The Chang-Łoś-Suszko theorem in a topological setting

Bankston, Paul

doi:10.1007/s00153-004-0238-y

Cited by 10 publications

(12 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…is an enforceable property and e.c. models of T are hereditarily indecomposable (see [1] again), we have the following corollary. The following corollary was the original motivation for this work.…”

Section: The Pseudoarcmentioning

confidence: 77%

“…The pseudoarc P is the unique metrizable continuum that is both hereditarily indecomposable and chainable. 8 In [1], it was shown that hereditary indecomposability is an ∀∃-property of models of T . 9 On the other hand, the main result of [7] shows that chainability is a sup inf-property.…”

Section: The Pseudoarcmentioning

confidence: 99%

“…, x n , namely ϕ(x) := sup{|ϕ(a)| : A is an L-structure and a ∈ A n }. By a restricted L-formula, we mean an L-formula constructed using only the unary connectives 1 2 and ¬ (here, ¬r := 1 − r ) and the binary connective (truncated addition). We will need the following fact (see [4,Theorem 6.3 and Proposition 6.6]).…”

Section: Introductionmentioning

confidence: 99%

“…We play this game for ω many steps. 1 At the end of this game, we have enumerated some countable, satisfiable set of expressions. Provided that the players behave, they can ensure that the final set of expressions yields complete information about all * -polynomials over the variables C (that is, for each * -polynomial p(c), there should be a unique r such that the play of the game implies that p(c) 2 = r ) and that these data describes a countable, dense * -subalgebra of a unique tracial von Neumann algebra, which is often called the compiled structure.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Enforceable Operator Algebras

Goldbring¹

2019

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

We adapt the classical notion of building models by games to the setting of continuous model theory. As an application, we study to what extent canonical operator algebras are enforceable models. For example, we show that the hyperfinite II $_{1}$ factor is an enforceable II $_{1}$ factor if and only if the Connes Embedding Problem has a positive solution. We also show that the set of continuous functions on the pseudoarc is an enforceable model of the theory of unital, projectionless, abelian $\text{C}^{\ast }$ -algebras and use this to show that it is the prime model of its theory.

show abstract

Section: The Pseudoarcmentioning

confidence: 77%

Section: The Pseudoarcmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Enforceable Operator Algebras

Goldbring¹

2019

J. Inst. Math. Jussieu

View full text Add to dashboard Cite

show abstract

“…Ultracoproducts both preserve and reflect unicoherence of continua [8,Theorem 5.1]; also if { ∈ : is not hereditarily unicoherent} ∈ , then it is easy to form two regular subcontinua of → with disconnected intersection. So hereditary unicoherence is reflected by the ultracoproduct construction, but we do not currently know whether it is also preserved.…”

Section: Remark 36mentioning

confidence: 99%

Ultracoproduct continua and their regular subcontinua

Bankston

2016

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

We continue our study of ultracoproduct continua, focusing on the role played by the regular subcontinua-those subcontinua which are themselves ultracoproducts. Regular subcontinua help us in the analysis of intervals, composants, and noncut points of ultracoproduct continua. Also, by identifying two points when they are contained in the same regular subcontinua, we naturally generalize the partition of a standard subcontinuum of ℍ � into its layers.

show abstract

On the first-order expressibility of lattice properties related to unicoherence in continua

Bankston

2011

Arch. Math. Logic

Self Cite

View full text Add to dashboard Cite

Abstract. Many properties of compacta have "textbook" definitions which are phrased in lattice-theoretic terms that, ostensibly, apply only to the full closed-set lattice of a space. We provide a simple criterion for identifying such definitions that may be paraphrased in terms that apply to all lattice bases of the space, thereby making model-theoretic tools available to study the defined properties. In this note we are primarily interested in properties of continua related to unicoherence; i.e., properties that speak to the existence of "holes" in a continuum and in certain of its subcontinua. the expressibility lemmaWe continue our study [4] of compacta (i.e., compact Hausdorff spaces), especially of continua (i.e., connected compacta), from the perspective of model theory. Many of the classic definitions of properties pertaining to compacta are easily phrased in finitistic lattice-theoretic terms involving closed sets. (It is convenient to use closed-rather than, say, open-set phraseology because, in the setting of compacta, closed sets are compacta themselves.) For any space X, let F (X) be the collection of all closed subsets of X, viewed as a bounded lattice; i.e., as a special structure for the first-order alphabet L = {⊔, ⊓, ⊥, ⊤}. As one would expect, ⊔ is interpreted as union, ⊓ as intersection, ⊥ as ∅, and ⊤ as X. A sublattice A of F (X) is a lattice base for X if every member of F (X) is an intersection of members of A.What makes the notion of lattice base so important in our study of compacta is the following Wallman-style representation theorem (see [11]): There is a sentence in the first-order language L ωω over L, whose models are precisely the lattices that are isomorphic to the lattice bases for compacta.

show abstract

The Chang-Łoś-Suszko theorem in a topological setting

Cited by 10 publications

References 12 publications

Enforceable Operator Algebras

Enforceable Operator Algebras

Ultracoproduct continua and their regular subcontinua

On the first-order expressibility of lattice properties related to unicoherence in continua

Contact Info

Product

Resources

About