Infinite Cardinals Research Articles

Killing The GCH Everywhere with a Single Real

AbstractShelah-Woodin [10] investigate the possibility of violating instances of GCH through the addition of a single real. In particular they show that it is possible to obtain a failure of CH by adding a single real to a model of GCH, preserving cofinalities. In this article we strengthen their result by showing that it is possible to violate GCH at all infinite cardinals by adding a single real to a model of GCH. Our assumption is the existence of an H(κ+3)-strong cardinal; by work of Gitik and Mitchell [6] it is known that more than an H(κ++)-strong cardinal is required.

A Unary Error Correction Code for the Near-Capacity Joint Source and Channel Coding of Symbol Values from an Infinite Set

A novel Joint Source and Channel Code (JSCC) is proposed, which we refer to as the Unary Error Correction (UEC) code. Unlike existing JSCCs, our UEC facilitates the practical encoding of symbol values that are selected from a set having an infinite cardinality. Conventionally, these symbols are conveyed using Separate Source and Channel Codes (SSCCs), but we demonstrate that the residual redundancy that is retained following source coding results in capacity loss. This loss is found to have a value of 1.11 dB in a particular practical scenario, where Quaternary Phase Shift Keying (QPSK) modulation is employed for transmission over an uncorrelated narrowband Rayleigh fading channel. By contrast, the proposed UEC code can eliminate this capacity loss, or reduce it to an infinitesimally small value. Furthermore, the UEC code has only a moderate complexity, facilitating its employment in practical low-complexity applications.

Testing microscopic discretization

What can we say about the spectra of a collection of microscopic variables when only their coarse-grained sums are experimentally accessible? In this paper, using the tools and methodology from the study of quantum nonlocality, we develop a mathematical theory of the macroscopic fluctuations generated by ensembles of independent microscopic discrete systems. We provide algorithms to decide which multivariate Gaussian distributions can be approximated by sums of finitely valued random vectors. We study non-trivial cases where the microscopic variables have an unbounded range, as well as asymptotic scenarios with infinitely many macroscopic variables. From a foundational point of view, our results imply that bipartite Gaussian states of light cannot be understood as beams of independent d-dimensional particle pairs. It is also shown that the classical description of certain macroscopic optical experiments, as opposed to the quantum one, requires variables with infinite cardinality spectra.

Picking up the pieces: Causal states in noisy data, and how to recover them

Automatic structure discovery is desirable in many Markov model applications where a good topology (states and transitions) is not known a priori. CSSR is an established pattern discovery algorithm for stationary and ergodic stochastic symbol sequences that learns a predictively optimal Markov representation consisting of so-called causal states. By means of a novel algebraic criterion, we prove that the causal states of a simple process disturbed by random errors frequently are too complex to be learned fully, making CSSR diverge. In fact, the causal state representation of many hidden Markov models, representing simple but noise-disturbed data, has infinite cardinality. We also report that these problems can be solved by endowing CSSR with the ability to make approximations. The resulting algorithm, robust causal states (RCS), is able to recover the underlying causal structure from data corrupted by random substitutions, as is demonstrated both theoretically and in an experiment. The algorithm has potential applications in areas such as error correction and learning stochastic grammars.

Automorphisms of infinite Johnson graphs

Let I be a set of infinite cardinality α. For every cardinality β≤α the Johnson graphs Jβ and Jβ are the graphs whose vertices are subsets X⊂I satisfying |X|=β, |I∖X|=α and |X|=α, |I∖X|=β (respectively) and vertices X,Y are adjacent if |X∖Y|=|Y∖X|=1. Note that Jα=Jα and Jβ is isomorphic to Jβ for every β<α. If β is finite then Jβ and Jβ are connected and it is not difficult to prove that their automorphisms are induced by permutations on I. In the case when β is infinite, these graphs are not connected and we determine the restrictions of their automorphisms to connected components.

The (λ,κ)-Freese-Nation Property for Boolean Algebras and Compacta

We study a two-parameter generalization of the Freese-Nation Property of boolean algebras and its order-theoretic and topological consequences. For every regular infinite κ, the (κ,κ)-FN, the (κ + ,κ)-FN, and the κ-FN are known to be equivalent; we show that the family of properties (λ, μ)-FN for λ > μ form a true two-dimensional hierarchy that is robust with respect to coproducts, retracts, and the exponential operation. The \((\kappa,\aleph_0)\)-FN in particular has strong consequences for base properties of compacta (stronger still for homogeneous compacta), and these consequences have natural duals in terms of special subsets of boolean algebras. We show that the \((\kappa,\aleph_0)\)-FN also leads to a generalization of the equality of weight and π-character in dyadic compacta. Elementary subalgebras and their duals, elementary quotient spaces, were originally used to define the (λ,κ)-FN and its topological dual, which naturally generalized from Stone spaces to all compacta, thereby generalizing Ščepin’s notion of openly generated compacta. We introduce a simple combinatorial definition of the (λ,κ)-FN that is equivalent to the original for regular infinite cardinals λ > κ.

Superatomic Boolean algebras constructed from strongly unbounded functions

Using Koszmider's strongly unbounded functions, we show the following consistency result: Suppose that κ, λ are infinite cardinals such that κ++ + ≤ λ, κ<κ = κ and 2κ = κ+, and η is an ordinal with κ+ ≤ η < κ++ and cf(η) = κ+. Then, in some cardinal-preserving generic extension there is a superatomic Boolean algebra \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathcal B$\end{document} such that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{ht}(\mathcal B) = \eta + 1$\end{document}, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{wd}_{\alpha }(\mathcal B) = \kappa$\end{document} for every α < η and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathrm{wd}_{\eta }(\mathcal B) = \lambda$\end{document}(i.e., there is a locally compact scattered space with cardinal sequence 〈κ〉η⌢︁〈λ〉). Especially, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\langle {\omega }\rangle _{{\omega }_1}{}^{\smallfrown } \langle {\omega }_3\rangle$\end{document} and \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\langle {\omega }_1\rangle _{{\omega }_2}{}^{\smallfrown } \langle {\omega }_4\rangle$\end{document} can be cardinal sequences of superatomic Boolean algebras.

The Rank and Generating Set for Iterated Wreath Products of Cyclic Groups

Let (n 1, n 2,…) be an infinite sequence of positive integers greater than 1. We prove that the topological rank of the corresponding infinitely iterated wreath product …≀C n 2 ≀C n 1 of finite cyclic groups is equal to k + 1, where k denotes the topological rank of the direct product C n 2 × C n 3 × … of finite cyclic groups (this rank may be of finite or infinite cardinality, that is, k is a finite number or k = ∞). We construct the minimal set generating topologically the above wreath product as a set of the so-called cyclic automorphisms of a spherically homogeneous rooted tree.

On α-regular archimedean f-rings

Abstract For a commutative ring A with identity, and for infinite cardinals α as well as the symbol ∞, which indicates the situation in which there are no cardinal restrictions, one defines A to be α-regular if for each subset D of A, with |D| &lt; α and de = 0, for any two distinct d, e ∈ D, there is an s ∈ A such that d 2 s = d, for each d ∈ D, and if xd = 0, for each d ∈ D, then xs = 0 This paper studies α-regular archimedean f-rings, relative to lateral α-completeness. The main result is that the operator l(α) that gives the lateral α-completion commutes with b, the reflection that closes an f-ring with respect to bounded inversion. An f-ring is α-regular if and only if it has bounded inversion and is laterally α-complete, and the operator that creates the α-regular hull is r(α) = b · l(α). It is shown that the space mr(α)A of all maximal ℓ-ideals of r(α)A is the same as that of the α-projectable hull. Finally, r(α)A contains the ring of α-quotients, and necessary conditions are given for them to coincide.

There exist subdirectly irreducible commutative basic algebras of an arbitrary infinite cardinality which are not MV-algebras

It was recently proved by P. Wojciechowski that for any infinite cardinal there exists a linearly ordered MV-algebra of this cardinality. Since basic algebras are a (non-associative) generalization of MV-algebras, there rises a natural question if this is true also for basic algebras which are not MV-algebras. Using the construction by P. Wojciechowski and the modified construction by the first author, we can set up certain defectors which enable us to prove the result of the title.

Chains of Group Localizations

We construct long sequences of localization functors L α in the category of abelian groups such that L α ≥ L β for infinite cardinals α < β less than some κ. For sufficiently large free abelian groups F and α < β we have proper inclusions L α F ⊊ L β F.

Infinite Combinatorial Issues Raised by Lifting Problems in Universal Algebra

The critical point between varieties $\mathcal{A}$ and $\mathcal{B}$ of algebras is defined as the least cardinality of the semilattice of compact congruences of a member of $\mathcal{A}$ but of no member of $\mathcal{B}$ , if it exists. The study of critical points gives rise to a whole array of problems, often involving lifting problems of either diagrams or objects, with respect to functors. These, in turn, involve problems that belong to infinite combinatorics. We survey some of the combinatorial problems and results thus encountered. The corresponding problematic is articulated around the notion of a k-ladder (for proving that a critical point is large), large free set theorems and the classical notation (κ,r,λ)→m (for proving that a critical point is small). In the middle, we find λ-lifters of posets and the relation $(\kappa,{<}\lambda)\leadsto P$ , for infinite cardinals κ and λ and a poset P.

Wide scattered spaces and morasses

We show that it is relatively consistent with ZFC that 2 ω is arbitrarily large and every sequence s = 〈 s α : α < ω 2 〉 of infinite cardinals with s α ⩽ 2 ω is the cardinal sequence of some locally compact scattered space.

Strong zero-dimensionality of hyperspaces

For a space X, 2 X denotes the collection of all non-empty closed sets of X with the Vietoris topology, and K ( X ) denotes the collection of all non-empty compact sets of X with the subspace topology of 2 X . The following are known: • 2 ω is not normal, where ω denotes the discrete space of countably infinite cardinality. • For every non-zero ordinal γ with the usual order topology, K ( γ ) is normal iff cf γ = γ whenever cf γ is uncountable. In this paper, we will prove: (1) 2 ω is strongly zero-dimensional. (2) K ( γ ) is strongly zero-dimensional, for every non-zero ordinal γ. In (2), we use the technique of elementary submodels.

Absolute E-rings

A ring R with 1 is called an E-ring if EndZR is ring-isomorphic to R under the canonical homomorphism taking the value 1σ for any σ∈EndZR. Moreover R is an absolute E-ring if it remains an E-ring in every generic extension of the universe. E-rings are an important tool for algebraic topology as explained in the introduction. The existence of an E-ring R of each cardinality of the form λℵ0 was shown by Dugas, Mader and Vinsonhaler (1987) [9]. We want to show the existence of absolute E-rings. It turns out that there is a precise cardinal-barrier κ(ω) for this problem: (The cardinal κ(ω) is the first ω-Erdős cardinal defined in the introduction. It is a relative of measurable cardinals.) We will construct absolute E-rings of any size λ<κ(ω). But there are no absolute E-rings of cardinality ⩾κ(ω). The non-existence of huge, absolute E-rings ⩾κ(ω) follows from a recent paper by Herden and Shelah (2009) [24] and the construction of absolute E-rings R is based on an old result by Shelah (1982) [31] where families of absolute, rigid colored trees (with no automorphism between any distinct members) are constructed. We plant these trees into our potential E-rings with the aim to prevent unwanted endomorphisms of their additive group to survive. Endomorphisms will recognize the trees which will have branches infinitely often divisible by primes. Our main result provides the existence of absolute E-rings for all infinite cardinals λ<κ(ω), i.e. these E-rings remain E-rings in all generic extensions of the universe (e.g. using forcing arguments). Indeed all previously known E-rings (Dugas, Mader and Vinsonhaler, 1987 [9]; Göbel and Trlifaj, 2006 [23]) of cardinality ⩾2ℵ0 have a free additive group R+ in some extended universe, thus are no longer E-rings, as explained in the introduction. Our construction also fills all cardinal-gaps of the earlier constructions (which have only sizes λℵ0). These E-rings are domains and as a by-product we obtain the existence of absolutely indecomposable abelian groups, compare Göbel and Shelah (2007) [22].

Making all cardinals almost Ramsey

We examine combinatorial aspects and consistency strength properties of almost Ramsey cardinals. Without the Axiom of Choice, successor cardinals may be almost Ramsey. From fairly mild supercompactness assumptions, we construct a model of ZF + \({\neg {\rm AC}_\omega}\) in which every infinite cardinal is almost Ramsey. Core model arguments show that strong assumptions are necessary. Without successors of singular cardinals, we can weaken this to an equiconsistency of the following theories: “ZFC + There is a proper class of regular almost Ramsey cardinals”, and “ZF + DC + All infinite cardinals except possibly successors of singular cardinals are almost Ramsey”.

A note on almost disjoint families

We give a short proof of the existence of arbitrarily large chro matic almost disjoint systems. With the same method we solve a problem of P. Szeptycki. One of the questions posed by P. Erdos and A. Hajnal in [2] was, do there exist arbitrarily large chromatic almost disjoint systems of countable sets? This was eventually proved by G. Elekes and G. Hoffmann in [1]. Later the result was extended to systems of sets of larger cardinality ([3]). These proofs applied a Baire category type argument in a topological setting. Here we give a short, direct proof of the Elekes-Hoffmann-Komjath theorem. With the same method we settle a problem of P. Szeptycki ([4]). He asked if it is consistent that for every family K of almost disjoint countable subsets of R there are sets Bo, B1,.... C R such that every H e KH has 1 ,> are infinite cardinals, then there exists an almost disjoint system Kt C [S] for some ISI = 2' with Chr (K) > i'. Proof. Set '1 = {f: a -+ r, a < s,+}. Clearly I (D = 2'. For some of the functions f E 1i we define H(f) c [4?] as follows. Let f: a -K i. If it is possible to find a set X C a of order type ,t, cofinal in a (and so cf(a) = cf(,u)) such that f is constant on X, then set f E 4I* and let H(f) = {ff13: /3 E X} for one such X. If no such set X can be found, make f V 4D* and leave H(f) undefined. Our system is K = {H(f): f E Lemma 1. KH is almost disjoint. Proof. Assume that IH(f ) n H(g)i = p Then H(f) = {f y: y E X} and H(g) = {gi6: 6 E Y} where f: a -* K, g 3 -s, X C a, Y C: are cofinal subsets of Received by the editors November 20, 2007. 2000 Mathematics Subject Classification. Primary 03E05.

Cardinal transfer properties in extender models

We prove that if L [ E ] is a Jensen extender model, then L [ E ] satisfies the Gap-1 morass principle. As a corollary to this and a theorem of Jensen, the model L [ E ] satisfies the Gap-2 Cardinal Transfer Property ( κ + + , κ ) → ( λ + + , λ ) for all infinite cardinals κ and λ .

First-order and counting theories ofω-automatic structures

AbstractThe logicextends first-order logic by a generalized form of counting quantifiers (“the number of elements satisfying … belongs to the setC”). This logic is investigated for structures with an injectivelyω-automatic presentation. If first-order logic is extended by an infinity-quantifier, the resulting theory of any such structure is known to be decidable [6]. It is shown that, as in the case of automatic structures [21], also modulo-counting quantifiers as well as infinite cardinality quantifiers (“there aremany elements satisfying …”) lead to decidable theories. For a structure of bounded degree with injectiveω-automatic presentation, the fragment ofthat contains only effective quantifiers is shown to be decidable and an elementary algorithm for this decision is presented. Both assumptions (ω-automaticity and bounded degree) are necessary for this result to hold.

Maximal irredundance and maximal ideal independence in Boolean algebras

Recall that a subset X of an algebra A is irredundant iff x ∉ 〈X∖{x}〉 for all x ϵ X, where 〈X∖{x}) is the subalgebra generated by X∖{x}. By Zorn's lemma there is always a maximal irredundant set in an algebra. This gives rise to a natural cardinal function Irrmm(A) = min{∣X∣: X is a maximal irredundant subset of A}. The first half of this article is devoted to proving that there is an atomless Boolean algebra A of size 2ω for which Irrmm(A) = ω.A subset X of a BA A is ideal independent iff x ∉ (X∖{x}〉id for all x ϵ X, where 〈X∖{x}〉id is the ideal generated by X∖{x}. Again, by Zorn's lemma there is always a maximal ideal independent subset of any Boolean algebra. We then consider two associated functions. A spectrum functionSspect(A) = {∣X∣: X is a maximal ideal independent subset of A}and the least element of this set, smm(A). We show that many sets of infinite cardinals can appear as Sspect(A). The relationship of Smm to similar “continuum cardinals” is investigated. It is shown that it is relatively consistent that Smm/fin) &lt; 2ω.We use the letter s here because of the relationship of ideal independence with the well-known cardinal invariant spread; see Monk [5]. Namely, sup{∣X∣: X is ideal independent in A} is the same as the spread of the Stone space Ult(A); the spread of a topological space X is the supremum of cardinalities of discrete subspaces.