A Consistency Result on Thin-Tall Superatomic Boolean Algebras

Martı́nez, Juan Carlos

doi:10.2307/2159270

Cited by 6 publications

(2 citation statements)

References 1 publication

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The proof of the following result simplifies the argument given in [13]. However, it is unknown whether V = L implies that for every regular cardinal κ and every ordinal α < κ ++ , there is a (κ, α)-LCS space.…”

Section: Constructions Of Uncountable Widthmentioning

confidence: 86%

Constructions of thin-tall Boolean spaces

Martı́nez

2003

Rev. Mat. Complut.

View full text Add to dashboard Cite

This is an expository paper about constructions of locally compact, Hausdorff, scattered spaces whose Cantor-Bendixson height has cardinality greater than their Cantor-Bendixson width.2000 Mathematics Subject Classification: 54A25, 54A35, 54G12,03E05, 06E15. Key words: Scattered space, thin-tall, partitions, Δ-functions, walks, forcing, constructibility IntroductionThe terminology used in this paper is standard. Undefined set-theoretic terms can be found in [6] or [12]. Undefined topological terms can be found in [5].The Cantor-Bendixson process for topological spaces is defined as follows. Suppose that X is a topological space. Then, for every ordinal α we define the α-derivative of X by: X 0 = X; if α = β + 1, X α is the set of accumulation points of X β ; and if α is a limit, X α = {X β : β < α}. We say that X is scattered, if X α = ∅ for some ordinal α.The Cantor-Bendixson process permits us to split a scattered space into levels. Suppose that X is a scattered space. We define the height of X by ht(X) = the least ordinal α such that X α is finite.For α < ht(X), we write I α (X) = X α \ X α+1 . If x ∈ I α (X), we say that α is the level of x and we write ρ(x) = α. Note that ρ(x) = α means that x is an 1 The preparation of this paper was supported by

show abstract

Section: Constructions Of Uncountable Widthmentioning

confidence: 86%

Constructions of thin-tall Boolean spaces

Martı́nez

2003

Rev. Mat. Complut.

View full text Add to dashboard Cite

show abstract

“…In this section we give partial answers to [M2, Problems 73, 77, 78] showing that, consistently, there is a superatomic Boolean algebra B such that s(B) = inc(B) < irr(B) = Id(B) < Sub(B). The forcing notion we use is a variant of the one of Martinez [Ma92], which in turn was a modification of the forcing notion used in Baumgartner Shelah [BaSh 254]. For more information on superatomic Boolean algebras we refer the reader to Koppelberg [Ko89], Roitman [Rt89] and Monk [M2].…”

Section: Forcing a Superatomic Boolean Algebramentioning

confidence: 99%

More on cardinal invariants of Boolean algebras

Roslanowski

Shelah

2000

Annals of Pure and Applied Logic

View full text Add to dashboard Cite

We address several questions of Donald Monk related to irredundance and spread of Boolean algebras, gaining both some ZFC knowledge and consistency results. We show in ZFC that irr(B0 × B1) = max{irr(B0), irr(B1)}. We prove consistency of the statement "there is a Boolean algebra B such that irr(B) < s(B ⊛ B)" and we force a superatomic Boolean algebra B * such that s(B * ) = inc(B * ) = κ, irr(B * ) = Id(B * ) = κ + and Sub(B * ) = 2 κ + . Next we force a superatomic algebra B0 such that irr(B0) < inc(B0) and a superatomic algebra B1 such that t(B1) > Aut(B1). Finally we show that consistently there is a Boolean algebra B of size λ such that there is no free sequence in B of length λ, there is an ultrafilter of tightness λ (so t(B) = λ) and λ / ∈ Depth Hs (B).

show abstract

On uncountable cardinal sequences for superatomic Boolean algebras

Martı́nez

1995

Arch Math Logic

View full text Add to dashboard Cite

A Consistency Result on Thin-Tall Superatomic Boolean Algebras

Cited by 6 publications

References 1 publication

Constructions of thin-tall Boolean spaces

Constructions of thin-tall Boolean spaces

More on cardinal invariants of Boolean algebras

On uncountable cardinal sequences for superatomic Boolean algebras

Contact Info

Product

Resources

About