bluuebunny/crossref_metadata_2025_split huggingface dataset:
Generalizations of Deissler's Minimality Rank
Authors: Kucera T. G. 
Published: May 2006

This is the first of two papers based on Chapter V of the author's Ph.D. thesis [K1]. The second paper [K3] will deal with a specific application of the ideas presented here to the study of modules, mostly to questions concerning the complexity of injective modules over a commutative Noetherian ring.In [D], R. Deissler introduced a “minimality rank” which I denote by “rk” here. This rank provides an ordinal measure on the difficulty of defining a given element b in a structure ℳ, allowing the use of parameters from definable subsets of ℳ. If A ⊂ M, rk(b, A, ℳ) = 0 if b is definable in ℳ by a formula with parameters from A. Roughly speaking, in the general case rk(b, A, ℳ) measures how hard we have to work at adding new parameters to A (from sets definable over A in ℳ) in order to be able to define b. “rk” is called a “minimality rank” because of the following: ℳ is a minimal model of the complete theory T = Th(ℳ) iff rk(b, Ø, ℳ) < ∞ for every b ∈ M. Deissler's rank was studied further by R. Woodrow and J. Knight [WK]. They improve on an example given by Deissler. The construction that they present illustrates well the difficulties and subtleties involved in exact calculations of rk.The central concept underlying Deissler's rank is that of a definable set. In §1 I introduce the idea of a context for definability Φ(x). A set B is Φ-definable over A if for some formula and . For Deissler's rank, Φ is the set of all formulas; for rk+ used in the study of modules, Φ+ is the set of all positive primitive formulas. Associated with each Φ is a relation ≺Φ between structures which says that Φ-definitions are preserved. In §1 I develop the basic properties of these two concepts and give a list of examples.
Positive deissler rank and the complexity of injective modules
Authors: Kucera T. G. 
Published: May 2006

This is the second of two papers based on Chapter V of the author's Ph.D. thesis [K1]. For acknowledgements please refer to [K3]. In this paper I apply some of the ideas and techniques introduced in [K3] to the study of a very specific example. I obtain an upper bound for the positive Deissler rank of an injective module over a commutative Noetherian ring in terms of Krull dimension. The problem of finding lower bounds is vastly more difficult and is not addressed here, although I make a few comments and a conjecture at the end.For notation, terminology and definitions, I refer the reader to [K3]. I also use material on ideals and injective modules from [N] and [Ma]. Deissler's rank was introduced in [D].For the most part, in this paper all modules are unitary left modules over a commutative Noetherian ring Λ but in §1 I begin with a few useful results on totally transcendental modules and the relation between Deissler's rank rk and rk+.Recall that if P is a prime ideal of Λ, then an ideal I of Λ is P-primary if I ⊂ P, λ ∈ P implies that λn ∈ I for some n and if λµ ∈ I, λ ∉ P, then µ ∈ I. The intersection of finitely many P-primary ideals is again P-primary, and any P-primary ideal can be written as the intersection of finitely many irreducible P-primary ideals since Λ is Noetherian. Every irreducible ideal is P-primary for some prime ideal P. In addition, it will be important to recall that if P and Q are prime ideals, I is P-primary, J is Q-primary, and J ⊃ I, then Q ⊃ P. (All of these results can be found in [N].)
Partitioning pairs of countable sets of ordinals
Authors: Velleman Dan 
Published: May 2006

In [3], Todorčević showed that ω1 ⇸ [ω1]ω12. In this paper we use similar methods to prove an analogous partition theorem for Pω1(λ), for certain uncountable cardinals λ.Recall that ω1 → [ω1]ω12, means that for every function f: [ω1]2 → ω1 there is a set A ∈ [ω1]ω1 such that f“[A]2 ≠ ω1, and of course ω1 ⇸ [ω1]ω12, is the negation of this statement. For partition relations on Pω1(→) it is customary to partition only those pairs of sets in which the first set is a subset of the second. Thus for A ⊆ Pω1(λ) we defineWe will write Pω1(λ) → [unbdd]λ2 to mean that for every function f: [Pω1(λ)]⊂2 → λ there is an unbounded set A ⊆ Pω1(λ) such that f“[A]⊂2 ≠ λ, and again Pω1(λ) ⇸ [unbdd]λ2 is the negation of this statement.
An existence theorem for recursion categories
Authors: Heller Alex 
Published: May 2006

Recursion theory has been dominated by one example. The notion of a dominical category was introduced (see [2] and [3]) at least in part in the hope of subverting this dominance. The structures of number theory, freed from this original context, led to much of modern algebra with a consequent enrichment of number theory itself: it seemed not unreasonable to attempt to contribute, however modestly, to a similar development of recursion theory. Thus (in the loosest sort of analogy) recursion categories are intended to stand in relation to the class of partial recursive functions as rings do to the rational integers.But if such a generalization is to serve its intended purpose, it ought to allow examples substantially different in spirit from the prototype. Recursion categories are, of course, the proper generalization of classical recursion theory. As a novice in this formidable field I am incapable of encompassing its full extent and must rely on the advice of others (I thank in particular R. DiPaola) in concluding that most of these earlier generalizations are not of this character. Indeed, the examples of recursion categories so far adduced (cf., in addition to the references above, [1] and [4]) are themselves closely tied to the classical one.We attempt here to open the door to radically disparate examples by proving an existence theorem allowing the construction of recursion categories in a wide, and so far unexplored, variety of contexts very distant from that of the natural or ordinal numbers and their subsets, the locus of the traditional theory.In order to do this, it has been necessary to introduce a number of unfamiliar notions—unfamiliar, that is, even in the context of the earlier discussion of dominical categories. This novelty is perhaps mitigated by the fact that, however unfamiliar, they cannot be said to be unusual, as the discussion later will show. We mention in particular those of a “formally free semigroup”, which retains the indexation function of a free semigroup, while giving up the characteristic universal property and that of “uniform generation”, which generalizes finite generation in the fashion assorted to our argument.
Des belles paires aux beaux uples
Authors: Bouscaren Elisabeth , Poizat Bruno 
Published: May 2006

Une bijection p de M2 dans M est dite sans cycles, ou encore localement libre, si pour tout terme t(x1,…,xn) formé à partir de p, et qui n'est pas une simple variable, on a, pour tous a1,…,an de M, t(a1,… an) ≠ a1; toutes ces “fonctions-paires” sont élémentairement équivalentes, à condition, bien sûr, que l'ensemble M soit non vide, et leur théorie (parfois traduite dans un autre langage) a été considérée par divers auteurs.Ainsi, A. Mal'cev [Mal'cev 1962] a décrit toutes les théories d'algèbre localement libres, associées à un langage fonctionnel arbitraire; O. Belegradek [Belegradek] a étudié dans le détail leurs propriétés de stabilité; F. Lucas [Lucas 1978] avait également considéré des fonctions-paires localement libres; c'est encore une fonction-paire qui a été utilisée par J. F. Pabion [Pabion 1982], pour ω1-saturer, mais pas davantage, les modèles de l'Arithmétique; il a été remarqué, dans [Poizat 1984], combien il était utile pour cela de disposer d'une théorie non superstable sans propriété de recouvrement fini; dans un tout autre contexte, et pour de tout autres raisons, cette théorie de fonctions-paires a également été utilisée dans [Poizat 1986]; tous ces emplois supposent qu'on y maîtrise bien l'élimination des quanteurs.
Strongly 2-dimensional theories
Authors: Tsuboi Akito 
Published: May 2006

In [8], we have shown the equivalence of almost strong minimality and strong unidimensionality. More precisely, we proved:Theorem [8]. Let T be a countable stable theory. Then the following two conditions are equivalent:(i) T is almost strongly minimal;(ii) T can be extended to a theory such that any two nonalgebraic types are not almost orthogonal.In the present paper, we define the notion of strong 2-dimensionality (of T). We show that if T is strongly 2-dimensional then T is ω-stable and its model has a simple structure. Roughly speaking, in a model of a strongly 2-dimensional theory, one of the following holds: (a) every element is in acl (δi is strongly minimal), or (b) every element is in acl (δ is strongly regular). Shelah's definition of 2-dimensionality does not imply even superstability. (See Exercise 5.5 in [6, Chapter V, §5].) We show also that condition (a) above implies strong 2-dimensionality of T. However condition (b) does not imply strong 2-dimensionality in general.Our notations and conventions are standard. T is always countable and stable. We work in . A,B,… are used to denote small subsets of . , … are used to denote finite sequences of elements in . δ, φ,… are used to denote formulas (with parameters), p, q, … are used to denote types (with parameters). The fact that p is a nonforking (forking) extension of q is denoted by p ⊃nfq(p ⊃fq). If p is stationary, p∣A denotes the type in S(A) which is parallel to p. (or ) denotes the set of realizations of p (or δ). The Morley rank of p is denoted by RM(p).
Some properties of large filters
Authors: Freiling Chris , Payne T. H. 
Published: May 2006

Dowker [1] raised the question of the existence of filters such that for every coloring (partition) of the underlying index set I with two colors there is a relation R on I which (i) is fat (in the sense that sets of the form {y Є I∣xRy} are in the filter) and (ii) has no bichromatic symmetric pairs (i.e., distinct indices x and y such that x R y and y R x). Additionally, he required that the filter have no anti-symmetric fat relation, for such a relation would vacuously satisfy (i) and (ii). The question of the existence of Dowker filters has been studied more recently by Rudin [3], [4], who conjectures [3] that such filters do not exist.For ZFC the problem remains open. However, Example 2 of this paper shows that one can construct a Dowker filter provided one drops the axiom of choice in favor of the Baire Property (BP) axiom which is known to be incompatible with ZFC but relatively consistent with ZF. In fact, the filter constructed is super-Dowker in the sense that (ii) can be replaced by the requirement that all components of all symmetric pairs have the same color. But, in ZFC the existence of a super-Dowker filter implies the existence of a measurable cardinal.Let F be a filter on an index set I. A set will be called big, small, or medium depending on whether F contains that set, its compliment, or neither, respectively. We define five cardinals associated with F:α denotes the smallest cardinal such that there is a family of α big sets whose intersection is not big.ν denotes the smallest cardinal such that there is a family of ν big sets whose intersection is small.
The false assumption underlying berry's paradox
Authors: French James D. 
Published: May 2006

This paper is divided into three sections. §1 consists of an argument against the validity of Berry's paradox; §2 consists of supporting arguments for the thesis presented in §1; and §3 examines the possibility of re-establishing the paradox.Berry's paradox, a semantic antinomy, is described on p. 4 of the textbook [4] as follows:For the sake of argument, let us admit that all the words of the English language are listed in some standard dictionary. Let T be the set of all thenatural numbers that can be described in fewer than twenty words of the English language. Since there are only a finite number of English words, there are only finitely many combinations of fewer than twenty such words—that is, T is a finite set. Quite obviously, then, there are natural numbers which are greater than all the elements of T; hence there is a least natural number which cannot be described in fewer than twenty words of the English language. By definition, this number is not in T; yet we have described it in sixteen words, hence it is in T.We are faced with a glaring contradiction; since the above argument would be unimpeachable if we admitted the existence of the set T, we are irrevocably led to the conclusion that a set such as T simply cannot exist.
The last word on elimination of quantifiers in modules
Authors: Gute Hans B. , Reuter K. K. 
Published: May 2006

In what follows, a coset is a subset of a group G of the form aH, where H is a subgroup of G; H can be recovered from the coset C: it is the only subgroup which is obtained from C by a left translation; we note in passing that these cosets, that we write systematically with the group to the right, are also of the form Ka, since aH = aHa−1a. A classical combinatorial lemma involving cosets appears in Neumann [1952]: If the coset C = aH is the union of the finite family of cosets C1 = a1H1,…,Cn = anHn, then it is the union of those Ci whose corresponding Hi has finite index in H.In a structure where a group G is defined, Boolean combinations of cosets modulo its definable subgroups form a family of definable sets (by definable, we mean “definable with parameters”). The situation when any definable set is of that kind has been characterized model-theoretically in Hrushovski and Pillay [1987]: A group G is one-based if and only if, for each n, every definable subset of the cartesian power Gn is a(finite!) Boolean combination of cosets modulo definable subgroups. One side is given by a beautiful lemma of Pillay, stating that, in a one-based group which is saturated enough, every type is a right translate of the generic of its left stabilizer.
An infinite superstable group has infinitely many conjugacy classes
Authors: Aguzarov I. , Farey R. E. , Goode J. B. 
Published: May 2006

We begin with some notes concerning the genesis of this paper. A preliminary version of it was written by the third author, who was moved by the desire to correct a mistake in Poizat [1987, p. 97], and to refresh some other minor results of the same book, concerning equations which are satisfied generically in a stable group. The book in question will be considered here as our basic reference on stable groups, and these other results will be discussed elsewhere.This preliminary version contained §§1, 2 and 3 of the present paper, restricted to the context of groups of finite Morley rank. It was observed that a counterexample of finite rank to the above theorem would be an extreme refutation of a conjecture by Zil'ber and Cherlin (cyrillic alphabet order), that may be not so solid as was believed some time ago, which states that a simple group of finite rank should be an algebraic group. Additional motivation for the problem was seen in Reineke's theorem (Reineke [1975]), stating that a connected group of rank one is abelian—the cornerstone for the study of superstable groups—whose proof rests on the fact that a group with two conjugacy classes either has only two elements, or has infinite chains of centralizers (a property that violates stability).
Some trivial considerations
Authors: Goode John B. 
Published: May 2006

At the source of what is now known as “geometric stability theory” was Zil'ber's intuition that the essential properties of an aleph-one-categorical theory were controlled by the geometries of its minimal types. (However, the situation is much more complex than was assumed in Zil'ber [1984], since the main conjecture of that paper has been disproved by Hrushovski.) This is not unnatural in this unidimensional case, where all these geometries have isomorphic contractions, but it was even realized later, in Cherlin, Harrington and Lachlan [1985] and Buechler [1986], that, for any superstable theory with finite ranks, a certain “local” property, i.e. a property satisfied by the geometry of each type of rank one (namely: to have a projective contraction), was equivalent to a “global” one (the theory is one-based, hence satisfies a coordinatization lemma). Then it was shown, in Pillay [1986], that this situation does not generalize to the infinite rank case, that, even for a theory of rank omega, the (local) assumption of projectivity for all the regular types of the theory does not have an exact global counterpart.To clarify this kind of phenomena, I suggest here the elimination of their geometrical aspect, considering only the case where all of the geometries are degenerate. I will study various notions of triviality, which make sense in a stable context, and turn out to be equivalent in the finite rank case; some of them have a definite global flavour, others are of local character.
A direct proof of the Feferman-Vaught theorem and other preservation theorems in products
Authors: Vourtsanis Yiannis 
Published: May 2006

Here we give short and direct proofs of the Feferman-Vaught theorem and other preservation theorems in products of structures. In 1952, Mostowski [5] first showed the preservation of ≡ωω by direct powers of structures. Subsequently, in 1959, Feferman and Vaught [2] proved the preservation of ≡ωω by arbitrary direct products and also by reduced products with respect to cofinite filters. In 1962, Frayne, Morel and Scott [3] noticed that the results extend to arbitrary reduced products. In 1970, Barwise and Eklof [1] showed the preservation of ≡∞λ by products and in 1971 Malitz [4] showed the preservation of ≡κλ with κ strongly inaccessible (or ∞) by products. Below, we give short proofs of the above results. The ideas used here have initiated, in [7], [8], [9], [10], [11], the introduction of several new methods in the theory of products, which on the one hand give new, direct proofs of the known results in the area, including generalizations or strengthenings of some of those, and, on the other hand, give several new results as well in the theory of products and related areas.Below, L denotes a (first order) language, and by a structure we mean an L-structure. 0 and 1 denote the logically valid and false sentence, respectively. We may write ā ∈ A for ā ∈ An for some n. Also, the values 1 and 2 of a parameter l in the definitions below express a duality corresponding to disjunctive and conjunctive forms.
Nonabsoluteness of elementary embeddings
Authors: Wehrung Friedrich 
Published: May 2006

Ifκis a measurable cardinal, let us say that a measure onκis aκ-complete nonprincipal ultrafilter onκ. IfUis a measure onκ, letjUbe the canonical elementary embedding ofVinto its Ultrapower UltU(V). Ifxis a set, say thatUmovesxwhenjU(x)≠x; say thatκmovesxwhen some measure onκmovesx. Recall Kunen's lemma (see [K]): “Every ordinal is moved only by finitely many measurable cardinals.” Kunen's proof (see [K]) and Fleissner's proof (see [KM, III, §10]) are essentially nonconstructive.The following proposition can be proved by using elementary facts about iterated ultrapowers.Proposition.Let ‹Un: n ∈ ω› be a sequence of measures on a strictly increasing sequence ‹κn: n ∈ ω› of measurable cardinals. Let U = ‹ Wα: α < ω2›, where Wωm + n= Um(m, n ∈ ω). Then, for each θ inUltU(V),if E is the (minimal) support of θ inUltU(V),then, for all m ∈ ω, Ummoves θ iff E ∩ [ωm, ω(m + 1))≠ ∅.
On the ranked points of a Π10 set
Authors: Cenzer Douglas , Smith Rick L. 
Published: May 2006

AbstractThis paper continues joint work of the authors with P. Clote, R. Soare and S. Wainer (Annals of Pure and Applied Logic, vol. 31 (1986), pp. 145–163). An element x of the Cantor space 2ω is said have rank α in the closed set P if x is in Dα(P)/Dα + 1(P), where Dα is the iterated Cantor-Bendixson derivative. The rank of x is defined to be the least α such that x has rank a in some set. The main result of the five-author paper is that for any recursive ordinal λ + n (where λ is a limit and n is finite), there is a point with rank λ + n which is Turing equivalent to O(λ + 2n) All ranked points constructed in that paper are singletons. We now construct a ranked point which is not a singleton. In the previous paper the points of high rank were also of high hyperarithmetic degree. We now construct points with arbitrarily high rank. We also show that every nonrecursive RE point is Turing equivalent to an RE point of rank one and that every nonrecursive point is Turing equivalent to a hyperimmune point of rank one. We relate Clote's notion of the height of a singleton in the Baire space with the notion of rank. Finally, we show that every hyperimmune point x is Turing equivalent to a point which is not ranked.
The strong anticupping property for recursively enumerable degrees
Authors: Cooper S. B. 
Published: May 2006

Following Soare [11] we say a recursively enumerable (r.e.) degree a has the anticupping (a.c.) property if there is a nonzero r.e. degree b < a such that for no r.e. c < a does a = b ∪ c.Cooper [2] and Yates showed that 0′ has the a.c. property, while Harrington (see Miller [6]) proved that every high r.e. degree a has the a.c. property.The recent paper by Ambos-Spies, Jockusch, Shore and Soare [1] describes a general theoretical framework for cupping and capping below 0′ which seems likely to be useful in a wider context.Definition. (1) We say b is strongly noncuppable belowa if 0 < b < a and, for each d < a, b ∪ d ≠ a.(2) We say an r.e. a has the strong anticupping property if there is an r.e. b which is strongly noncuppable below a.The main results on cupping in (≤ 0′) are due to Epstein, Posner and Robinson. For instance it is known (Posner and Robinson [8]) that the s.a.c. property fails for 0′.We prove below that r.e. degrees with the s.a.c. property do exist, hence obtaining a nonzero r.e. degree a such that (≤ a) ≢e(≤h) for any high r.e. degree h. This result, obtained by means of an infinite injury construction in (≤ 0′), extends Theorem 2 of [3], proved using a finite injury construction in (≤ 0′).Our main source of notation and terminology is [3].
The elementary class of products of totally ordered abelian group
Authors: Gluschankof Daniel 
Published: May 2006

A basic goal in model-theoretic algebra is to obtain the classification of the complete extensions of a given (first-order) algebraic theory.Results of this type, for the theory of totally ordered abelian groups, were obtained first by A. Robinson and E. Zakon [5] in 1960, later extended by Yu. Gurevich [4] in 1964, and further clarified by P. Schmitt in [6].Within this circle of ideas, we give in this paper an axiomatization of the first-order theory of the class of all direct products of totally ordered abelian groups, construed as lattice-ordered groups (l-groups)—see the theorem below. We think of this result as constituing a first step—undoubtedly only a small one—towards the more general goal of classifying the first-order theory of abelian l-groups.We write groups for abelian l-groups construed as structures in the language 〈 ∨, ∧, +, −, 0〉 (“−” is an unary operation). For unproved statements and unexplicated definitions, the reader is referred to [1].