bluuebunny/crossref_metadata_2025_split huggingface dataset:
Sacks forcing sometimes needs help to produce a minimal upper bound
Authors: Lubarsky Robert S. 
Published: May 2006

Does every countable set of hyperdegrees have a minimal upper bound?This question remains unanswered. In this paper, we extend the known results.The standard way to construct minimal upper bounds for degrees is to force with pointed perfect trees. This works for hyperdegrees, in the right context. Sacks [Sa] showed that if an admissible set A satisfies Σ1 DC, then forcing with its uniformly hyperarithmetically pointed perfect trees yields a minimal upper bound for the degrees in A.A next question is whether Σ1DC is necessary. Abramson [A] built an admissible set such that Sacks forcing, or anything like it, would not produce a minimal upper bound. He left open the question, though, whether there is such a bound for his set.We answer this question affirmatively.In §II we summarize the previous relevant results, including Steel forcing. In §III we give a construction different from Abramson's of an admissible set for which Sacks forcing does not produce the desired bound. We present this alternative because it is different (although still based on Steel forcing), simpler than the original, and fully illustrates the technique of finding the bound, which applies equally well to the earlier example. §IV describes the construction of the bound. §V closes with questions.I thank Professor Sy Friedman for bringing this problem to my attention. This paper is dedicated to Professor Alexander Kechris on the occasion of his fortieth birthday, and to the Los Angeles VIGOL on its tenth anniversary.
Powers of the ideal of Lebesgue measure zero sets
Authors: Burke Maxim R. 
Published: May 2006

AbstractWe investigate the cofinality of the partial order κ of functions from a regular cardinal κ into the ideal of Lebesgue measure zero subsets of R. We show that when add () = κ and the covering lemma holds with respect to an inner model of GCH, then cf (κ) = max{cf(κκ), cf([cf()]κ)}. We also give an example to show that the covering assumption cannot be removed.
Embedding the diamond in the Σ2 enumeration degrees
Authors: Ahmad Seema 
Published: May 2006

Lachlan [5] has shown that it is not possible to embed the diamond lattice in the r.e. Turing degrees while preserving least and greatest elements; that is, there do not exist incomparable r.e. Turing degrees a and b such that a ∧ b = 0 and a ∨ b = 0′. Cooper [3] has compared the r.e. Turing degrees to the enumeration degrees below 0e′ and has asked if the two structures are elementarily equivalent.In this paper we show that such an embedding is possible in the Σ2enumeration degrees, which implies a negative answer to Cooper's question.Theorem. There are low enumeration degreesaandbsuch thata ∧ b = 0eanda ∨ b = 0e′.Lower case italic letters denote elements of ω while upper case italic letters denote subsets of ω. D, E and F are reserved for finite sets, and K for ′. If D = {x0, x1, …, xn} then the canonical index of D is , and the canonical index of is ∅. Dx denotes the set with canonical index x. {Wi}i∈ω is any fixed standard listing of the r.e. sets, and <·, ·> is any fixed recursive bijection from ω × ω to ω.Intuitively, A is enumeration reducible to B if there is an effective algorithm for producing an enumeration of A from any enumeration of B. There is a natural one-to-one correspondence between all such algorithms and the r.e. sets.
Isomorphisms and nonisomorphisms of graph models
Authors: Schellinx Harold 
Published: May 2006

AbstractIn this paper the existence or nonexistence of isomorphic mappings between graph models for the untyped lambda calculus is studied. It is shown that Engeler's is DA completely determined, up to isomorphism, by the cardinality of its ‘atom-set’ A. A similar characterization is given for a collection of graph models of the -type; from this some propositions regarding automorphisms are obtained. Also we give an indication of the complexity of the first-order theory of graph models by showing that the second-order theory of first-order definable elements of a graph model is first-order expressable in the model.
Remarks on the Church-Rosser Property
Authors: López-Escobar E. G. K. 
Published: May 2006

AbstractA reduction algebra is defined as a set with a collection of partial unary functions (called reduction operators). Motivated by the lambda calculus, the Church-Rosser property is defined for a reduction algebra and a characterization is given for those reduction algebras satisfying CRP and having a measure respecting the reductions. The characterization is used to give (with 20/20 hindsight) a more direct proof of the strong normalization theorem for the impredicative second order intuitionistic propositional calculus.
Partitions of products
Authors: Di Prisco Carlos A. , Henle James M. 
Published: May 2006

We will consider some partition properties of the following type: given a function F: ωω →2, is there a sequence H0, H1, … of subsets of ω such that F is constant on ΠiεωHi? The answer is obviously positive if we allow all the Hi's to have exactly one element, but the problem is nontrivial if we require the Hi's to have at least two elements. The axiom of choice contradicts the statement “for all F: ωω→ 2 there is a sequence H0, H1, H2,… of subsets of ω such that {i|(Hi) ≥ 2} is infinite and F is constant on ΠHi”, but the infinite exponent partition relation ω(ω)ω implies it; so, this statement is relatively consistent with an inaccessible cardinal. (See [1] where these partition properties were considered.)We will also consider partitions into any finite number of pieces, and we will prove some facts about partitions into ω-many pieces.Given a partition F: ωω → k, we say that H0, H1…, a sequence of subsets of ω, is homogeneous for F if F is constant on ΠHi. We say the sequence H0, H1,… is nonoverlapping if, for all i ∈ ω, ∪Hi > ∩Hi+1.The sequence 〈Hi: i ∈ ω〉 is of type 〈α0, α1,…〉 if, for every i ∈ ω, ∣Hi∣ = αi.We will adopt the usual notation for polarized partition relations due to Erdös, Hajnal, and Rado.means that for every partition F: κ1 × κ2 × … × κn→δ there is a sequence H0, H1,…, Hn such that Hi ⊂ κi and ∣Hi∣ = αi for every i, 1 ≤ i ≤ n, and F is constant on H1 × H2 × … × Hn.
Logic, sheaves, and factorization systems
Authors: Monro G. P. 
Published: May 2006

In this paper we extend the models for the “logic of categories” to a wider class of categories than is usually considered. We consider two kinds of logic, a restricted first-order logic and the full higher-order logic of elementary topoi.The restricted first-order logic has as its only logical symbols ∧, ∃, Τ, and =. We interpret this logic in a category with finite limits equipped with a factorization system (in the sense of [4]). We require to satisfy two additional conditions: ⊆ Monos, and any pullback of an arrow in is again in . A category with a factorization system satisfying these conditions will be called an EM-category.The interpretation of the restricted logic in EM-categories is given in §1. In §2 we give an axiomatization for the logic, and in §§3 and 5 we give two completeness proofs for this axiomatization. The first completeness proof constructs an EM-category out of the logic, in the spirit of Makkai and Reyes [8], though the construction used here differs from theirs. The second uses Boolean-valued models and shows that the restricted logic is exactly the ∧, ∃-fragment of classical first-order logic (adapted to categories). Some examples of EM-categories are given in §4.The restricted logic is powerful enough to handle relations, and in §6 we assign to each EM-category a bicategory of relations Rel() and a category of “functional relations” fr. fr is shown to be a regular category, and it turns out that Rel( and Rel(fr) are biequivalent bicategories. In §7 we study complete objects in an EM-category where an object of is called complete if every functional relation into is yielded by a unique morphism into . We write c for the full subcategory of consisting of the complete objects. Complete objects have some, but not all, of the properties that sheaves have in a category of presheaves.
Amenable versus hyperfinite Borel equivalence relations
Authors: Kechris Alexander S. 
Published: May 2006

LetXbe a standard Borel space (i.e., a Polish space with the associated Borel structure), and letEbe acountableBorel equivalence relation onX, i.e., a Borel equivalence relationEfor which every equivalence class [x]Eis countable. By a result of Feldman-Moore [FM],Eis induced by the orbits of a Borel action of a countable groupGonX.The structure of general countable Borel equivalence relations is very little understood. However, a lot is known for the particularly important subclass consisting of hyperfinite relations. A countable Borel equivalence relation is calledhyperfiniteif it is induced by a Borel ℤ-action, i.e., by the orbits of a single Borel automorphism. Such relations are studied and classified in [DJK] (see also the references contained therein). It is shown in Ornstein-Weiss [OW] and Connes-Feldman-Weiss [CFW] that for every Borel equivalence relationEinduced by a Borel action of a countable amenable groupGonXand for every (Borel) probability measure μ onX, there is a Borel invariant setY⊆Xwith μ(Y) = 1 such thatE↾Y(= the restriction ofEtoY) is hyperfinite. (Recall that a countable group G isamenableif it carries a finitely additive translation invariant probability measure defined on all its subsets.) Motivated by this result, Weiss [W2] raised the question of whether everyEinduced by a Borel action of a countable amenable group is hyperfinite. Later on Weiss (personal communication) showed that this is true forG= ℤn. However, the problem is still open even for abelianG. Our main purpose here is to provide a weaker affirmative answer for general amenableG(and more—see below). We need a definition first. Given two standard Borel spacesX, Y, auniversally measurableisomorphism betweenXandYis a bijection ƒ:X→Ysuch that both ƒ, ƒ-1are universally measurable. (As usual, a mapg:Z→W, withZandWstandard Borel spaces, is calleduniversally measurableif it is μ-measurable for every probability measure μ onZ.) Notice now that to assert that a countable Borel equivalence relation onXis hyperfinite is trivially equivalent to saying that there is a standard Borel spaceYand a hyperfinite Borel equivalence relationFonY, which isBorelisomorphic toE, i.e., there is a Borel bijection ƒ:X→YwithxEy⇔ ƒ(x)Fƒ(y). We have the following theorem.
Quasi-endomorphisms in small stable groups
Authors: Wagner Frank O. 
Published: May 2006

AbstractWe generalise various properties of quasiendomorphisms from groups with regular generic to small abelian groups. In particular, for a small stable abelian group such that no infinite definable quotient is connected-by-finite, the ring of quasi-endomorphisms is locally finite. Under some additional assumptions, it decomposes modulo some nil ideal into a sum of finitely many matrix rings.
Les automorphismes d'un ensemble fortement minimal
Authors: Lascar Daniel 
Published: May 2006

AbstractLet be a countable saturated structure, and assume that D(v) is a strongly minimal formula (without parameter) such that is the algebraic closure of D(). We will prove the two following theorems:Theorem 1. If G is a subgroup of Aut() of countable index, there exists a finite set A in such that every A-strong automorphism is in G.Theorem 2. Assume that G is a normal subgroup of Aut() containing an element g such that for all n there exists X ⊆ D() such that Dim(g(X)/X) > n. Then every strong automorphism is in G.
Lattice of algebraically closed sets in one-based theories
Authors: Low Lee Fong 
Published: May 2006

AbstractLet T be a one-based theory. We define a notion of width, in the case of T having the finiteness property, for the lattice of finitely generated algebraically closed sets and proveTheorem. Let T be one-based with the finiteness property. If T is of bounded width, then every type in T is nonorthogonal to a weight one type. If T is countable, the converse is true.
On the ring of differentially-algebraic entire functions
Authors: Rubel Lee A. 
Published: May 2006

Let be the ring of all entire functions of one complex variable, and let DA be the subring of those entire functions that are differentially algebraic (DA); that is, they satisfy a nontrivial algebraic differential equation.where P is a non-identically-zero polynomial in its n + 2 variables. It seems not to be known whether DA is elementarily equivalent to . This would mean that DA and have exactly the same true statements about them, in the first-order language of rings. (Roughly speaking, a sentence about a ring R is first-order if it has finite length and quantifies only over elements (i.e., not subsets or functions or relations) of R.) It follows from [NAN] that DA and are not isomorphic as rings, but this does not answer the question of elementary equivalence.
A Spector-Gandy theorem for cPCd() classes
Authors: Lavine Shaughan 
Published: May 2006

AbstractLet be an admissible structure. A cPCd() class is the class of all models of a sentence of the form , where is an -r.e. set of relation symbols and Φ is an -r.e. set of formulas of ℒ∞,ω that are in . The main theorem is a generalization of the following: Let be a pure countable resolvable admissible structure such that is not Σ-elementarily embedded in HYP(). Then a class K of countable structures whose universes are sets of urelements is a cPCd() class if and only if for some Σ formula σ (with parameters from ), is in K if and only if is a countable structure with universe a set of urelements and σ, where , the smallest admissible set above relative to , is a generalization of HYP to structures with similarity type Σ over that is defined in this article. Here we just note that when Lα is admissible, HYPLα() is Lβ() for the least β ≥ α such that Lβ() is admissible, and so, in particular, that is just HYP() in the usual sense when has a finite similarity type.The definition of is most naturally formulated using Adamson's notion of a +-admissible structure (1978). We prove a generalization from admissible to +-admissible structures of the well-known truncation lemma. That generalization is a key theorem applied in the proof of the generalized Spector-Gandy theorem.
Imaginary modules
Authors: Kucera T. G. , Prest M. 
Published: May 2006

It was a fundamental insight of Shelah that the equivalence classes of a definable equivalence relation on a structure M often behave like (and indeed must be treated like) the elements of the structure itself, and that these so-called imaginary elements are both necessary and sufficient for developing many aspects of stability theory. The expanded structure Meq was introduced to make this insight explicit and manageable. In Meq we have “names” for all things (subsets, relations, functions, etc.) “definable” inside M. Recent results of Bruno Poizat allow a particularly simple and more or less algebraic modification of Meq in the case that M is a module. It is the purpose of this paper to describe this nearly algebraic structure in such a way as to make the usual algebraic tools of the model theory of modules readily available in this more general context. It should be pointed out that some of our discussion has been part of the “folklore” of the subject for some time; it is certainly time to make this “folklore” precise, correct, and readily available.Modules have proved to be good examples of stable structures. Not, we mean, in the sense that they are well-behaved (which, in the main, they are), but in the sense that they are straightforward enough to provide comprehensible illustrations of concepts while, at the same time, they have turned out to be far less atypical than one might have supposed. Indeed, a major feature of recent stability theory has been the ubiquitous appearance of modules or more general “abelian structures” in abstract stable structures.
Quasiunitriangular groups
Authors: Belegradek O. V. 
Published: May 2006

For a ring with unit R, which need not be associative, denote the group of upper unitriangular 3 × 3 matrices over R by UT3(R). Let e1 = (1,0,0), e2 = (0,1,0), where (α, β, γ) denotes the matrixDenote the expanded group (UT3(R), e1, e2) by (R). A. 1. Mal′cev [M] gave an algebraic characterization of the expanded groups of the form (R) as follows. Let h1, h2 be elements of a group H; then (H, h1, h2) is isomorphic to (R), for some R, if and only if(i) H is 2-step nilpotent;(ii) CH(hi) are abelian, i = 1,2;(iii) CH(h1) ∩ CH(h2) = Z(H);(iv) [CH(h1),h2] = [h1, CH(h2)] = Z(H);(v) Z(H) is a direct summand in both CH(hi).(In [M] condition (v) is a bit stronger; the version above is presented in [B2].)A pair (h1, h2) of elements of a group H is said to be a base if (H, h1, h2) satisfies the conditions (i)–(iv). A. I. Mal′cev [M] found a uniform way of first order interpreting a ring Ring(H, h1, h2) in any group with a base (H, h1, h2); in particular, Ring((R)) ≃ R.
Every n-generic degree is a minimal cover of an n-generic degree
Authors: Kumabe Masahiro 
Published: May 2006

The notions of forcing and generic set were introduced by Cohen in 1963 to prove the independence of the Axiom of Choice and the Continuum Hypothesis in set theory. Let ω be the set of natural numbers, i.e., {0,1,2,3,…}. A string is a mapping from an initial segment of ω into {0,1}. We identify a set A ⊆ ω to with its characteristic function.We now consider a set generic over the arithmetic sets. A set A ⊆ ω is called n-generic if it is Cohen-generic for n-quantifier arithmetic. This is equivalent to saying that for every -set of strings S, there is a σ ⊂ A such that σ ∈ S or (∀v ≥ σ)(v ∉ S). By degree we mean Turing degree (of unsolvability). We call a degree n-generic if it has an n-generic representative. For a degree a, let D(≤a) denote the set of degrees which are recursive in a.Before Cohen's work, there was a precursor of the notion of forcing in recursion theory. Friedberg showed that for every degree b above the complete degree 0', i.e., the degree of a complete r.e. set, there is a degree a such that a′ = a ⋃ 0′ = b. He actually proved this result by using the notion of forcing for statements.
Algebraic types and automorphism groups
Authors: Tsuboi Akito 
Published: May 2006

Galois theory states that if L is a certain algebraic extension (called a Galois extension) of a field K, then there is a one-to-one correspondence (called a Galois correspondence) between subfields M, K ⊂ M ⊂ L and subgroups of the automorphism groups of L fixing the elements in K.A subfield of a field L can be considered as a substructure of L in general model theory. However, a substructure is a subset closed under functions which are interpretations of function symbols in a given language, so the notion of substructure may change if we expand the language by adding definable notions. On the other hand a definably closed substructure is a subset which is closed under all definable functions, and it does not change by such expansions. If we are interested in subfields of an algebraically closed field of characteristic 0, these two notions are the same. But in a field of prime characteristic they are not equal. Speaking roughly, a Galois extension of a field K is an extension whose subfields are relatively definably closed. Poizat [4] showed that if a structure M has elimination of imaginaries there is a kind of Galois correspondence between definably closed substructures and subgroups of bijective elementary mappings of M.In this paper, using Poizat's result we study algebraic types. As is well known, one motivation for developing the Galois theory was to show the unsolvability of equations with degree ≥ 5. We want to take this unsolvability as a special case of general phenomena. For this purpose, we introduce several notions which are stronger than mere algebraicity and study relations between these notions and groups of bijective elementary mappings. (See Theorems 3.7 and 3.9.)
Minimal upper bounds for arithmetical degrees
Authors: Kumabe Masahiro 
Published: May 2006

This paper was inspired by Lerman [14] in which he proved various properties of upper bounds for the arithmetical degrees. The degrees which are upper bounds for the arithmetical degrees were first studied by Hodes [5] and Enderton and Putnam [5]. Enderton and Putnam [5] showed that if a is an upper bound for the arithmetical degrees, then a(2) ≥ 0(ω). This area was further studied by Hodes [5], Knight, Lachlan and Soare [12], Jockusch and Simpson [12], and Sacks [17].Enderton and Putnam [5] and Sacks [17] have combined to show that 0(ω) is the least degree in {a(2): a is an upper bound for the arithmetical degrees}. So there seem to be some analogies between the degrees of upper bounds for the arithmetical degrees and the degrees below 0(2). But Sacks [17] showed an important difference between these two structures; namely, the Turing jumps of upper bounds for the arithmetical degrees have no least element. In Lerman [14], a systematic investigation of properties of the jumps of upper bounds for the arithmetical degrees was suggested, which could lead to a definition of the jump operator over the elementary theory of the partial ordering of the degrees. Although Cooper [5] found a degree-theoretic definition of the jump operator, Lerman's plan is still interesting.We say a degree a is generic if there is a representative A of a such that A is Cohen generic for the arithmetical sentences. By Jockusch [5], the statement that A is Cohen generic for the arithmetical sentences is equivalent to saying that for any arithmetical set S of binary strings, there is a σ extended by A such that σ is in S or no extension of σ is in S. First, we investigate the relation between generic degrees and upper bounds for the arithmetical degrees. In the case D(≤ 0′), the set of degrees below 0′, it is well known that any nonrecursive r.e. degree bounds a 1-generic degree (Cohen generic for Σ1 sentences). Jockusch and Ponser [12] showed that any degree a with a″ >T (a ∪ 0′)′ bounds a 1-generic degree.
On one-based theories
Authors: Bouscaren E. , Hrushovski E. 
Published: May 2006

We know from [H1], [H2] that in a stable theory, given a nontrivial locally modular regular type q, one can define a group with generic domination equivalent to q, and that the dependence relation on q can be analyzed in terms of this group. In a stable one-based theory, every regular type is locally modular; hence, this result holds for every nontrivial regular type. We show here that, in fact, in a stable one-based theory, a similar type of construction can be done without the assumption of regularity. More precisely, we show that for any type q, the nontrivial part of q can be analyzed by generics of groups and that any nontrivial relation can be described by affine relations (Theorem A).This construction is then used to answer a question about homogeneity in pairs of models which is still open in the case of arbitrary stable theories (Theorem C).
Axiomatizing a category of categories
Authors: McLarty Colin 
Published: May 2006

AbstractElementary axioms describe a category of categories. Theorems of category theory follow, including some on adjunctions and triples. A new result is that associativity of composition in categories follows from cartesian closedness of the category of categories. The axioms plus an axiom of infinity are consistent iff the axioms for a well-pointed topos with separation axiom and natural numbers are. The theory is not finitely axiomatizable. Each axiom is independent of the others. Further independence and definability results are proved. Relations between categories and sets, the latter defined as discrete categories, are described, and applications to foundations are discussed.