Superatomic Boolean algebras constructed from morasses

Koepke, Peter; Martı́nez, Juan Carlos

doi:10.2307/2275766

Cited by 18 publications

(16 citation statements)

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“…If κ and λ are cardinals, then ℘κ(λ) = {X ⊆ λ : |X| < κ}. 2 Formally, in the original terminology of [59] and [58] a (κ, κ + )-cardinal is a neat simplified (κ, 1)-morass, however in many following papers e.g., [36], [22], [16] a (κ, 1)-morass is what formally Velleman called an expanded neat simplified morass. This shift towards the expanded version (already present in the above papers of Velleman) is justified by the fact that the above authors do all the calculations with the expanded versions i.e., use maps rather than sets.…”

Section: Introductionmentioning

confidence: 99%

“…In fact, the heart of the philosophy of (κ, κ + )-cardinals is to view κ + as built from fragments of sizes less than κ so that a recursive construction of a structure of size κ + does not have to deal with the case of an intermediate construction having size κ. As the main example we propose a direct construction of a κ-thin tall Boolean algebra due to Koepke and Martinez [22], instead of using the morass version of Martin's axiom developed by Velleman in [58] and [59]. In a sense the claim of this section is that we can do very well without this version of Martin's axiom, if we represent appropriately the intermediate structures.…”

Section: Introductionmentioning

confidence: 99%

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On constructions with 2-cardinals

Koszmider

2017

Arch. Math. Logic

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We propose developing the theory of consequences of morasses relevant in mathematical applications in the language alternative to the usual one, replacing commonly used structures by families of sets originating with Velleman's neat simplified morasses called 2-cardinals. The theory of related trees, gaps, colorings of pairs and forcing notions is reformulated and sketched from a unifying point of view with the focus on the applicability to constructions of mathematical structures like Boolean algebras, Banach spaces or compact spaces.The paper is dedicated to the memory of Jim Baumgartner whose seminal joint paper [5] with Saharon Shelah provided a critical mass in the theory in question.A new result which we obtain as a side product is the consistency of the existence of a function f : [λ ++ ] 2 → [λ ++ ] ≤λ with the appropriate λ + -version of property ∆ for regular λ ≥ ω satisfying λ <λ = λ.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On constructions with 2-cardinals

Koszmider

2017

Arch. Math. Logic

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“…The interested reader will find a list of consistency results on this type of cardinal sequences in [5,8,9] and in the survey paper [11]. However, no general consistency result is known for this type of cardinal sequences.…”

Section: Introductionmentioning

confidence: 99%

A consistency result on cardinal sequences of scattered Boolean spaces

Martı́nez

2005

MLQ - Math. Log. Quart.

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“…We focus on the case of κ = ω because it is the most interesting and also because we would like to avoid discussing here constructions that require additional set-theoretic assumptions. For example, the only known constructions of commutative κ-thin-tall algebras for κ > ω use such assumptions ( [38]) and for κ-short-wide algebras, it is known that such assumptions are necessary (Theorem 3.4 of [3]). Also it is known that already the existence of a thin-very tall commutative C * -algebra is independent of the usual axioms ( [37,4]).…”

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confidence: 99%

Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras

Ghasemi

Koszmider

2018

Topology and its Applications

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We analyze the sequence obtained by consecutive applications of the Cantor-Bendixson derivative for a noncommutative scattered C * -algebra A, using the ideal I At (A) generated by the minimal projections of A. With its help, we present some fundamental results concerning scattered C * -algebras, in a manner parallel to the commutative case of scattered compact or locally compact Hausdorff spaces and superatomic Boolean algebras. It also allows us to formulate problems which have motivated the "cardinal sequences" programme in the classical topology, in the noncommutative context. This leads to some new constructions of noncommutative scattered C * -algebras and new open problems. In particular, we construct a type I C * -algebra which is the inductive limit of stable ideals Aα, along an uncountable limit ordinal λ, such that A α+1 /Aα is * -isomorphic to the algebra of all compact operators on a separable Hilbert space and A α+1 is σ-unital and stable for each α < λ, but A is not stable and where all ideals of A are of the form Aα. In particular, A is a nonseparable C * -algebra with no ideal which is maximal among the stable ideals. This answers a question of M. Rørdam in the nonseparable case. All the above C * -algebras Aαs and A satisfy the following version of the definition of an AF algebra: any finite subset can be approximated from a finite-dimensional subalgebra. Two more complex constructions based on the language developed in this paper are presented in separate papers [25,26].Proof. We prove J α = I α ⊗ K(ℓ 2 ) by induction on α ≤ ht(A). For α = 1 this follows from Lemma 5.2. At a successor ordinal by Lemma 5.2 and the inductive assumption we have J α+1 /J α =

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Superatomic Boolean algebras constructed from morasses

Cited by 18 publications

References 10 publications

On constructions with 2-cardinals

On constructions with 2-cardinals

A consistency result on cardinal sequences of scattered Boolean spaces

Noncommutative Cantor–Bendixson derivatives and scattered C⁎-algebras

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