Cardinality Of The Continuum Research Articles

Order-isomorphic [formula omitted]-orderings in Cohen extensions

In this paper we prove that, in the Cohen extension (adding ℵ 2 -generic reals) of a model M of ZFC+CH containing a simplified ( ω 1 , 1 ) -morass, η 1 -orderings without endpoints having cardinality of the continuum, and satisfying specified technical conditions, are order-isomorphic. Furthermore, any order-isomorphism in M between countable subsets of the η 1 -orderings can be extended to an order-isomorphism between the η 1 -orderings in the Cohen extension of M . We use the simplified ( ω 1 , 1 ) -morass, and commutativity conditions with morass maps on terms in the forcing language, to extend countable partial functions on terms in the forcing language that are forced in all generic extensions to be order-preserving injections. This technique provides for the construction of functions in Cohen extensions adding ℵ 2 generic reals for which the only known arguments require transfinite constructions of order type no greater than ω 1 in models of ZFC+CH. The specific example presented in this paper is an extension of Tarski’s classic result that in models of ZFC+CH, η 1 -orderings are order-isomorphic.

Some topolocial properties of a random subset of Lebesgue measure zero

The set of fast points associated with the process Y(s) = inf &ldet; x(t) &rdet; t&#8805 8 z ) where x(t) is a standard Brownian motion, is considered. This random subset of exceptional time set is shown to be everywhere dense of the second category and has the cardinality of the continuum. Journal of the Nigerian Association of Mathematical Physics Vol. 9 2005: pp. 57-60

Ordered fields and $${{{\rm \L}\Pi\frac{1}{2}}}$$ -algebras

In this work we further explore the connection between $${{{\rm \L}\Pi\frac{1}{2}}}$$-algebras and ordered fields. We show that any two $${{{\rm \L}\Pi\frac{1}{2}}}$$-chains generate the same variety if and only if they are related to ordered fields that have the same universal theory. This will yield that any $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain generates the whole variety if and only if it contains a subalgebra isomorphic to the $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain of real algebraic numbers, that consequently is the smallest $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain generating the whole variety. We also show that any two different subalgebras of the $${{{\rm \L}\Pi\frac{1}{2}}}$$-chain over the real algebraic numbers generate different varieties. This will be exploited in order to prove that the lattice of subvarieties of $${{{\rm \L}\Pi\frac{1}{2}}}$$-algebras has the cardinality of the continuum. Finally, we will also briefly deal with some model-theoretic properties of $${{{\rm \L}\Pi\frac{1}{2}}}$$-chains related to real closed fields, proving quantifier-elimination and related results.

On classifying monotone complete algebras of operators

We give a classification of “small” monotone complete C *-algebras by order properties. We construct a corresponding semigroup. This classification filters out von Neumann algebras; they are mapped to the zero of the classifying semigroup. We show that there are 2 c distinct equivalence classes (where c is the cardinality of the continuum). This remains true when the classification is restricted to special classes of monotone complete C *-algebras e.g. factors, injective factors, injective operator systems and commutative algebras which are subalgebras of l∞. Some examples and applications are given.

On periodic groups of odd period n ≥ 1003

In the paper, using the Adyan-Lysenok theorem claiming that, for any odd number n ≥ 1003, there is an infinite group each of whose proper subgroups is contained in a cyclic subgroup of order n, it is proved that the set of groups with this property has the cardinality of the continuum (for a given n). Further, it is proved that, for m ≥ k ≥ 2 and for any odd n ≥ 1003, the m-generated free n-periodic group is residually both a group of the above type and a k-generated free n-periodic group, and it does not satisfy the ascending and descending chain conditions for normal subgroups either.

Dependent sets of a family of relations of full measure on a probability space

For a probability space (X, B, µ) a subfamily F of the σ-algebra B is said to be a regular base if every B ∈ B can be arbitrarily approached by some member of F which contains B in the sense of the measure theory. Assume that {R γ } γ∈Γ is a countable family of relations of the full measure on a probability space (X, B, µ), i.e. for every γ ∈ Γ there is a positive integer s γ such that R γ ⊂ \(X^{s_\gamma } \) with \(\mu ^{s_\gamma } \) (R γ ) = 1. In the present paper we show that if (X, B, µ) has a regular base, the cardinality of which is not greater than the cardinality of the continuum, then there exists a set K ⊂ X with µ*(K) = 1 such that (x 1, …, \(x_{^{s_\gamma } } \)) ∈ R γ for any γ ∈ Γ and for any s γ distinct elements x 1, …, \(x_{^{s_\gamma } } \) of K, where µ* is the outer measure induced by the measure µ. Moreover, an application of the result mentioned above is given to the dynamical systems determined by the iterates of measure-preserving transformations.

UNCOUNTABLY MANY VARIETIES OF COMPLETELY SIMPLE SEMIGROUPS WITH METABELIAN SUBGROUPS

It has been proved by D. E. Cohen [1] that the lattice of all varieties of metabelian groups is countable. In this paper, we show that the lattice of all varieties of completely simple semigroups with metabelian subgroups has the cardinality of the continuum. M. Petrich and N. R. Reilly have introduced in [6] the notion of near varieties of idempotent generated completely simple semigroups. The mapping assigning to every variety is a complete lattice homomorphism of the lattice of all varieties of completely simple semigroups onto the lattice of all near varieties of idempotent generated completely simple semigroups. In this paper we show that, in fact, the lattice of all near varieties of idempotent generated completely simple semigroups with metabelian subgroups has itself the cardinality of the continuum.

Non‐equivalent greedy and almost greedy bases in lp

For 1 < p < 8 and p ? 2 we construct a family of mutually non‐equivalent greedy bases in lp having the cardinality of the continuum. In fact, no basis from this family is equivalent to a rearranged subsequence of any other basis thereof. We are able to extend this statement to the spaces Lp and H1. Moreover, the technique used in the proof adapts to the setting of almost greedy bases where similar results are obtained.

Forcing hereditarily separable compact-like group topologies on Abelian groups

Let c denote the cardinality of the continuum. Using forcing we produce a model of ZFC + CH with 2 c “arbitrarily large” and, in this model, obtain a characterization of the Abelian groups G (necessarily of size at most 2 c ) which admit: (i) a hereditarily separable group topology, (ii) a group topology making G into an S-space, (iii) a hereditarily separable group topology that is either precompact, or pseudocompact, or countably compact (and which can be made to contain no infinite compact subsets), (iv) a group topology making G into an S-space that is either precompact, or pseudocompact, or countably compact (and which also can be made without infinite compact subsets if necessary). As a by-product, we completely describe the algebraic structure of the Abelian groups of size at most 2 c which possess, at least consistently, a countably compact group topology (without infinite compact subsets, if desired). We also put to rest a 1980 problem of van Douwen about the cofinality of the size of countably compact Abelian groups.

Continuum-many Boolean algebras of the form Borel

Abstract.We examine the question of how many Boolean algebras, distinct up to isomorphism, that are quotients of the powerset of the naturals by Borel ideals, can be proved to exist in ZFC alone. The maximum possible value is easily seen to be the cardinality of the continuum ; earlier work by Ilijas Farah had shown that this was the value in models of Martin's Maximum or some similar forcing axiom, but it was open whether there could be fewer in models of the Continuum Hypothesis.We develop and apply a new technique for constructing many ideals whose quotients must be nonisomorphic in any model of ZFC. The technique depends on isolating a kind of ideal, called shallow, that can be distinguished from the ideal of all finite sets even after any isomorphic embedding, and then piecing together various copies of the ideal of all finite sets using distinct shallow ideals. In this way we are able to demonstrate that there are continuum-many distinct quotients by Borel ideals, indeed by analytic P-ideals, and in fact that there is in an appropriate sense a Borel embedding of the Vitali equivalence relation into the equivalence relation of isomorphism of quotients by analytic P-ideals. We also show that there is an uncountable definable wellordered collection of Borel ideals with distinct quotients.

On Homeomorphisms of Effective Topological Spaces

We study effective presentations and homeomorphisms of effective topological spaces. By constructing a functor from the category of computable models into the category of effective topological spaces, we show in particular that there exist homeomorphic effective topological spaces admitting no hyperarithmetical homeomorphism between them and there exist effective topological spaces whose autohomeomorphism group has the cardinality of the continuum but whose only hyperarithmetical autohomeomorphism is trivial. It is also shown that if the group of autohomeomorphisms of a hyperarithmetical topological space has cardinality less than 2ω then this group is hyperarithmetical. We introduce the notion of strong computable homeomorphism and solve the problem of the number of effective presentations of T0-spaces with effective bases of clopen sets with respect to strong homeomorphisms.

Infinite Independent Systems of Identities of an Associative Algebra over an Infinite Field of Characteristic Two

Let $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ be the variety of associative (special Jordan, respectively) algebras over an infinite field of characteristic 2 defined by the identity ((((x 1,x 2),x 3), ((x 4,x 5),x 6)), (x 7,x 8)) = 0 (((x 1 x 2 · x 3)(x 4 x 5 · x 6))(x 7 x 8) = 0, respectively). In this paper, we construct infinite independent systems of identities in the variety $$\mathfrak{B}$$ ( $$\mathfrak{D}$$ , respectively). This implies that the set of distinct nonfinitely based subvarieties of the variety $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ has the cardinality of the continuum and that there are algebras in $$\mathfrak{B}$$ $$\user2{(}\mathfrak{D}\user2{)}$$ with undecidable word problem.

Small Opaque Sets

A set in a separable metric space is called \emph{Borel--opaque} if it meets every Borel set of positive topological dimension. We show that if there is a set of reals with cardinality of the continuum and universal measure zero, then each separable space contains a Borel--opaque set that is of universal measure zero. Similar results hold for opaque sets that are perfectly meager, \la sets, \lap sets etc., and can be extended to some nonseparable spaces. On the other hand, we show that a \s set is zero--dimensional. Using opacity we also construct universal measure zero sets of positive Hausdorff dimension.

A finitely generated variety of combinatorial inverse semigroups having uncountably many subvarieties

A finite combinatorial inverse semigroup Θ of moderate size is presented such that the variety of combinatorial inverse semigroups generated by Θ possesses the following properties. The lattice of all subvarieties of this variety has the cardinality of the continuum. Moreover, this semigroup Θ, and hence also the variety it generates and its subvarieties, all have E-unitary covers over any non-trivial variety of groups. This indicates that the mentioned uncountable sublattice appears quite near the bottom of the lattice of all varieties of combinatorial inverse semigroups.

Open colorings, the continuum and the second uncountable cardinal

The purpose of this article is to analyze the cardinality of the continuum using Ramsey theoretic statements about open colorings or open coloring axioms. In particular it will be shown that the conjunction of two well-known axioms, OCA [ARS] and OCA [T] , implies that the size of the continuum is N 2 .

Markets with Many More Agents than Commodities: Aumann's “Hidden” Assumption

We address a question posed by J.-F. Mertens and show that, indeed, R. J. Aumann's classical existence and equivalence theorems depend on there being “many more agents than commodities.” We show that for an arbitrary atomless measure space of agents there is a fixed non-separable infinite dimensional commodity space in which one can construct an economy that satisfies all the standard assumptions but which has no equilibrium, a core allocation that is not Walrasian, and a Pareto efficient allocation that is not a valuation equilibrium. We identify the source of the failure as the requirement that allocations be strongly measurable. Our main example is set in a commodity–measure space pair that displays an “acute scarcity” of strongly measurable allocations—where strong measurability necessitates that consumer choices be closely correlated no matter the prevailing prices. This makes the core large since there may not be any strongly measurable improvements even though there are many weakly measurable strict improvements. Moreover, at some prices the aggregate demand correspondence is empty since disaggregated demand has no strongly measurable selections, though it does have weakly measurable selections. We note that our example can be constructed in any vector space whose dimension is greater than the cardinality of the continuum—that is, whenever there are at least as many commodities as agents. We also prove a positive core equivalence result for economies in non-separable commodity spaces. Journal of Economic Literature Classification Numbers: C62, C71, D41, D50.

Pseudo-finite dimensional representations of $ sl(2,k) $

Let k be an algebraically closed field of characteristic zero and L = sl(2,k) the Lie algebra of 2 × 2 traceless matrices over k. It is shown that there exists a von Neumann regular extension \( U(L) \subseteq U'(L) \) of the universal enveloping algebra, which is an epimorphism in the category of rings. The article is devoted to the study of the simple representations of U'(L), which may be topologized via the Ziegler topology on the set of injective indecomposable representations of U'(L) or via the Jacobson topology on the set of primitive ideals. These two topologies coincide and the finite dimensional simple representations of L form a dense, discrete and open subset. The field of fractions K(L) of the universal enveloping algebra is another simple representation of U'(L). If the point K(L) is removed from the Ziegler spectrum of U'(L), one obtains a compact totally disconnected topological space, which has the cardinality of the continuum. It is also shown that the lattice of ideals of U'(L) is isomorphic to the lattice of open subsets. The epimorphic ring extension \( U(L) \subseteq U'(L) \) is used to find an axiomatization of the finite dimensional representations of L in the language of left U(L)-modules. A representation V of L is called pseudo-finite dimensional if it satisfies these axioms. It is shown that a representation V of L is pseudo-finite dimensional if and only if for every central idempotent \( e \in U'(L) \) for which \( eK(L) \neq 0 \), whenever the subrepresentation eV is nonzero, then it has a nonzero highest weight space.

Properties of the Absolute That Affect Smoothness of Flows on Closed Surfaces

Let \(M_g^2\) be a closed orientable surface of genus \(g \geqslant 2\), endowed with the structure of a Riemann manifold of constant negative curvature. For the universal covering \(\Delta\), there is the notion of absolute, each of whose points determines an asymptotic direction of a bundle of parallel equidirected geodesics. In the paper it is proved that there is a set \(U_g\) on the absolute having the cardinality of the continuum and such that if an arbitrary flow on \(M_g^2\) has a semitrajectory whose covering has asymptotic direction defined by a point from \(U_g\), then this flow is not analytical and has infinitely many stationary points.

ON I-ASYMMETRY

Sets of approximative asymmetry in the sense of category are introduced. The following theorem is proved. If $f : \mathbb{R}\to \mathbb{R}$ is a function, then the set of $\mathcal{I}$-asymmetry points of $f$ is of the type $F_{\sigma \delta \sigma}$ and is $\sigma$-well-porous. This illustrates the difference between measure and category. We give an example of a function with the set of $\mathcal{I}-$asymmetry points of the cardinality of the continuum.

A Graph-Theoretic Approach to the Unique Midset Property of Metric Spaces

A metric space X has the unique midset property if there is a topology-preserving metric d on X such that for every pair of distinct points x, y there is one and only one point p such that d(x, p) = d(y, p). The following are proved. (1) The discrete space with cardinality n has the unique midset property if and only if n ≠ 2, 4 and n ⩽ c, where c is the cardinality of the continuum. (2) If D is a discrete space with cardinality not greater than c, then the countable power DN of D has the unique midset property. In particular, the Cantor set and the space of irrational numbers have the unique midset property. A finite discrete space with n points has the unique midset property if and only if there is an edge colouring φ of the complete graph Kn such that for every pair of distinct vertices x, y there is one and only one vertex p such that φ(xp) = φ(yp). Let ump(Kn) be the smallest number of colours necessary for such a colouring of Kn. The following are proved. (3) For each k ⩾ 0, ump(K2k+1) = k. (4) For each k ⩾ 3, k ⩽ ump(K2k) ⩽ 2k−1.