No CrossRef data available.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 13 September 2021
Let 
$\mathcal {I}$
be an ideal on 
$\omega $
. For 
$f,\,g\in \omega ^{\omega }$
we write 
$f \leq _{\mathcal {I}} g$
if 
$f(n) \leq g(n)$
for all 
$n\in \omega \setminus A$
with some 
$A\in \mathcal {I}$
. Moreover, we denote 
$\mathcal {D}_{\mathcal {I}}=\{f\in \omega ^{\omega }: f^{-1}[\{n\}]\in \mathcal {I} \text { for every } n\in \omega \}$
(in particular, 
$\mathcal {D}_{\mathrm {Fin}}$
denotes the family of all finite-to-one functions).
We examine cardinal numbers 
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))$
and 
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}}\times \mathcal {D}_{\mathrm {Fin}}))$
describing the smallest sizes of unbounded from below with respect to the order 
$\leq _{\mathcal {I}}$
sets in 
$\mathcal {D}_{\mathrm {Fin}}$
and 
$\mathcal {D}_{\mathcal {I}}$
, respectively. For a maximal ideal 
$\mathcal {I}$
, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers.
We show that 
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathrm {Fin}} \times \mathcal {D}_{\mathrm {Fin}})) =\mathfrak {b}$
for all ideals 
$\mathcal {I}$
with the Baire property and that 
$\aleph _1 \leq \mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) \leq \mathfrak {b}$
for all coanalytic weak P-ideals (this class contains all 
$\bf {\Pi ^0_4}$
ideals). What is more, we give examples of Borel (even 
$\bf {\Sigma ^0_2}$
) ideals 
$\mathcal {I}$
with 
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}}))=\mathfrak {b}$
as well as with 
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {I}} \times \mathcal {D}_{\mathcal {I}})) =\aleph _1$
.
We also study cardinals 
$\mathfrak {b}(\geq _{\mathcal {I}}\cap (\mathcal {D}_{\mathcal {J}} \times \mathcal {D}_{\mathcal {K}}))$
describing the smallest sizes of sets in 
$\mathcal {D}_{\mathcal {K}}$
not bounded from below with respect to the preorder 
$\leq _{\mathcal {I}}$
by any member of 
$\mathcal {D}_{\mathcal {J}}\!$
. Our research is partially motivated by the study of ideal-QN-spaces: those cardinals describe the smallest size of a space which is not ideal-QN.
$\omega$
—their ideals and their cardinal characteristics. Transactions of the American Mathematical Society, vol. 351 (1999), no. 7, 2643–2674.Google Scholar
${wQ}{N}_{\ast }$
and
${wQ}{N}^{\ast }$
spaces. Topology and its Applications, vol. 156 (2008), no. 1, 24.CrossRefGoogle Scholar
${QN}$
-spaces,
${wQN}$
-spaces and covering properties. Topology and its Applications, vol. 154 (2007), no. 4, 848–858.CrossRefGoogle Scholar
$\mathcal{I}$
-quasinormal convergence. Commentationes Mathematicae Universitatis Carolinae, vol. 54 (2013), no. 1, 83–96.Google Scholar
$P$
-ideals. Commentationes Mathematicae Universitatis Carolinae, vol. 50 (2009), no. 2, 281–295.Google Scholar
$\mathcal{I}$
-Baire classes. Journal of Mathematical Analysis and Applications, vol. 391 (2012), no. 1, 1–9.CrossRefGoogle Scholar
$P$
-ideals. Canadian Journal of Mathematics, vol. 59 (2007), no. 3, 575–595.CrossRefGoogle Scholar
$\mathcal{F}$
-limits of filter sequences. Journal of Mathematical Analysis and Applications, vol. 398 (2013), no. 2, 872–878.CrossRefGoogle Scholar
${C}_p(X)$
. Topology and its Applications, vol. 154 (2007), no. 3, 552–560.CrossRefGoogle Scholar
${S}_1\left(\mathbf{\mathcal{P}},\mathbf{\mathcal{R}}\right)$
: ideals and functions. Topology and its Applications, vol. 258 (2019), 282–304.CrossRefGoogle Scholar