · · , xn) holds in M if and only if it does in W. We usually think of W as being V, the universe of all sets, but for technical reasons officially deal with W = H(θ), the collection of all sets of hereditary cardinality less than θ, a “sufficiently large” regular uncountable cardinal and rather than dealing with ZFC, we deal with sufficiently large fragments of it. (For more on these technical reasons, see [JW].) The non-logician reader will not lose much by thinking of elementary submodels of V. Elementary submodels have been used in set-theoretic topology with increasing frequency and depth over the past 20 years (see e.g. [D]). As often happens in mathematics, one’s tools become objects of study; thus in [JT] we inaugurated a systematic investigation of the topological spaces induced by elementary submodels. This paper is a continuation of that study, although it is mainly independent of [JT]. The Downward Lowenheim-Skolem Theorem of Logic implies that, given any set X ∈ H(θ) and an infinite cardinal κ ≤ |H(θ)| , there is an elementary submodel M of H(θ) with X ∈ M and |M | = κ. Given a topological space ∈ M, we define XM to be the space X∩M with topology TM generated by {U ∩M : U ∈ T ∩M}. The Downward Lowenheim-Skolem Theorem yields XM ’s with X ∩ M having any infinite cardinality ≤ |X | ; a natural question is whether an Upward Lowenheim-Skolem Theorem holds in this context, i.e. 1Research supported by NSERC grant A-7354. AMS Mathematics Subject Classification. Primary 03C62, 03E35, 54A35. Secondary 54F65.

In this chapter we will see how one can learn something about a structure by using symmetries of its elementary extensions. We will examine the specific example of the ordering of the natural numbers, and we will prove that the structure \(({\mathbb {N}},<)\) is minimal. After so many pages, the reader will probably find it hard to believe that this example was my original motivation to write this book. Initially, it seemed that not much technical preparation was needed.

We give a method for explicitly constructing an elementary cubic extension L over which an elliptic curve ED:y2+Dy=x3 (D∈Q∗) has Mordell-Weil rank of at least a given positive integer by finding a close connection between a 3-isogeny of ED and a generic polynomial for cyclic cubic extensions. In our method, the extension degree [L:Q] often becomes small.

The other way around—true ⇒ derivable—is studied in a separate chapter. Here Gödel’s completeness theorem: Truth = derivability, is proved following the method of Leon Henkin. Here a bit of set theory comes in, e.g. classes of structures, extensions, etc. The student need not worry, all prerequisites are lined up—consistent, maximally consistent, conservative extension, witness. The key lemma, the model existence lemma, turns out to be a gentle piecing together of structures. The techniques are used to prove a number of results that take us to model theory, such as the compactness theorem: if all finite parts of a theory are consistent, so is the theory itself. The Skolem-Löwenheim theorems provide (for most structures) larger or smaller structures that are logically indistinguishable from the original one (elementary equivalence, elementary extension). Questions like: Is a particular class of structures (say, algebraically close fields) axiomatizable? Which theories are decidable? are handled by model theoretic means. Skolem considered the possibility of introducing a function picking suitable elements in a structure, once ∀x∃yφ(x,y) has been shown. The basic facts of these Skolem functions are discussed. The famous Herbrand theorem is found in the exercise section. The chapter ends with a new section on ultraproducts, i.e. structures produced by a clever product construction from given structures. An example is a non-standard extension of the natural number system. The topic is on the miraculous side of our logic course, as it shows us that it is perfectly possible to obtain logical results in a logic-free way.KeywordsPropositional LogicCompactness TheoremCompleteness TheoremConservative ExtensionExistential QuantifierThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Given an o-minimal structure M which expands a field, we define, for each positive integer d, a real-valued additive measure on a Boolean algebra of subsets of Md and we prove that all the definable sets included in the finite part Fin(Md) of Md are measurable. When the domain of M is R we obtain Lebesgue measure, but restricted to a proper subalgebra of that of the Lebesgue measurable sets (the Jordan measurable sets). Our measure has good logical properties, being invariant under elementary extensions and under expansions of the language. In the final part of the paper we consider the problem of defining an analogue of the Haar measure for definably compact groups.

Recall that a theory is said to be almost strongly minimal if in every model every element is in the algebraic closure of a strongly minimal set. In 1970 Hodges and Macintyre conjectured that there is a natural number n such that every ℵ0-categorical almost strongly minimal theory is Σn axiomatizable. Recently Ahlbrandt and Baldwin [A-B] proved that if T is ℵ0-categorical and almost strongly minimal, then T is Σn axiomatizable for some n. This result also follows from Ahlbrandt and Ziegler's results on quasifinite axiomatizability [A-Z]. In this paper we will refute Hodges and Macintyre's conjecture by showing that for each n there is an ℵ0-categorical almost strongly minimal theory which is not Σn axiomatizable.Before we begin we should note that in all these examples the complexity of the theory arises from the complexity of the definition of the strongly minimal set. It is still open whether the conjecture is true if we allow a predicate symbol for the strongly minimal set.We will prove the following result.Theorem. For every n there is an almost strongly minimal ℵ0-categorical theory T with models M and N such that N is Σn elementary but not Σn + 1 elementary.To show that these theories yield counterexamples to the conjecture we apply the following result of Chang [C].Theorem. If T is a Σn axiomatizable theory categorical in some infinite power, M and N are models of T and N is a Σn elementary extension of M, then N is an elementary extension of M.

For T any completion of Peano Arithmetic and for n any positive integer, there is a model of T of size 2, with no (n + 1)-length sequence of indiscernibles. Hence the Hanf number for omitting over T, H(T), is at least 2, (Now, using an upper bound previously obtained by Julia Knight H (true arithmetic) is exactly 2b,.) If TX true arithmetic, then H(T) = 2,. If 8 -A (p)'-, then any completion of Peano Arithmetic has a model of size 8 with no set of indiscernibles of size p. There are similar results for theories strongly resembling Peano Arithmetic, e.g., ZF + V = L. ?0. Introduction. The main accomplishment of the research presented here is the lifting of the Specker-MacDowell-Gaifman technology for over arithmetic to a technique for building models from carefully constructed n-types. In Specker and MacDowell [1959] it is shown that every model of Peano Arithmetic, (PA), has a proper elementary extension. Gaifman [1965], [1968] shows how to construct end extension 1-types which give rise to such extensions, investigates the iteration of such extension techniques, and gives a construction of independent extension types which is the direct forebear of the generic types we use here. It is implicit in these papers (and, in fact, is a special case of the M(I) in Gaifman [1976]) that a model of PA generated by 21-many mutually generic will have no two distinct elements with the same 1-type.3 Here we give a direct exposition of such a model, in a way that clearly suggests how to construct a model of PA with no (n + 1)-length sequence of indiscernibles. Carrying through this suggestion involves far more complicated combinatorics than in the case of 2i. What is needed is a certain generalization of finitary Ramsey's Theorem proved independently by Neset'ril and Rodl [1976] and by the present authors. Our original exposition also involved some complicated machinery specially designed for our particular results. We owe a large debt to Haim Gaifman who pointed out some structural properties of our models which are of interest in their own right and which can be used directly to obtain our results. The approach of the present paper is that suggested to us by Gaifman, and we will clearly indicate the specific results which are due to him. Received April 19, 1976; revised September 12, 1977. University of Wisconsin-Milwaukee. This author's research was supported, in part, by an American Mathematical Society Postdoctoral Research Fellowship and by NSF Grant Number

which represent all positive integers, were first obtained by Jacobi (5), Liouville (9), and Pepin (13). Rainanujan (14) proved that there are only 54 sets of positive integers a, b, c, d such that (1. 1) represents all positive integers. Dickson (2) called such forms utniversal. Universal quaternary quadratic forms with cross products were studied by Dickson (2) and Morrow (11). In the above mentioned paper Ramanujan proposed another problem, viz., the problem of determining the conditions under which positive quadratic forms (1. 1) represent all except a finite number of integers. Kloosterman (7), employing the methods of Hardy-Littlewood succeeded, save for a finite number of exceptions, in solving that problem. It is natural to ask Ramanujan's question concerning general positive quaternary quadratic forms. Should Tartakowsky's theorem (19) concerning the representation of large integers by positive quadratic forms in n ? 5 variables hold also for n = 4, then one would expect the answer to that question to be found as an elementary corollary of this theorem and to be expressed in terms of the generic characters of quadratic forms. It is of interest to note that, although Tartakowsky's theorem does not carry over unconditionally to forms in four variables, still for forms of odd determinants and certain orders of even determinants, the answer to Ramanujan's question may be obtained as an elementary extension of the results of Kloosterman and some other elementary considerations, and that, moreover, save for a finite number (of classes) of exceptions the conditions are given in terms of the generic characters. The results here obtained suggest conditions which the generic characters of a genus of quaternary forms should fulfil in order that all forms of that genus should represent the same large integers. The method employed may be summarized as follows: Through the use of the canonical form of Section 3, the problem of the representation of in-

Let Q ( − d ) \mathbf {Q}(\sqrt {-d}) and Q ( 3 d ) \mathbf {Q}(\sqrt {3d}) be quadratic fields with d ≡ d \equiv 2 (mod 3) a positive integer. Let λ − , λ + \lambda ^-, \lambda ^+ be the respective Iwasawa λ \lambda -invariants of the cyclotomic Z 3 \mathbf {Z}_3 -extension of these fields. We show that if λ − = 1 \lambda ^- =1 , then 3 does not divide the class number of Q ( 3 d ) \mathbf {Q}(\sqrt {3d}) and λ + = 0 \lambda ^+ = 0 .

0. Let p be a prime number or zero and let g be a non-negative integer. Then there is a coarse moduli space Mg for complete non-singular irreducible curves of genus g defined over fields of characteristic p, which is an irreducible variety over the algebraic closure F̅p of the prime field Fp. (Especially, F0 is also denoted by Q as usual.) ([8], [2]). The curve corresponding to a generic point of Mg over F̅p is called a generic curve of genus g.

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.

Classification of ℵ0-categorical C-minimal pure C-sets

If L is a first order language and n is a natural number, then Ln is the set of formulas which only make use of the variables x1,…,xn. While every finite structure is determined up to isomorphism by its theory in L, the same is no longer true in Ln. This simple observation is the source of a number of intriguing questions. For example, Poizat [2] has asked whether a complete theory in Ln which has at least two nonisomorphic finite models must necessarily also have an infinite one. The purpose of this paper is to present some counterexamples to this conjecture.Theorem. For each n ≤ 3 there are complete theories in L2n−2andL2n−1having exactly n + 1 models.In our notation and definitions, we follow Poizat [2]. To test structures for elementary equivalence in Ln, we shall use the modified Ehrenfeucht-Fraïssé games of Immerman [1]. For convenience, we repeat his definition here.Suppose that L is a purely relational language, each of the relations having arity at most n. Let and ℬ be two structures for L. Define the Ln game on and ℬ as follows. There are two players, I and II, and there are n pairs of counters a1, b1, …, an, bn. On each move, player I picks up any of the counters and places it on an element of the appropriate structure.