Combinatorics of dense subsets of the rationals

B. Balcar, F. Hernández-Hernández, and M. Hrušák. "Combinatorics of dense subsets of the rationals." Fundamenta Mathematicae 183.1 (2004): 59-80. <http://eudml.org/doc/283218>.

@article{B2004,
abstract = {We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants $_\{ℚ\}$, $_\{ℚ\}$, $_\{ℚ\}$, $_\{ℚ\}$, $_\{ℚ\}$, $_\{ℚ\}$ describing properties of Dense(ℚ). These invariants satisfy $_\{ℚ\}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$. We compare them with their analogues in the well studied Boolean algebra (ω)/fin. We show that $ℚ = p$, $ℚ = t$ and $ℚ = i$, whereas $ℚ > h$ and $ℚ > r$ are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of ,ℚ. In particular, we show that $non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.},
author = {B. Balcar, F. Hernández-Hernández, M. Hrušák},
journal = {Fundamenta Mathematicae},
keywords = {rational numbers; nowhere dense ideal; distributivity of Boolean algebras; cardinal invariants of the continuum},
language = {eng},
number = {1},
pages = {59-80},
title = {Combinatorics of dense subsets of the rationals},
url = {http://eudml.org/doc/283218},
volume = {183},
year = {2004},
}

TY - JOUR
AU - B. Balcar
AU - F. Hernández-Hernández
AU - M. Hrušák
TI - Combinatorics of dense subsets of the rationals
JO - Fundamenta Mathematicae
PY - 2004
VL - 183
IS - 1
SP - 59
EP - 80
AB - We study combinatorial properties of the partial order (Dense(ℚ),⊆). To do that we introduce cardinal invariants $_{ℚ}$, $_{ℚ}$, $_{ℚ}$, $_{ℚ}$, $_{ℚ}$, $_{ℚ}$ describing properties of Dense(ℚ). These invariants satisfy $_{ℚ}$ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ ≤ ℚ$. We compare them with their analogues in the well studied Boolean algebra (ω)/fin. We show that $ℚ = p$, $ℚ = t$ and $ℚ = i$, whereas $ℚ > h$ and $ℚ > r$ are both shown to be relatively consistent with ZFC. We also investigate combinatorics of the ideal nwd of nowhere dense subsets of ,ℚ. In particular, we show that $non(M)=min||: ⊆ Dense(R) ∧ (∀I ∈ nwd(R))(∃D ∈ )(I ∩ D = ∅) and cof(M) = min||: ⊆ Dense(ℚ) ∧ (∀I ∈ nwd)(∃D ∈ )(I ∩ = ∅). We use these facts to show that cof(M) ≤ i, which improves a result of S. Shelah.
LA - eng
KW - rational numbers; nowhere dense ideal; distributivity of Boolean algebras; cardinal invariants of the continuum
UR - http://eudml.org/doc/283218
ER -