Uncountable Chains Research Articles

More about divisibility in βN

AbstractWe continue the research of an extension of the divisibility relation to the Stone‐Čech compactification . First we prove that ultrafilters we call prime actually possess the algebraic property of primality. Several questions concerning the connection between divisibilities in and nonstandard extensions of are answered, providing a few more equivalent conditions for divisibility in . Results on uncountable chains in are proved and used in a construction of a well‐ordered chain of maximal cardinality. Probably the most interesting result is the existence of a chain of type in . Finally, we consider ultrafilters without divisors in and among them find the greatest class.

Closed ideals of operators on the Tsirelson and Schreier spaces

Let B(X) denote the Banach algebra of bounded operators on X, where X is either Tsirelson's Banach space or the Schreier space of order n for some n∈N. We show that the lattice of closed ideals of B(X) has a very rich structure; in particular B(X) contains at least continuum many maximal ideals.Our approach is to study the closed ideals generated by the basis projections. Indeed, the unit vector basis is an unconditional basis for each of the above spaces, so there is a basis projection PN∈B(X) corresponding to each non-empty subset N of N. A closed ideal of B(X) is spatial if it is generated by PN for some N. We can now state our main conclusions as follows:•the family of spatial ideals lying strictly between the ideal of compact operators and B(X) is non-empty and has no minimal or maximal elements;•for each pair I⫋J of spatial ideals, there is a family {ΓL:L∈Δ}, where the index set Δ has the cardinality of the continuum, such that ΓL is an uncountable chain of spatial ideals, ⋃ΓL is a closed ideal that is not spatial, andI⫋L⫋JandL+M‾=J whenever L,M∈Δ are distinct and L∈ΓL, M∈ΓM.

Hyperbolic structures on wreath products

Abstract The study of the poset of hyperbolic structures on a group G was initiated by C. Abbott, S. Balasubramanya and D. Osin, [Hyperbolic structures on groups, Algebr. Geom. Topol. 19 2019, 4, 1747–1835, 10.2140/agt.2019.19.1747]. However, this poset is still very far from being understood, and several questions remain unanswered. In this paper, we give a complete description of the poset of hyperbolic structures on the lamplighter groups wr ⁡ ℤ n ℤ {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} and obtain some partial results about more general wreath products. As a consequence of this result, we answer two open questions regarding quasi-parabolic structures: we give an example of a group G with an uncountable chain of quasi-parabolic structures and prove that the lamplighter groups wr ⁡ ℤ n ℤ {\mathbb{Z}_{n}\mathbin{\mathrm{wr}}\mathbb{Z}} all have finitely many quasi-parabolic structures.

Sequence-singular operators

In this paper we study ℓp-related collections of operators introduced by Beanland and Freeman in [6], on the subject of forming operator ideals. We show that these collections are not always closed under addition, and hence do not form operator ideals. Nevertheless, they allow us to construct an uncountable chain of closed ideals in each of the operator algebras L(ℓ1⊕ℓq), 1<q<∞, and L(ℓ1⊕c0). This finishes answering a longstanding question of Pietsch.

Krull-dimension of the power series ring over a nondiscrete valuation domain is uncountable

Let V be a rank-one nondiscrete valuation domain with maximal ideal M. We prove that the Krull-dimension of V〚X〛V∖(0) is uncountable, and hence the Krull-dimension of V〚X〛 is uncountable. This corresponds to the well-known fact that the Krull-dimension of the ring of entire functions is uncountable. In fact we construct an uncountable chain of prime ideals inside M〚X〛 such that all the members contract to (0) in V. Our method provides a new proof that the Krull-dimension of the ring of entire functions is uncountable. It is also shown that V〚X〛V∖(0) is not even a Prüfer domain, while the ring of entire functions is a Bezout domain. These are answers to Eakin and Sathayeʼs questions. Applying the above results, we show that the Krull-dimension of V〚X〛 is uncountable if V is a nondiscrete valuation domain.

Q-Density points and q-density topologies

Using the notion of q-complete convergence for q>0, of a sequence of measurable functions, we introduce the notion of q-density point of a measurable set, which classifies the density points according to the rate of thickness of the set around them. Also using this notion we create an uncountable chain of topologies between the ordinary and density topology.

Subsemivarieties of 𝑄-algebras

A variety is a class of Banach algebras V V , for which there exists a family of laws { ‖ P ‖ ≤ K p } P \{\|P\|\le K_p\}_P such that V V is precisely the class of all Banach algebras A A which satisfies all of the laws (i.e. for all P P , ‖ P ‖ A ≤ K p ) \|P\|_A\le K_p) . We say that V V is an H H -variety if all of the laws are homogeneous. A semivariety is a class of Banach algebras W W , for which there exists a family of homogeneous laws { ‖ P ‖ ≤ K P } P \{\|P\|\le K_P\}_P such that W W is precisely the class of all Banach algebras A A , for which there exists K &gt; 0 K&gt;0 such that for all homogeneous polynomials P P , ‖ P ‖ A ≤ K i ⋅ K P \|P\|_A\le K^i\cdot K_P , where i = deg ⁡ ( P ) i=\deg (P) . However, there is no variety between the variety of all I Q IQ -algebras and the variety of all I R IR -algebras, which can be defined by homogeneous laws alone. So the theory of semivarieties and the theory of varieties differ significantly. In this paper we shall construct uncountable chains and antichains of semivarieties which are not varieties.

Finite Semigroups Whose Varieties Have Uncountably Many Subvarieties

For several large classes of semigroups we provide a description of all semigroups which generate varieties with uncountably many subvarieties. These include the class of all Rees quotients of free monoids, the class of finite orthodox monoids, the class of monoids of index greater than two, and the class of finite inherently not finitely based semigroups. The first example of a finite, finitely based semigroup generating a variety with uncountably many subvarieties is presented and a number of related results are obtained. All varieties found with uncountably many subvarieties contain uncountable chains of subvarieties with the same ordering as the real numbers.

Metrizability, monotone normality, and other strong properties in trees

Metrizability is an extremely strong property where trees are concerned, and it turns out that in many ways, monotone normality is the appropriate generalization when the trees have uncountable chains. We show that monotone normality is equivalent to the tree being the topological direct sum of ordinal spaces, each of which is a convex chain in the tree. Several metrization theorems are proven, some in ZFC, some just assuming ZF or “ZF + Countable AC”, and still others assuming ZFC-independent axioms, as well as theorems in a similar spirit with monotone normality of the tree as a conclusion. The property of being collectionwise Hausdorff plays a key role, and we obtain partial results on the still unsolved problems of whether it is consistent that every collectionwise Hausdorff tree or every normal tree is monotone normal.

Uncountable chains and antichains of varieties of Banach algebras

Uncountable chains and antichains of varieties of Banach algebras

A Boolean Algebra with Few Subalgebras, Interval Boolean Algebras and Retractiveness

Using ${\diamondsuit _{{\aleph _1}}}$ we construct a Boolean algebra $B$ of power ${\aleph _1}$, with the following properties: (a) $B$ has just ${\aleph _1}$ subalgebras. (b) Every uncountable subset of $B$ contains a countable independent set, a chain of order type $\eta$, and three distinct elements $a,b$ and $c$, such that $a \cap b = c$. (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. $B$ is embeddable in $P(\omega )$. A variant of the construction yields an almost Jónson Boolean algebra. We prove that every subalgebra of an interval algebra is retractive. This answers affirmatively a conjecture of $\text {B}$. Rotman. Assuming $\text {MA}$ or the existence of a Suslin tree we find a retractive $\text {BA}$ not embeddable in an interval algebra. This refutes a conjecture of B. Rotman. We prove that an uncountable subalgebra of an interval algebra contains an uncountable chain or an uncountable antichain. Assuming $\text {CH}$ we prove that the theory of Boolean algebras in Magidor’s and Malitz’s language is undecidable. This answers a question of M. Weese.

A Boolean algebra with few subalgebras, interval Boolean algebras and retractiveness

Using ♢ ℵ 1 {\diamondsuit _{{\aleph _1}}} we construct a Boolean algebra B B of power ℵ 1 {\aleph _1} , with the following properties: (a) B B has just ℵ 1 {\aleph _1} subalgebras. (b) Every uncountable subset of B B contains a countable independent set, a chain of order type η \eta , and three distinct elements a , b a,b and c c , such that a ∩ b = c a \cap b = c . (a) refutes a conjecture of J. D. Monk, (b) answers a question of R. McKenzie. B B is embeddable in P ( ω ) P(\omega ) . A variant of the construction yields an almost Jónson Boolean algebra. We prove that every subalgebra of an interval algebra is retractive. This answers affirmatively a conjecture of B \text {B} . Rotman. Assuming MA \text {MA} or the existence of a Suslin tree we find a retractive BA \text {BA} not embeddable in an interval algebra. This refutes a conjecture of B. Rotman. We prove that an uncountable subalgebra of an interval algebra contains an uncountable chain or an uncountable antichain. Assuming CH \text {CH} we prove that the theory of Boolean algebras in Magidor’s and Malitz’s language is undecidable. This answers a question of M. Weese.

Chains and antichains in

Consider the following propositions:(A) Every uncountable subset of contains an uncountable chain or antichain (with respect to ⊆).(B) Every uncountable Boolean algebra contains an uncountable antichain (i.e., an uncountable set of pairwise incomparable elements).Until quite recently, relatively little was known about these propositions. The oldest result, due to Kunen [4] and the author independently, asserts that if the Continuum Hypothesis (CH) holds, then (A) is false. In fact there is a counter-example 〈Aα: α &lt; ω1〉 such that α &lt; β implies Aβ −Aα is finite. Kunen also observed that Martin's Axiom (MA) + ¬CH implies that no such counterexample 〈Aα: α &lt; ω1〉 exists.Much later, Komjáth and the author [2] showed that ◊ implies the existence of several kinds of uncountable Boolean algebras with no uncountable chains or antichains. Similar results (but motivated quite differently) were obtained independently by Rubin [5]. Berney [3] showed that CH implies that (B) is false, but his algebra has uncountable chains. Finally, Shelah showed very recently that CH implies the existence of an uncountable Boolean algebra with no uncountable chains or antichains.Except for Kunen's result cited above, the only result in the other direction was the theorem, due also to Kunen, that MA + ¬CH implies that any uncountable subset of with no uncountable antichains must have both ascending and decending infinite sequences under ⊆.

Chain Conditions for Modular Lattices with Finite Group Actions

This paper establishes the following combinatorial result concerning the automorphisms of a modular lattice.THEOREM. Let M be a modular lattice and let G be a finite subgroup of the automorphism group of M. If the sublattice, MG, of (common) fixed points (under G) satisfies any of a large class of chain conditions, then M satisfies the same chain condition. Some chain conditions in this class are the following: the ascending chain condition; the descending chain condition; Krull dimension; the property of having no uncountable chains, no chains order-isomorphic to the rational numbers; etc.

SOME QUESTIONS OF SPECTRAL SYNTHESIS ON SPHERES

This paper considers the Banach algebra with the usual norm and convolution as multiplication. A characterization is given for closed ideals of which are rotation invariant and have as spectrum, in terms of annihilators of certain collections of pseudomeasures. The main result of the paper is connected with a construction which yields an uncountable chain of closed ideals intermediate between neighboring invariant closed ideals with spectrum . This construction associates an ideal with a closed subset . It is shown that if then . Another result is the lack of a continuous projection from the largest to the smallest ideal when , and when , from an invariant ideal onto the neighboring smaller invariant ideal. A certain algebra of functions on the sphere which arises naturally in the construction of the intermediate ideals is also studied.Bibliography: 18 items.