Cardinal sequences of length < ω₂ under GCH

István Juhász, Lajos Soukup, and William Weiss. "Cardinal sequences of length < ω₂ under GCH." Fundamenta Mathematicae 189.1 (2006): 35-52. <http://eudml.org/doc/282701>.

@article{IstvánJuhász2006,
abstract = {Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put $_\{λ\}(α) = \{s ∈ (α): s(0) = λ = min[s(β): β < α]\}$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $λ₀i > λ₁ > ... > λ_\{n-1\}$ and ordinals $α₀,...,α_\{n-1\}$ such that $α = α₀ + ⋯ +α_\{n-1\}$ and $f = f₀⏜f₁⏜...⏜f_\{n-1\}$ where each $f_i ∈ _\{λ_i\}(α_i)$. Under GCH we prove that if α < ω₂ then (i) $_\{ω\}(α) = \{s ∈ ^\{α\}\{ω,ω₁\}: s(0) = ω\}$; (ii) if λ > cf(λ) = ω, $_\{λ\}(α) = \{s ∈ ^\{α\}\{λ,λ⁺\}: s(0) = λ, s^\{-1\}\{λ\} is ω₁-closed in α\}$; (iii) if cf(λ) = ω₁, $_\{λ\}(α) = \{s ∈ ^\{α\}\{λ,λ⁺\}: s(0) = λ, s^\{-1\}\{λ\} is ω-closed and successor-closed in α\}$; (iv) if cf(λ) > ω₁, $_\{λ\}(α) = ^\{α\}\{λ\}$. This yields a complete characterization of the classes (α) for all α < ω₂, under GCH.},
author = {István Juhász, Lajos Soukup, William Weiss},
journal = {Fundamenta Mathematicae},
keywords = {compact scattered space; cardinal sequence; superatomic Boolean algebra; GCH},
language = {eng},
number = {1},
pages = {35-52},
title = {Cardinal sequences of length < ω₂ under GCH},
url = {http://eudml.org/doc/282701},
volume = {189},
year = {2006},
}

TY - JOUR
AU - István Juhász
AU - Lajos Soukup
AU - William Weiss
TI - Cardinal sequences of length < ω₂ under GCH
JO - Fundamenta Mathematicae
PY - 2006
VL - 189
IS - 1
SP - 35
EP - 52
AB - Let (α) denote the class of all cardinal sequences of length α associated with compact scattered spaces (or equivalently, superatomic Boolean algebras). Also put $_{λ}(α) = {s ∈ (α): s(0) = λ = min[s(β): β < α]}$. We show that f ∈ (α) iff for some natural number n there are infinite cardinals $λ₀i > λ₁ > ... > λ_{n-1}$ and ordinals $α₀,...,α_{n-1}$ such that $α = α₀ + ⋯ +α_{n-1}$ and $f = f₀⏜f₁⏜...⏜f_{n-1}$ where each $f_i ∈ _{λ_i}(α_i)$. Under GCH we prove that if α < ω₂ then (i) $_{ω}(α) = {s ∈ ^{α}{ω,ω₁}: s(0) = ω}$; (ii) if λ > cf(λ) = ω, $_{λ}(α) = {s ∈ ^{α}{λ,λ⁺}: s(0) = λ, s^{-1}{λ} is ω₁-closed in α}$; (iii) if cf(λ) = ω₁, $_{λ}(α) = {s ∈ ^{α}{λ,λ⁺}: s(0) = λ, s^{-1}{λ} is ω-closed and successor-closed in α}$; (iv) if cf(λ) > ω₁, $_{λ}(α) = ^{α}{λ}$. This yields a complete characterization of the classes (α) for all α < ω₂, under GCH.
LA - eng
KW - compact scattered space; cardinal sequence; superatomic Boolean algebra; GCH
UR - http://eudml.org/doc/282701
ER -