# Ideals generated by families of sequences of natural numbers

Pratulananda Das  Department of Mathematics
Jadavpur University
Kolkata - 700032
West Bengal
India<EMAIL_ADDRESS>,  Szymon Gła̧b  Institute of Mathematics
Łódź University of Technology, Wólczańska 215, 93-005 Łódź, Poland
<EMAIL_ADDRESS>http://im0.p.lodz.pl/ sglab ,  Rafał Filipów  Institute
of Mathematics
Faculty of Mathematics, Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>http://mat.ug.edu.pl/ rfilipow and  Jacek Tryba
Institute of Mathematics
Faculty of Mathematics, Physics and Informatics
University of Gdańsk
ul. Wita Stwosza 57
80-308 Gdańsk
Poland<EMAIL_ADDRESS>
###### Abstract.

We consider ideals on $\omega\times\omega$ generated by subsets of
$\omega^{\omega}$. Namely, for $\mathcal{F}\subseteq\omega^{\omega}$ we define

$\mathcal{I}(\mathcal{F})=\\{A\subseteq\omega\times\omega:\exists
f\in\mathcal{F}\,\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in A\\}|\leq
f(n))\\}.$

These ideals encompass other well known ideals e.g.
$\mathrm{Fin}\otimes\mathrm{Fin}$ and $\operatorname{\mathcal{ED}}$. We
examine some properties of these ideals including topological complexity and
cardinal characteristics associated with them.

###### Key words and phrases:

ideal, Borel ideal, $F_{\sigma}$ ideal, filter, cardinal characteristics of an
ideal, cardinal characteristics of the continuum.

###### 2010 Mathematics Subject Classification:

Primary: 40A35, 03E17. Secondary: 26A03, 03E15, 28A05, 03E05.

##  1\. Introduction

All the notions and notations used in the introduction are defined in Section
2.

Ideals (and the dual notion – filters) are extensively used in various parts
of set theory, topology, real analysis and so on (see e.g. [1, 2, 8, 9, 17,
16, 22, 24, 30, 31, 36, 38, 39, 44, 46] to mention only a few recent
publications). Ideals can play an important role in characterizing other
objects. For instance, Laczkovich and Recław [38], and independently Debs and
Saint Raymond [22], used the ideal

$\mathrm{Fin}\otimes\mathrm{Fin}=\\{A\subseteq\omega\times\omega:\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in
A\\}|<\aleph_{0}))\\}$

to characterize the family of all functions of Baire class $1$, whereas Hrušák
[31] used the ideal

$\operatorname{\mathcal{ED}}=\\{A\subseteq\omega\times\omega:\exists
m\in\omega\,\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in A\\}|<m))\\}$

to characterize selective ideals.

These two ideals, i.e. $\mathrm{Fin}\otimes\mathrm{Fin}$ and
$\operatorname{\mathcal{ED}}$, are examples of ideals defined in the following
manner: for a set $\mathcal{F}\subseteq\omega^{\omega}$ of sequences of
natural numbers we define

$\mathcal{I}(\mathcal{F})=\\{A\subseteq\omega\times\omega:\exists
f\in\mathcal{F}\,\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in A\\}|\leq
f(n))\\}.$

It is easy to see that if $\mathcal{F}=\omega^{\omega}$, then
$\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\mathrm{Fin}$, and if
$\mathcal{F}$ consists of all constant functions, then
$\mathcal{I}(\mathcal{F})=\operatorname{\mathcal{ED}}$.

It seems that ideals of the form $\mathcal{I}(\mathcal{F})$ have not been
investigated before in general. In this paper we study some of the properties
of these ideals. Below is an outline of the results we obtained.

In Section 3 we prove some basic properties of ideals
$\mathcal{I}(\mathcal{F})$. For instance, we study the relationships between
ideals generated by different families (Proposition 3.1), we prove a necessary
and sufficient condition for $\mathcal{I}(\mathcal{F})$ to be an ideal
(Proposition 3.2), we give a necessary and sufficient condition for
$\mathcal{I}(\mathcal{F})$ to be one of the three well known ideals i.e. to be
$\mathrm{Fin}\otimes\mathrm{Fin}$, $\mathrm{Fin}\otimes\emptyset$ and
$\operatorname{\mathcal{ED}}$ (Proposition 3.5), we show that
$\mathcal{I}(\mathcal{F})$ is never a P-ideal (Proposition 3.6), we prove a
necessary and sufficient condition for $\mathcal{I}(\mathcal{F})$ to be tall
(Proposition 3.7), and we show that $\mathcal{I}(\mathcal{F})$ (except
$\mathrm{Fin}\otimes\emptyset$) is never nowhere tall (Proposition 3.9).

In Section 4 we study topological complexity of the ideals of the form
$\mathcal{I}(\mathcal{F})$. First, we show that $\mathcal{I}(\mathcal{F})$ has
the Baire property for every $\mathcal{F}$ (Proposition 4.1) even when
$\mathcal{F}$ does not have the Baire property. We notice that for any
Bernstein set $\mathcal{F}$ (hence a set without the Baire property)
$\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\mathrm{Fin}$ (Proposition 4.2),
and prove that it is consistent that there is a set $\mathcal{F}$ without the
Baire property such that
$\mathcal{I}(\mathcal{F})\neq\mathrm{Fin}\otimes\mathrm{Fin}$. The question if
such an example can be constructed in ZFC is still open (Question 1).

Next we study Borel complexity of ideals $\mathcal{I}(\mathcal{F})$. We show
that for a small family $\mathcal{F}$ the ideal $\mathcal{I}(\mathcal{F})$ is
equal or contained in an $F_{\sigma}$ ideal (Theorem 4.5(3),(2)), and for a
large family the ideal $\mathcal{I}(\mathcal{F})$ is $F_{\sigma\delta\sigma}$
(Theorem 4.5(1)). Moreover if a family $\mathcal{F}$ is Borel or analytic then
$\mathcal{I}(\mathcal{F})$ is analytic (Theorem 4.5(4)). We also show that
there are Borel ideals of the form $\mathcal{I}(\mathcal{F})$ of arbitrarily
high Borel complexity (Theorem 4.9). Finally, we show that there is
$\mathcal{F}$ such that $\mathcal{I}(\mathcal{F})$ is not Borel (Theorem
4.10).

It is not difficult to see that there are infinitely many (even $\mathfrak{c}$
many) pairwise distinct ideals of the form $\mathcal{I}(\mathcal{F})$ (Example
5.1). In Section 5 we prove that there are uncountably many pairwise
nonisomorphic ideals of the form $\mathcal{I}(\mathcal{F})$ (Theorem 5.2) and
that there are infinitely many pairwise nonisomorphic $F_{\sigma}$ ideals of
the form $\mathcal{I}(\mathcal{F})$ (Theorem 5.3). However, we do not know if
there are $\mathfrak{c}$ many pairwise nonisomorphic ideals of the form
$\mathcal{I}(\mathcal{F})$ (Question 2).

In Section 6 we study cardinal characteristics (additivity, uniformity,
cofinality and covering) of tall ideals of the form
$\mathcal{I}(\mathcal{F})$. In the case of additivity and uniformity we notice
that the value of these cardinals equals $\aleph_{0}$ for every ideal of the
form $\mathcal{I}(\mathcal{F})$ (Propositions 6.3 and 6.4).

Cofinality of all ideals of the form $\mathcal{I}(\mathcal{F})$ is at least
$\mathfrak{d}$ (Proposition 6.5). For ideals generated by one function which
takes infinitely many nonzero values, cofinality goes up to $\mathfrak{c}$
(Theorem 6.6). We do not know if the same holds for ideals generated by
countably many functions (Question 4).

Covering of all ideals of the form $\mathcal{I}(\mathcal{F})$ is at least
$\mathfrak{b}$ and at most $\operatorname{non}(\mathcal{M})$ (Proposition
6.8). If $\mathcal{F}$ is small, the covering of $\mathcal{I}(\mathcal{F})$ is
equal to $\operatorname{non}(\mathcal{M})$ (Proposition 6.12).

##  2\. Preliminaries

By $\omega$ we denote the set of all natural numbers including zero.

For a formula $\phi(x)$ and a set $X$ we write $\forall^{\infty}x\in
X\,(\phi(x))$ to abbreviate that $\phi(x)$ holds for all but finitely many
$x\in X$, and $\exists^{\infty}x\in X\,(\phi(x))$ to abbreviate that $\phi(x)$
holds for infinitely many $x\in X$.

For $f,g\in\omega^{\omega}$ we write $f\leq^{*}g$ if
$\forall^{\infty}n\in\omega\,(f(n)\leq g(n))$. If $f\leq^{*}g$ and
$g\leq^{*}f$, we write $f=^{*}g$ (in this case
$\forall^{\infty}n\in\omega\,(f(n)=g(n))$). For $A,B\subseteq\omega$ we write
$A\subseteq^{*}B$ if $A\setminus B$ is finite.

Let $\mathcal{F},\mathcal{G}\subseteq\omega^{\omega}$. We say that
$\mathcal{F}$ is _cofinal_ in $\mathcal{G}$ if for every $g\in\mathcal{G}$
there is $f\in\mathcal{F}$ such that $g\leq^{*}f$.

By $\mathfrak{b}$ we denote the _bounding number_ i.e. the smallest
cardinality of an unbounded subset in $(\omega^{\omega},\leq^{*})$. By
$\mathfrak{d}$ we denote the _dominating number_ i.e. the smallest cardinality
of a cofinal subset in $(\omega^{\omega},\leq^{*})$.

When we write about topological properties of sets $A\subseteq\omega^{\omega}$
we assume that $\omega^{\omega}$ is equipped with the product topology with
the discrete topology on $\omega$ (i.e. $\omega^{\omega}$ is the so called
_Baire space_).

By $\operatorname{non}(\mathcal{M})$ we denote the smallest cardinality of
non-meager sets in $\omega^{\omega}$, whereas
$\operatorname{cov}(\mathcal{M})$ denotes the smallest cardinality of families
$\mathcal{F}$ of meager sets in $\omega^{\omega}$ such that
$\bigcup\mathcal{F}=\omega^{\omega}$. It is known that
$\operatorname{non}(\mathcal{M})<\mathfrak{d}$ is consistent with ZFC (see
e.g. [4, p. 382]) and $\operatorname{cov}(\mathcal{M})\leq\mathfrak{d}$ holds
in ZFC (see e.g. [4, p. 35]).

For a set $A$ and $n\in\omega$ we write $[A]^{n}=\\{B\subseteq A:|B|=n\\}$,
$[A]^{<\omega}=\\{B\subseteq A:|B|<\aleph_{0}\\}$ and
$[A]^{\omega}=\\{B\subseteq A:|B|=\aleph_{0}\\}$.

For $A\subseteq X\times Y$ and $x\in X$ we write $A_{x}=\\{y\in Y:(x,y)\in
A\\}$ (i.e. $A_{x}$ is the vertical section of $A$ at the point $x$).

A set $B$ in a topological space has the _Baire property_ if $B=(U\setminus
M)\cup N$ for some open set $U$ and meager sets $M,N$.

Let $X$ be a topological space. Let $\mathbf{\Sigma}^{0}_{1}(X)$ be the family
of all open subsets of $X$, $\mathbf{\Pi}^{0}_{\alpha}(X)$ be the family of
complements of sets from $\mathbf{\Sigma}^{0}_{\alpha}(X)$
($1\leq\alpha<\omega_{1}$) and $\mathbf{\Sigma}^{0}_{\alpha}(X)$ be the family
of countable unions of sets from
$\bigcup_{\beta<\alpha}\mathbf{\Pi}^{0}_{\beta}(X)$ ($\alpha>1$).

###  2.1. Ideals

######  Definition 2.1.

A family $\mathcal{I}\subseteq\mathcal{P}(X)$ is called an _ideal on $X$_ if

  1. (1)

$A,B\in\mathcal{I}\implies A\cup B\in\mathcal{I}$,

  2. (2)

$A\subseteq B\land B\in\mathcal{I}\implies A\in\mathcal{I}$,

  3. (3)

$\mathcal{I}$ contains all finite subsets of $X$,

  4. (4)

$X\notin\mathcal{I}$.

For and ideal $\mathcal{I}$ on $X$, we write $\mathcal{I}^{*}=\\{X\setminus
A:A\in\mathcal{I}\\}$ and call it the _filter dual to $\mathcal{I}$_.

######  Example 2.2.

The ideal of all finite subsets of an infinite set $X$ is denoted by
$\mathrm{Fin}(X)$ (or $\mathrm{Fin}$ for short).

######  Definition 2.3.

Ideals $\mathcal{I}$ and $\mathcal{J}$ on $X$ and $Y$ respectively are
_isomorphic_ if there exists a bijection $\phi:X\to Y$ such that
$A\in\mathcal{I}\iff\phi[A]\in\mathcal{J}$ for every $A\subseteq X$.

######  Notation.

For an ideal $\mathcal{I}$ on $\omega$ and $A\subseteq\omega$ we define
$\mathcal{I}\restriction A=\\{B\cap A:B\in\mathcal{I}\\}$. It is easy to see
that $\mathcal{I}\restriction A$ is an ideal on $A$ $\iff$
$A\notin\mathcal{I}$.

######  Definition 2.4.

An ideal $\mathcal{I}$ on $X$ is

  1. (1)

_tall_ if for every infinite $A\subseteq X$ there is an infinite
$B\in\mathcal{I}$ such that $B\subseteq A$ (some authors use the name a _dense
ideal_ in this case),

  2. (2)

_nowhere tall_ if $\mathcal{I}\restriction A$ is not tall for every
$A\notin\mathcal{I}$,

  3. (3)

a _P-ideal_ if for every countable family $\mathcal{A}\subseteq\mathcal{I}$
there is $B\in\mathcal{I}$ such that $A\setminus B$ is finite for every
$A\in\mathcal{A}$.

######  Remark.

By identifying sets of natural numbers with their characteristic functions, we
equip $\mathcal{P}(\omega)$ with the topology of the Cantor space
$\\{0,1\\}^{\omega}$ and therefore we can assign topological complexity to
ideals on $\omega$. In particular, an ideal $\mathcal{I}$ is $F_{\sigma}$
(Borel resp.) if it is an $F_{\sigma}$ (Borel resp.) subset of the Cantor
space.

######  Remark.

While considering an ideal on an infinite countable set we can see it as an
ideal on $\omega$ by identifying this set with $\omega$ via a fixed bijection.
In fact this paper is about ideals on $\omega\times\omega$.

######  Example 2.5.

The following families are ideals on $\omega\times\omega$ (see e.g. [31]).

  1. (1)

$\emptyset\otimes\mathrm{Fin}=\\{A\subseteq\omega\times\omega:\forall
n\in\omega\,(|\\{k:(n,k)\in A\\}|<\aleph_{0}))\\}$.

  2. (2)

$\mathrm{Fin}\otimes\emptyset=\\{A\subseteq\omega\times\omega:\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in
A\\}|=0))\\}$.

  3. (3)

$\mathrm{Fin}\otimes\mathrm{Fin}=\\{A\subseteq\omega\times\omega:\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in
A\\}|<\aleph_{0}))\\}$.

  4. (4)

$\operatorname{\mathcal{ED}}=\\{A\subseteq\omega\times\omega:\exists
m\in\omega\,\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in A\\}|<m))\\}$.

###  2.2. Ideals generated by families of sequences of natural numbers

######  Definition 2.6.

For $\mathcal{F}\subseteq\omega^{\omega}$ we define

$\mathcal{I}(\mathcal{F})=\\{A\subseteq\omega\times\omega:\exists
f\in\mathcal{F}\,\forall^{\infty}n\in\omega\,(|\\{k:(n,k)\in A\\}|\leq
f(n))\\}.$

######  Example 2.7.

If

  1. (1)

$\mathcal{F}=\emptyset$, then $\mathcal{I}(\mathcal{F})=\emptyset$;

  2. (2)

$\mathcal{F}$ consists of all constant functions, then
$\mathcal{I}(\mathcal{F})=\operatorname{\mathcal{ED}}$;

  3. (3)

$\mathcal{F}=\omega^{\omega}$, then
$\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\mathrm{Fin}$;

  4. (4)

$\mathcal{F}=\\{f\in\omega^{\omega}:\forall^{\infty}n\in\omega\,(f(n)=0)\\}$,
then $\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\emptyset$.

##  3\. Basic properties

######  Proposition 3.1.

Let $\mathcal{F},\mathcal{G}\subseteq\omega^{\omega}$.

  1. (1)

$\mathcal{F}\subseteq\mathcal{G}\implies\mathcal{I}(\mathcal{F})\subseteq\mathcal{I}(\mathcal{G})$.

  2. (2)

$\mathcal{F}$ is cofinal in $\mathcal{G}$ $\iff$
$\mathcal{I}(\mathcal{F})\supseteq\mathcal{I}(\mathcal{G})$.

  3. (3)

$\mathcal{F}$ is cofinal in $\mathcal{G}$ and $\mathcal{G}$ is cofinal in
$\mathcal{F}$ $\iff$ $\mathcal{I}(\mathcal{F})=\mathcal{I}(\mathcal{G})$.

  4. (4)

$\mathcal{F}\subseteq\mathcal{G}$ and $\mathcal{F}$ is cofinal in
$\mathcal{G}$ $\implies$ $\mathcal{I}(\mathcal{F})=\mathcal{I}(\mathcal{G})$.

###### Proof.

Straightforward. ∎

######  Notation.

For $\mathcal{F}\subseteq\omega^{\omega}$ we denote by $\widehat{\mathcal{F}}$
the smallest superset of $\mathcal{F}$ that is closed under taking sums of its
elements i.e.
$\widehat{\mathcal{F}}=\\{f_{1}+\dots+f_{n}:f_{1},\dots,f_{n}\in\mathcal{F}\land
n\in\omega\\}$.

######  Proposition 3.2.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty. The following
conditions are equivalent.

  1. (1)

$\mathcal{I}(\mathcal{F})$ is an ideal on $\omega\times\omega$.

  2. (2)

$\forall f,g\in\mathcal{F}\,\exists h\in\mathcal{F}\,(f+g\leq^{*}h)$.

  3. (3)

$\mathcal{F}$ is cofinal in $\widehat{\mathcal{F}}$.

  4. (4)

$\mathcal{I}(\mathcal{F})=\mathcal{I}(\widehat{\mathcal{F}})$.

###### Proof.

$(\ref{prop:ideal-on-NtimesN-generated-by-functions-
ideal})\implies(\ref{prop:ideal-on-NtimesN-generated-by-functions-bounded})$
Let $f,g\in\mathcal{F}$. Let $A=\\{(n,k):k\leq f(n)\\}$ and
$B=\\{(n,k):f(n)<k\leq f(n)+g(n)\\}$. Since $A,B\in\mathcal{I}(\mathcal{F})$,
so $A\cup B\in\mathcal{I}(\mathcal{F})$. Hence there is $h\in\mathcal{F}$ such
that $|\\{k:(n,k)\in A\cup B\\}|\leq h(n)$ for all but finitely many $n$. Then
$f+g\leq^{*}h$.

$(\ref{prop:ideal-on-NtimesN-generated-by-functions-
bounded})\implies(\ref{prop:ideal-on-NtimesN-generated-by-functions-cofinal})$
It is enough to show that for every $n\in\omega$ and
$f_{1},\dots,f_{n}\in\mathcal{F}$ there is $f\in\mathcal{F}$ with
$g=f_{1}+\dots+f_{n}\leq^{*}f$. We proceed by induction on $n$. For $n=1$,
$g\leq f_{1}\in\mathcal{F}$, so we are done. Suppose that it is true for $n$.
Then there is $h\in\mathcal{F}$ with $f_{1}+\dots+f_{n}\leq^{*}h$, so there is
also $f\in\mathcal{F}$ with $h+f_{n+1}\leq^{*}f$. This shows that
$f_{1}+\dots+f_{n}+f_{n+1}\leq^{*}f$ and the proof is finished.

$(\ref{prop:ideal-on-NtimesN-generated-by-functions-
cofinal})\implies(\ref{prop:ideal-on-NtimesN-generated-by-functions-equal})$
Since $\mathcal{F}\subseteq\widehat{\mathcal{F}}$, by Proposition 3.1,
$\mathcal{I}(\mathcal{F})\subseteq\mathcal{I}(\widehat{\mathcal{F}})$. Let
$A\in\mathcal{I}(\widehat{\mathcal{F}})$. Then there are
$f_{1},\dots,f_{k}\in\mathcal{F}$ such that $|A_{n}|\leq g(n)$ for all but
finitely many $n\in\omega$, where $g=f_{1}+\dots+f_{k}$. Since $\mathcal{F}$
is cofinal in $\widehat{\mathcal{F}}$, there is $f\in\mathcal{F}$ with
$g\leq^{*}f$. Thus $|A_{n}|\leq f(n)$ for all but finitely many $n\in\omega$,
so $A\in\mathcal{I}(\mathcal{F})$.

$(\ref{prop:ideal-on-NtimesN-generated-by-functions-
equal})\implies(\ref{prop:ideal-on-NtimesN-generated-by-functions-ideal})$
Take any $f\in\mathcal{F}$. Since $\\{k:(n,k)\in\emptyset\\}=\emptyset$ and
$0\leq f(n)$ for every $n$, $\emptyset\in\mathcal{I}(\mathcal{F})$.

Since $\\{k:(n,k)\in\omega\times\omega\\}=\omega$ for every $n\in\omega$,
there is no $f\in\omega^{\omega}$ such that
$|\\{k:(n,k)\in\omega\times\omega\\}|\leq f(n)$ for all but finitely many
$n\in\omega$. Thus $\omega\times\omega\notin\mathcal{I}(\mathcal{F})$.

Let $A\in\mathcal{I}(\mathcal{F})$ and $B\subseteq A$. Let $f\in\mathcal{F}$
be such that $|A_{n}|\leq f(n)$ for all but finitely many $n$. Clearly,
$B_{n}\subseteq A_{n}$ and so $|B_{n}|\leq|A_{n}|\leq f(n)$ for all but
finitely many $n$. Thus $B\in\mathcal{I}(\mathcal{F})$.

Let $A,B\in\mathcal{I}(\mathcal{F})$. Let $f,g\in\mathcal{F}$ be such that
$|A_{n}|\leq f(n)$ and $|B_{n}|\leq g(n)$ for all but finitely many $n$. Let
$h=f+g$. Then $\\{k:(n,k)\in A\cup B\\}=A_{n}\cup B_{n}$, so $|\\{k:(n,k)\in
A\cup B\\}|\leq|A_{n}|+|B_{n}|\leq f(n)+g(n)=h(n)$ for all but finitely many
$n$. Since $h\in\widehat{\mathcal{F}}$, $A\cup
B\in\mathcal{I}(\widehat{\mathcal{F}})=\mathcal{I}(\mathcal{F})$.

Let $F\subseteq\omega\times\omega$ be finite. Take any $f\in\mathcal{F}$. Then
$\\{n:F_{n}\not=\emptyset\\}$ is finite, so $|\\{k:(n,k)\in F\\}|=0\leq f(n)$
for all but finitely many $n$. Thus $F\in\mathcal{I}(\mathcal{F})$. ∎

######  Corollary 3.3.

If a nonempty family $\mathcal{F}\subseteq\omega^{\omega}$ is closed under
taking sums of its elements (i.e. $f,g\in\mathcal{F}\implies
f+g\in\mathcal{F}$), then $\mathcal{I}(\mathcal{F})$ is an ideal.

######  Notation.

Let $\mathcal{A}\subseteq\mathcal{P}(\omega)$. If $\omega$ cannot be covered
by finitely many members of $\mathcal{A}$ then $\operatorname{i}(\mathcal{A})$
denotes the smallest ideal containing $\mathcal{A}$ i.e.
$\operatorname{i}(\mathcal{A})=\\{B\subseteq\omega:\exists
n\in\omega\,\exists\mathcal{B}\in[\mathcal{A}]^{n}\,(B\subseteq^{*}\bigcup\mathcal{B})\\}$.

######  Proposition 3.4.

If $\mathcal{F}\subseteq\omega^{\omega}$ is nonempty, then
$\operatorname{i}(\mathcal{I}(\mathcal{F}))=\mathcal{I}(\widehat{\mathcal{F}})$.

###### Proof.

$(\subseteq)$ Since $\mathcal{F}\subseteq\widehat{\mathcal{F}}$ then
$\mathcal{I}(\mathcal{F})\subseteq\mathcal{I}(\widehat{\mathcal{F}})$. Thus
$\operatorname{i}(\mathcal{I}(\mathcal{F}))\subseteq\operatorname{i}(\mathcal{I}(\widehat{\mathcal{F}}))=\mathcal{I}(\widehat{\mathcal{F}})$.

$(\supseteq)$ Let $A\in\mathcal{I}(\widehat{\mathcal{F}})$. Then there is
$f\in\widehat{\mathcal{F}}$ such that $|A_{n}|\leq f(n)$ for all but finitely
many $n\in\omega$. Let $f_{1},\dots,f_{m}\in\mathcal{F}$ be such that
$f=f_{1}+\dots+f_{m}$. We partition the set $A$ into sets $B_{1},\dots,B_{m}$
such that $|\\{k:(n,k)\in B_{i}\\}|\leq f_{i}(n)$ for all but finitely many
$n\in\omega$ and all $i=1,\dots,m$. Then
$B_{1},\dots,B_{m}\in\mathcal{I}(\mathcal{F})$, so $A=B_{1}\cup\dots\cup
B_{m}\in\operatorname{i}(\mathcal{I}(\mathcal{F}))$. ∎

######  Proposition 3.5.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$.

  1. (1)

$\mathrm{Fin}\otimes\emptyset\subseteq\mathcal{I}(\mathcal{F})\subseteq\mathrm{Fin}\otimes\mathrm{Fin}$.

  2. (2)

$\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\mathrm{Fin}$ $\iff$
$\mathcal{F}$ is cofinal in $\omega^{\omega}$.

  3. (3)

$\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\emptyset$ $\iff$ $\forall
f\in\mathcal{F}\,\forall^{\infty}n\in\omega\,(f(n)=0)$.

  4. (4)

$\operatorname{\mathcal{ED}}\subseteq\mathcal{I}(\mathcal{F})$ $\iff$ $\exists
f\in\mathcal{F}\,\forall^{\infty}n\,(f(n)\neq 0)$.

  5. (5)

$\mathcal{I}(\mathcal{F})=\operatorname{\mathcal{ED}}$ $\iff$

    1. (a)

$\forall f\in\mathcal{F}\,\exists
k\in\omega\,\forall^{\infty}n\in\omega\,(f(n)\leq k)$,

    2. (b)

$\exists f\in\mathcal{F}\,\forall^{\infty}n\,(f(n)\neq 0)$.

###### Proof.

(1) Straightforward.

(2) Follows from Proposition 3.1(3) and Example 2.7(3).

(3,$\impliedby$) Straightforward.

(3,$\implies$) If there was $f\in\mathcal{F}$ such that
$A=\\{n\in\omega:f(n)\not=0\\}$ is infinite, then
$A\times\\{1\\}\in\mathcal{I}(\mathcal{F})\setminus\mathrm{Fin}\otimes\emptyset$.

(4, $\implies$) Since
$A=\omega\times\\{0\\}\in\operatorname{\mathcal{ED}}\subseteq\mathcal{I}(\mathcal{F})$,
there is $f\in\mathcal{F}$ such that $1=|A_{n}|\leq f(n)$ for all but finitely
many $n\in\omega$.

(4, $\impliedby$) Let $A\in\operatorname{\mathcal{ED}}$. Then there is $m$
such that $|A_{n}|\leq m$ for all but finitely many $n$. Let $f\in\mathcal{F}$
be such that $f(n)\neq 0$ for all but finitely many $n$. Since $\mathcal{F}$
is cofinal in $\widehat{\mathcal{F}}$, there is $g\in\mathcal{F}$ with $m\cdot
f\leq^{*}g$. Then $|A_{n}|\leq m\leq m\cdot f(n)\leq g(n)$ for all but
finitely many $n$. Thus $A\in\mathcal{I}(\mathcal{F})$.

(5,$\implies$)

(a) By Proposition 3.1(3), the family of all constant functions is cofinal in
$\mathcal{F}$. Hence for every $f\in\mathcal{F}$ there is $k\in\omega$ with
$f(n)\leq k$ for all but finitely many $n\in\omega$.

(b) Follows from (4).

(5, $\impliedby$)

($\subseteq$) Let $A\in\mathcal{I}(\mathcal{F})$. There is $f\in\mathcal{F}$
with $|A_{n}|\leq f(n)$ for all but finitely many $n\in\omega$. Let
$k\in\omega$ be such that $f(n)\leq k$ for all but finitely many $n\in\omega$.
Then $|A_{n}|\leq k$ for all but finitely many $n\in\omega$, so
$A\in\operatorname{\mathcal{ED}}$.

($\supseteq$) Follows from (4). ∎

######  Proposition 3.6.

If $\mathcal{F}\subseteq\omega^{\omega}$ is nonempty and cofinal in
$\widehat{\mathcal{F}}$, then the ideal $\mathcal{I}(\mathcal{F})$ is not a
P-ideal.

###### Proof.

Let $A_{n}=\\{n\\}\times\omega$ and note that
$A_{n}\in\mathcal{I}(\mathcal{F})$ for every $n$. Let
$B\subseteq\omega\times\omega$ be such that $A_{n}\setminus B$ is finite for
every $n$. Then $\\{k:(n,k)\in B\\}$ is infinite for every $n$, hence
$B\notin\mathcal{I}(\mathcal{F})$. ∎

######  Proposition 3.7.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$. The ideal $\mathcal{I}(\mathcal{F})$ is tall $\iff$
$\forall A\in[\omega]^{\omega}\,\exists B\in[A]^{\omega}\,\exists
f\in\mathcal{F}\,\forall n\in B\,(f(n)>0)$.

###### Proof.

$(\implies)$ Let $A\in[\omega]^{\omega}$. Since $C=A\times\\{0\\}$ is
infinite, there is an infinite set $D\in\mathcal{I}(\mathcal{F})$ such that
$D\subseteq C$. Let $f\in\mathcal{F}$ and $n_{0}\in\omega$ such that
$|D_{n}|\leq f(n)$ for every $n\geq n_{0}$. Then $B=\\{n\geq n_{0}:(n,0)\in
D\\}$ is infinite and $B\subseteq A$. Moreover, $f(n)\geq|D_{n}|=1>0$ for
every $n\in B$.

($\impliedby$) Let $C\subseteq\omega\times\omega$ be infinite. If there is $n$
such that $C_{n}$ is infinite, then $D=\\{n\\}\times
C_{n}\in\mathcal{I}(\mathcal{F})$ and $D\subseteq C$. Assume now that $C_{n}$
is finite for every $n\in\omega$. Since $C$ is infinite, the set
$A=\\{n\in\omega:C_{n}\neq\emptyset\\}$ is infinite. Let $B\in[A]^{\omega}$
and $f\in\mathcal{F}$ be such that $f(n)>0$ for every $n\in B$. Then for every
$n\in B$ there is $k_{n}$ such that $(n,k_{n})\in C$. Observe that
$D=\\{(n,k_{n}):n\in B\\}$ is infinite, belongs to $\mathcal{I}(\mathcal{F})$
and $D\subseteq C$. ∎

######  Corollary 3.8.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$. If there exists $f\in\mathcal{F}$ such that $f(n)\neq
0$ for all but finitely many $n$, then the ideal $\mathcal{I}(\mathcal{F})$ is
tall.

###### Proof.

Apply Proposition 3.7. ∎

######  Proposition 3.9.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$. The following conditions are equivalent.

  1. (1)

$\mathcal{I}(\mathcal{F})$ is nowhere tall (i.e.
$\mathcal{I}(\mathcal{F})\restriction B$ is not a tall ideal for any
$B\notin\mathcal{I}(\mathcal{F})$).

  2. (2)

$\mathcal{I}(\mathcal{F})\restriction(A\times\omega)$ is not a tall ideal for
any $A\in[\omega]^{\omega}$.

  3. (3)

$\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\emptyset$.

###### Proof.

$(1)\implies(2)$ Since $B\notin\mathcal{I}(\mathcal{F})$, the set
$A=\\{n\in\omega:B_{n}\not=\emptyset\\}$ is infinite. Since
$\mathcal{I}(\mathcal{F})\restriction B$ is tall, by Proposition 3.7 we have
that for all $C\in[A]^{\omega}$ there is $D\in[C]^{\omega}$ and
$f\in\mathcal{F}$ such that $f(n)>0$ for all $n\in D$. By the same
characterization, $\mathcal{I}(\mathcal{F})\restriction(A\times\omega)$ is
tall.

$(2)\implies(3)$ Let $B=A\times\\{1\\}$. Then there is some infinite
$C\subseteq B$ such that $C\in\mathcal{I}(\mathcal{F})$. However,
$C\not\in\mathrm{Fin}\otimes\emptyset$.

$(3)\implies(1)$ By Proposition 3.5(2) there is $f\in\mathcal{F}$ such that
the set $A=\\{n\in\omega:f(n)\not=0\\}$ is infinite. Then
$B=A\times\omega\not\in\mathcal{I}(\mathcal{F})$ and
$\mathcal{I}(\mathcal{F})\restriction B$ is tall by Proposition 3.7. ∎

##  4\. Topological complexity

###  4.1. Baire property

######  Proposition 4.1.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$. The ideal $\mathcal{I}(\mathcal{F})$ has the Baire
property.

###### Proof.

For every $i\in\omega$ we define
$F_{i}=\\{(n,k)\in\omega\times\omega:n+k=i\\}$. Then $\\{F_{i}:i\in\omega\\}$
is a partition of $\omega\times\omega$ into finite sets. Moreover, it is not
difficult to see that if $A\subseteq\omega\times\omega$ contains infinitely
many sets $F_{i}$ then all vertical sections of $A$ are infinite, so
$A\notin\mathcal{I}(\mathcal{F})$. Thus, by Talagrand characterization [48,
Théorème 21] (see also [4, Theorem 4.1.2]), the ideal
$\mathcal{I}(\mathcal{F})$ has the Baire property. ∎

######  Remark.

Proposition 4.1 shows that even if $\mathcal{F}$ does not have the Baire
property in $\omega^{\omega}$, the ideal $\mathcal{I}(\mathcal{F})$ has the
Baire property. So the natural question arises whether there is a nonempty set
$\mathcal{F}$ such that $\mathcal{F}$ is cofinal in $\widehat{\mathcal{F}}$
and $\mathcal{F}$ does not have the Baire property. Below we show (Proposition
4.2) that the answer is positive.

Let $X$ be a topological space. A set $B\subseteq X$ is a _Bernstein set_ if
both $B$ and $X\setminus B$ are totally imperfect (i.e. neither $B$ nor
$X\setminus B$ contain a nonempty perfect set). It is known that Bernstein
sets exist in the Baire space $\omega^{\omega}$ and they do not have the Baire
property (see e.g. [13, Theorem 7.20 and 7.22]).

######  Proposition 4.2.

Any Bernstein set $B\subseteq\omega^{\omega}$ is cofinal in $\omega^{\omega}$.
Hence $B$ is cofinal in $\widehat{B}$ and
$\mathcal{I}(B)=\mathrm{Fin}\otimes\mathrm{Fin}$.

###### Proof.

First we show that $B$ is cofinal in $\omega^{\omega}$. Let
$f\in\omega^{\omega}$. Let $A_{n}=\\{f(n)+1,f(n)+2\\}$ for every $n\in\omega$.
Since $P=\prod_{n\in\omega}A_{n}$ is a nonempty and perfect subset of
$\omega^{\omega}$, $B\cap P\neq\emptyset$. Moreover, $f\leq g$ for any $g\in
B\cap P$.

Since $\widehat{B}\subseteq\omega^{\omega}$, $B$ is cofinal in $\widehat{B}$.

Then $\mathcal{I}(B)=\mathrm{Fin}\otimes\mathrm{Fin}$ follows from Proposition
3.1(4) and Example 2.7 (3). ∎

Another question is if there is a set $\mathcal{F}$ without the Baire property
such that $\mathcal{F}$ is cofinal in $\widehat{\mathcal{F}}$ but
$\mathcal{F}$ is not cofinal in $\omega^{\omega}$. Below we show (Proposition
4.3) that the positive answer is consistent with the ZFC set theory. However
we do not know if such an example exists in ZFC (see Question 1).

######  Proposition 4.3.

Assume $\operatorname{non}(\mathcal{M})<\mathfrak{d}$. There is a set
$\mathcal{F}\subseteq\omega^{\omega}$ without the Baire property such that
$\mathcal{F}$ is cofinal in $\widehat{\mathcal{F}}$, but $\mathcal{F}$ is not
cofinal in $\omega^{\omega}$. Hence
$\mathcal{I}(\mathcal{F})\neq\mathrm{Fin}\otimes\mathrm{Fin}$.

###### Proof.

Let $A\subseteq\omega^{\omega}$ be a non-meager set of cardinality
$\operatorname{non}(\mathcal{M})$. Since
$|A|=\operatorname{non}(\mathcal{M})<\mathfrak{d}\leq\mathfrak{c}$, $A$ does
not have the Baire property (otherwise, as non-meager, it should have the
cardinality $\mathfrak{c}$). Let $\mathcal{F}=\widehat{A}$. Since
$\widehat{\mathcal{F}}=\mathcal{F}$, $\mathcal{F}$ is cofinal in
$\widehat{\mathcal{F}}$. On the other hand,
$|\mathcal{F}|\leq|[A]^{<\omega}|=\operatorname{non}(\mathcal{M})<\mathfrak{d}$
from which it follows that $\mathcal{F}$ is not cofinal in $\omega^{\omega}$
and it does not have the Baire property. ∎

######  Question 1.

Does there exist (in ZFC) a set $\mathcal{F}\subseteq\omega^{\omega}$ without
the Baire property such that $\mathcal{F}$ is cofinal in
$\widehat{\mathcal{F}}$, but $\mathcal{F}$ is not cofinal in
$\omega^{\omega}$?

###  4.2. Borel complexity

######  Lemma 4.4.

  1. (1)

The set
$B=\\{(A,f)\in\mathcal{P}(\omega\times\omega)\times\omega^{\omega}:\forall^{\infty}n\in\omega|\\{k:(n,k)\in
A\\}|\leq f(n)\\}$ is $F_{\sigma}$.

  2. (2)

$\mathcal{I}(\mathcal{F})=\operatorname{proj}_{1}[X]$, where
$\operatorname{proj}_{1}$ is the projection onto the first coordinate,
$X=(\mathcal{P}(\omega\times\omega)\times\mathcal{F})\cap B$ and
$\mathcal{F}\subseteq\omega^{\omega}$.

###### Proof.

(1) It follows from the fact that the sets
$B_{f,n}=\\{(A,f)\in\mathcal{P}(\omega\times\omega)\times\omega^{\omega}:|\\{k:(n,k)\in
A\\}|\leq f(n)\\}$ are closed and
$B=\bigcup_{m\in\omega}\bigcap_{n>m}B_{f,n}$.

(2) Straightforward. ∎

######  Theorem 4.5.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$.

  1. (1)

If $\mathcal{F}$ is cofinal in $\omega^{\omega}$, then
$\mathcal{I}(\mathcal{F})$ is an $F_{\sigma\delta\sigma}$ ideal.

  2. (2)

If $\mathcal{F}$ is $\sigma$-compact in $\omega^{\omega}$, then
$\mathcal{I}(\mathcal{F})$ is a $\sigma$-compact ideal (and in particular it
is an $F_{\sigma}$ ideal).

  3. (3)

If $\mathcal{F}$ is bounded in $\omega^{\omega}$, then
$\mathcal{I}(\mathcal{F})$ is contained in a $\sigma$-compact ideal (and in
particular it is contained in an $F_{\sigma}$ ideal).

  4. (4)

If $\mathcal{F}$ is a Borel (or even analytic) set, then
$\mathcal{I}(\mathcal{F})$ is an analytic ideal.

###### Proof.

(1) By Proposition 2,
$\mathcal{I}(\mathcal{F})=\mathrm{Fin}\otimes\mathrm{Fin}$ and it is not
difficult to see that $\mathrm{Fin}\otimes\mathrm{Fin}$ is
$F_{\sigma\delta\sigma}$.

(2) Let $B$ and $X$ be as in Lemma 4.4. If $\mathcal{F}$ is $\sigma$-compact
in the Baire space $\omega^{\omega}$, then $X$ is $\sigma$-compact (as the
intersection of the $\sigma$-compact set
$\mathcal{P}(\omega\times\omega)\times\mathcal{F}$ and the $F_{\sigma}$ set
$B$). Consequently $\mathcal{I}(\mathcal{F})$ is $\sigma$-compact as a
continuous image of $X$.

(3) It follows from (2) as every bounded set in $\omega^{\omega}$ is contained
in a $\sigma$-compact set.

(4) Let $B$ and $X$ be as in Lemma 4.4. If $\mathcal{F}$ is Borel (resp.
analytic) in the Baire space $\omega^{\omega}$, then $X$ is Borel (resp.
analytic) (as the intersection of the Borel (resp. analytic) set
$\mathcal{P}(\omega\times\omega)\times\mathcal{F}$ and the $F_{\sigma}$ set
$B$). Consequently $\mathcal{I}(\mathcal{F})$ is analytic set as a continuous
image of $X$.∎

######  Corollary 4.6.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$.

  1. (1)

If $\mathcal{F}$ is of cardinality less than $\mathfrak{b}$, then
$\mathcal{I}(\mathcal{F})$ is contained in an $F_{\sigma}$ ideal.

  2. (2)

If $\mathcal{F}$ is countable, then $\mathcal{I}(\mathcal{F})$ is an
$F_{\sigma}$ ideal.

###### Proof.

(1) It follows from Theorem 4.5(3) because every family of cardinality less
than $\mathfrak{b}$ is bounded in $(\omega^{\omega},\leq^{*})$.

(2) It follows from Theorem 4.5(2) because every countable family is
$\sigma$-compact. ∎

######  Lemma 4.7.

Let $\mathcal{A}\subseteq\mathcal{P}(\omega)$ be an ideal on $\omega$. Let
$\mathcal{F}_{\mathcal{A}}=\\{k\cdot\operatorname{\mathbf{1}}_{A}:A\in\mathcal{A}\land
k\in\omega\\}$, where $\operatorname{\mathbf{1}}_{A}$ is the characteristic
function of a set $A$.

  1. (1)

$\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is an ideal on $\omega\times\omega$.

  2. (2)

There exists a continuous function
$\Phi:\mathcal{P}(\omega)\to\mathcal{P}(\omega\times\omega)$ such that
$\Phi^{-1}[\mathcal{I}(\mathcal{F}_{\mathcal{A}})]=\mathcal{A}$.

  3. (3)

There exists a Borel function
$\Psi:\mathcal{P}(\omega\times\omega)\rightarrow\mathcal{P}(\omega)\times(\omega+1)$
such that
$\mathcal{I}(\mathcal{F}_{\mathcal{A}})=\Psi^{-1}[\mathcal{A}\times\omega]$.
(We consider the order topology on the set $\omega+1$.)

###### Proof.

(1) Let $A,B\in\mathcal{A}$. Since $A\cup B\in\mathcal{A}$ and
$k\operatorname{\mathbf{1}}_{A}+m\operatorname{\mathbf{1}}_{B}\leq(k+m)\operatorname{\mathbf{1}}_{A\cup
B}$, the family $\mathcal{F}$ is cofinal in $\widehat{\mathcal{F}}$ and
consequently $\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is an ideal (by
Proposition 3.2).

(2) Let $\Phi:\mathcal{P}(\omega)\to\mathcal{P}(\omega\times\omega)$ be given
by $\Phi(A)=A\times\\{0\\}$ for every $A\in\mathcal{P}(\omega)$. It is not
difficult to see that $\Phi$ is a continuous function. Let us show
$\Phi^{-1}[\mathcal{I}(\mathcal{F}_{\mathcal{A}})]=\mathcal{A}$.

$(\supseteq)$ Let $A\in\mathcal{A}$. Then
$|(\Phi(A))_{n}|=|(A\times\\{0\\})_{n}|=\operatorname{\mathbf{1}}_{A}(n)$ for
every $n\in\omega$ and
$\operatorname{\mathbf{1}}_{A}\in\mathcal{F}_{\mathcal{A}}$. Thus
$\Phi(A)\in\mathcal{I}(\mathcal{F}_{\mathcal{A}})$.

$(\subseteq)$ Let $B\in\Phi^{-1}[\mathcal{I}(\mathcal{F}_{\mathcal{A}})]$.
Then $\Phi(B)\in\mathcal{I}(\mathcal{F}_{\mathcal{A}})$, so there is
$A\in\mathcal{A}$ and $k\in\omega$ such that $|(\Phi(B))_{n}|\leq
k\operatorname{\mathbf{1}}_{A}(n)$ for all but finitely many $n\in\omega$. On
the other hand,
$|(\Phi(B))_{n}|=|(B\times\\{0\\})_{n}|=\operatorname{\mathbf{1}}_{B}(n)$ for
every $n\in\omega$. Thus,
$\operatorname{\mathbf{1}}_{B}(n)\leq\operatorname{\mathbf{1}}_{A}(n)$ for all
but finitely many $n\in\omega$. Hence $B\subseteq^{*}A$. Since
$A\in\mathcal{A}$ and $\mathcal{A}$ is an ideal, $B\in\mathcal{A}$.

(3) Let
$\Psi:\mathcal{P}(\omega\times\omega)\rightarrow\mathcal{P}(\omega)\times(\omega+1)$
be given by

$\Psi(B)=\left(\operatorname{proj}_{1}[B],\
\sup\\{\min\\{k,|B_{k}|\\}:k\in\omega\\}\right)$

where $\operatorname{proj}_{1}$ is the projection onto the first coordinate.

Let $\Psi_{1}:\mathcal{P}(\omega\times\omega)\to\mathcal{P}(\omega)$ be given
by $\Psi_{1}(B)=\operatorname{proj}_{1}[B]$ and
$\Psi_{2}:\mathcal{P}(\omega\times\omega)\to\omega+1$ be given by
$\Psi_{2}(B)=\sup\\{\min\\{k,|B_{k}|\\}:k\in\omega\\}$. If we show that
$\Psi_{1}$ and $\Psi_{2}$ are Borel, then $\Psi=(\Psi_{1},\Psi_{2})$ will be
Borel as well.

($\Psi_{1}$) Recall that the topology on $\mathcal{P}(\omega)$ and
$\mathcal{P}(\omega\times\omega)$ is inherited from the products
$\\{0,1\\}^{\omega}$ and $\\{0,1\\}^{\omega\times\omega}$ respectively by
identifying sets with its characteristic functions. Thus it is enough to show
that the function $F:\\{0,1\\}^{\omega\times\omega}\to\\{0,1\\}^{\omega}$
given by $F(y)=\operatorname{\mathbf{1}}_{\operatorname{proj}_{1}[z^{-1}(1)]}$
is Borel.

Since the sets of the form $[s]=\\{x\in\\{0,1\\}^{\omega}:s\subseteq x\\}$
with $n\in\omega$ and $s\in\\{0,1\\}^{n}$ form a base of the topology, it is
enough to show that $F^{-1}[[s]]$ is Borel.

But $F^{-1}[[s]]=\\{z\in\\{0,1\\}^{\omega\times\omega}:s\subseteq
F(z)\\}=\\{z\in\\{0,1\\}^{\omega\times\omega}:s\subseteq\operatorname{\mathbf{1}}_{\operatorname{proj}_{1}[z^{-1}(1)]}\\}=\\{z\in\\{0,1\\}^{\omega\times\omega}:\left(\forall{i\in
s^{-1}(0)}\forall k\in\omega\,z(i,k)=0\right)\land\left(\forall{i\in
s^{-1}(1)}\exists k\in\omega\,z(i,k)=1\right)\\}$, so $F^{-1}[[s]]$ is a
$G_{\delta}$ set, hence Borel.

($\Psi_{2}$) For every $n\in\omega$ we define the function
$\psi_{n}:\mathcal{P}(\omega\times\omega)\to\omega+1$ by
$\psi_{n}(B)=\max\\{\min\\{k,|B_{k}\cap\\{0,1,\dots,n\\}|\\}:k\leq n\\}$. It
is not difficult to see that the functions $\psi_{n}$ are continuous and
$\Psi_{2}$ is a pointwise limit of the sequence $(\psi_{n})$. Thus $\Psi_{2}$
is a Baire-one function, hence it is a Borel function as well.

Now we will show that
$\mathcal{I}(\mathcal{F}_{\mathcal{A}})=\Psi^{-1}[\mathcal{A}\times\omega]$,
which will complete the proof.

$(\subseteq)$ Let $B\in\mathcal{I}(\mathcal{F}_{\mathcal{A}})$. Then there are
$l\in\omega$ and $A\in\mathcal{A}$ such that $|B_{n}|\leq
l\cdot\operatorname{\mathbf{1}}_{A}(n)$ for all $n\geq l$. It follows that
$\Psi_{1}(B)=\operatorname{proj}_{1}[B]\subseteq
A\cup\\{1,\ldots,l\\}\in\mathcal{A}$ and
$\Psi_{2}(B)=\sup\\{\min\\{k,|B_{k}|\\}:k\in\omega\\}\leq l$. Therefore,
$\Psi(B)\in\mathcal{A}\times\omega$.

$(\supseteq)$ Let $B\in\Psi^{-1}[\mathcal{A}\times\omega]$. It means that
there is $A\in\mathcal{A}$ such that $\operatorname{proj}_{1}[B]=A$ and there
is $l\in\omega$ such that $|B_{n}|\leq l$ for all $n\geq l$. Thus,
$|B_{n}|\leq l\cdot\operatorname{\mathbf{1}}_{A}(n)$ for all $n\geq l$, so
$B\in\mathcal{I}(\mathcal{F}_{\mathcal{A}})$. ∎

######  Corollary 4.8.

If $\mathcal{A}$ is a Borel ideal on $\omega$, then
$\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is a Borel ideal on
$\omega\times\omega$.

###### Proof.

By Lemma 4.7(1), $\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is an ideal and by
Lemma 4.7(3), $\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is Borel. ∎

######  Theorem 4.9.

There are Borel ideals of the form $\mathcal{I}(\mathcal{F})$ of arbitrarily
high Borel complexity i.e. for every $\alpha<\omega_{1}$ there exists
$\mathcal{F}\subseteq\omega^{\omega}$ such that the ideal
$\mathcal{I}(\mathcal{F})$ is Borel but not in $\mathbf{\Sigma}^{0}_{\alpha}$.

###### Proof.

Let $\alpha<\omega_{1}$. Let $\mathcal{A}$ be a Borel ideal on $\omega$ such
that $\mathcal{A}\notin\mathbf{\Sigma}^{0}_{\alpha}$ (for the existence of
such ideals see e.g. [14], [51] or [50]). Let
$\mathcal{F}_{\mathcal{A}}=\\{k\cdot\operatorname{\mathbf{1}}_{A}:A\in\mathcal{A}\land
k\in\omega\\}$. By Corollary 4.8, $\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is
a Borel ideal. On the other hand, Lemma 2(2) implies that
$\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is not in
$\mathbf{\Sigma}^{0}_{\alpha}$. ∎

######  Theorem 4.10.

There exists $\mathcal{F}\subseteq\omega^{\omega}$ such that
$\mathcal{I}(\mathcal{F})$ is not Borel.

###### Proof.

Let $\mathcal{A}\subseteq\mathcal{P}(\omega)$ be an ideal on $\omega$ which is
not Borel (for instance a maximal ideal – see e.g. [4, Theorem 4.1.1]). Let
$\mathcal{F}_{\mathcal{A}}=\\{k\cdot\operatorname{\mathbf{1}}_{A}:A\in\mathcal{A}\land
k\in\omega\\}$. Then $\mathcal{I}(\mathcal{F}_{\mathcal{A}})$ is an ideal by
Lemma 4.7(1) and it is not a Borel ideal by Lemma 4.7(2). ∎

##  5\. How many?

######  Example 5.1.

There are $\mathfrak{c}$ many pairwise distinct ideals of the form
$\mathcal{I}(\mathcal{F})$. Indeed, for $A\subseteq\omega$ we define
$\mathcal{F}_{A}=\\{f\cdot\operatorname{\mathbf{1}}_{A}:f\in\omega^{\omega}\\}$
where $\operatorname{\mathbf{1}}_{A}$ is the characteristic function of the
set $A$. Then it is easy to see that $\mathcal{I}(\mathcal{F}_{A})$ are
pairwise distinct for distinct $A$. However, every
$\mathcal{I}(\mathcal{F}_{A})$ is isomorphic to either
$\mathrm{Fin}\otimes\mathrm{Fin}$ or $\mathrm{Fin}\otimes\emptyset$ or
$\mathrm{Fin}\otimes\mathrm{Fin}\restriction(2\omega\times\omega)\cup\mathrm{Fin}\otimes\emptyset\restriction((2\omega+1)\times\omega)$.

######  Theorem 5.2.

There are $\aleph_{1}$ pairwise nonisomorphic ideals of the form
$\mathcal{I}(\mathcal{F})$.

###### Proof.

Since isomorphic ideals have the same Borel complexity, the proof follows from
Theorem 4.9. ∎

######  Question 2.

Are there $\mathfrak{c}$ many pairwise nonisomorphic ideals of the form
$\mathcal{I}(\mathcal{F})$?

We have presented uncountably many pairwise nonisomorphic ideals of the form
$\mathcal{I}(\mathcal{F})$. However, each of them had a different Borel class.
Below we show that there are also infinitely many pairwise nonisomorphic
ideals of the form $\mathcal{I}(\mathcal{F})$ with the same Borel class.

######  Theorem 5.3.

There are infinitely many pairwise nonisomorphic $F_{\sigma}$ ideals of the
form $\mathcal{I}(\mathcal{F})$.

###### Proof.

Let $(p_{1},p_{2},\ldots)$ be the enumeration of all prime numbers. For every
$m\in\omega$, define the function $f_{m}\in\omega^{\omega}$ by
$f_{m}(l)=(p_{m}^{l})!$. It is easy to observe the following two properties of
the functions $f_{m}$.

######  Claim.

  1. (1)

$\forall m_{1}\neq m_{2}\,\forall l_{1},l_{2}\in\omega\,(f_{m_{1}}(l_{1})\neq
f_{m_{2}}(l_{2}))$.

  2. (2)

$\forall m_{1}\neq m_{2}\,\forall k\in\omega\,\exists l\in\omega\,\forall
l_{1},l_{2}>l\,\left(\frac{f_{m_{1}}(l_{1})}{f_{m_{2}}(l_{2})}>k\lor\frac{f_{m_{2}}(l_{2})}{f_{m_{1}}(l_{1})}>k\right)$.

Next, we define the family $\mathcal{F}_{m}=\\{k\cdot f_{m}:k\in\omega\\}$ for
each $m\in\omega$. Clearly, every $\mathcal{F}_{m}$ is cofinal in
$\widehat{\mathcal{F}}_{m}$, hence
$\mathcal{I}(\mathcal{F}_{m})=\mathcal{I}(\widehat{\mathcal{F}}_{m})$ by
Proposition 3.2. Since the families $\mathcal{F}_{m}$ are countable, ideals
$\mathcal{I}(\mathcal{F}_{m})$ are $F_{\sigma}$ (we have seen this in
Corollary 4.6). If we show that $\mathcal{I}(\mathcal{F}_{m_{1}})$ and
$\mathcal{I}(\mathcal{F}_{m_{2}})$ are not isomorphic for distinct $m_{1}$ and
$m_{2}$, the proof will be finished.

Suppose to the contrary that $\mathcal{I}(\mathcal{F}_{m_{1}})$ and
$\mathcal{I}(\mathcal{F}_{m_{2}})$ are isomorphic for some $m_{1}\neq m_{2}$
and that $g:\omega\times\omega\rightarrow\omega\times\omega$ is a bijection
such that $A\in\mathcal{I}(\mathcal{F}_{m_{1}})\iff
g[A]\in\mathcal{I}(\mathcal{F}_{m_{2}})$ for every
$A\subseteq\omega\times\omega$. Then for every $i\in\omega$ we have
$g^{-1}[\\{i\\}\times\omega]\in\mathrm{Fin}\otimes\mathrm{Fin}$, because the
set $\\{i\\}\times\omega$ belongs to every ideal of the form
$\mathcal{I}(\mathcal{F})$ and
$\mathcal{I}(\mathcal{F})\subseteq\mathrm{Fin}\otimes\mathrm{Fin}$ for every
family $\mathcal{F}\subseteq\omega^{\omega}$ by Proposition 3.1.

Moreover, we show that for all but finitely many $i\in\omega$ we have
$g[\\{i\\}\times\omega]\in\mathrm{Fin}\otimes\emptyset$. Suppose that it is
not the case. Then there exists a sequence of natural numbers
$i_{1}<i_{2}<\ldots$ such that
$g[\\{i_{n}\\}\times\omega]\not\in\mathrm{Fin}\otimes\emptyset$ for every
$n\in\omega$. Therefore, for every $n\in\omega$ we may find a sequence
$((a_{k}^{n},b_{k}^{n}))_{k\in\omega}$ with $a_{k}^{n}<a_{k+1}^{n}$ and
$(a_{k}^{n},b_{k}^{n})\in g[\\{i_{n}\\}\times\omega]$ for every $k\in\omega$.
We can now define the set $A=\\{(a_{k}^{n},b_{k}^{n}):n\in\omega,k>n\\}$. Then
each $A\cap(\\{i\\}\times\omega)$ has at most $i$ elements, hence
$A\in\mathcal{I}(\mathcal{F}_{m_{2}})$ as $f_{m_{2}}(i)\geq i$. On the other
hand, $g^{-1}[A]\cap(\\{i_{n}\\}\times\omega)$ is infinite for every
$n\in\omega$, thus $g^{-1}[A]\not\in\mathcal{I}(\mathcal{F}_{m_{1}})$, a
contradiction.

Since for all but finitely many $i\in\omega$ we have
$g[\\{i\\}\times\omega]\in\mathrm{Fin}\otimes\emptyset$ and
$\\{i\\}\times\omega$ is an infinite set, it follows that
$g[\\{i\\}\times\omega]\not\in\emptyset\otimes\mathrm{Fin}$ for almost all
$i\in\omega$. Combining it with the fact that
$g^{-1}[\\{i\\}\times\omega]\in\mathrm{Fin}\otimes\mathrm{Fin}$ for all
$i\in\omega$, we conclude that we can find sequences
$(a_{i})_{i\in\omega},(b_{i})_{i\in\omega}$ such that
$a_{i}\not=a_{j},b_{i}\not=b_{j}$ for $i\not=j$ and
$g[\\{a_{i}\\}\times\omega]\cap(\\{b_{i}\\}\times\omega)$ is infinite for
every $i\in\omega$. Now, by the first observation from the Claim, we have two
cases:

  1. (1)

$A=\\{i\in\omega:f_{m_{1}}(a_{i})<f_{m_{2}}(b_{i})\\}$ is infinite;

  2. (2)

$B=\\{i\in\omega:f_{m_{1}}(a_{i})>f_{m_{2}}(b_{i})\\}$ is infinite.

In the first case, we create the set $C\subseteq\omega\times\omega$ by taking
$f_{m_{2}}(b_{i})$ elements from every
$g[\\{a_{i}\\}\times\omega]\cap(\\{b_{i}\\}\times\omega)$ for each $i\in A$.
Clearly, $C\in\mathcal{I}(\mathcal{F}_{m_{2}})$. However, by the second
observation from the Claim, we have $\lim_{i\in
A}\frac{f_{m_{2}}(b_{i})}{f_{m_{1}}(a_{i})}=\infty$, hence
$g^{-1}[C]\not\in\mathcal{I}(\mathcal{F}_{m_{1}})$, which contradicts the
assumption that $g$ is an isomorphism.

In the second case, we create the set $C\subseteq\omega\times\omega$ by taking
$f_{m_{1}}(a_{i})$ elements from every
$g[\\{a_{i}\\}\times\omega]\cap(\\{b_{i}\\}\times\omega)$ for each $i\in B$.
Clearly, $g^{-1}[C]\in\mathcal{I}(\mathcal{F}_{m_{1}})$. However, by the
second observation from the Claim, we have $\lim_{i\in
B}\frac{f_{m_{1}}(a_{i})}{f_{m_{2}}(b_{i})}=\infty$, hence
$C\not\in\mathcal{I}(\mathcal{F}_{m_{2}})$, which contradicts the assumption
that $g$ is an isomorphism. ∎

######  Question 3.

Are there uncountably many pairwise nonisomorphic $F_{\sigma}$ ideals of the
form $\mathcal{I}(\mathcal{F})$?

##  6\. Cardinal characteristics

Some properties of ideals can be described by cardinal characteristics
associated with them. There are four well known characteristics called
additivity, covering, uniformity and cofinality defined in the following way.

######  Definition 6.1 (see e.g. [4]).

Let $\mathcal{I}$ be an ideal on $X$.

$\begin{split}\operatorname{add}(\mathcal{I})&=\min\left\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\land\bigcup\mathcal{A}\notin\mathcal{I}\right\\}\\\
\operatorname{cov}(\mathcal{I})&=\min\left\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\land\bigcup\mathcal{A}=X\right\\}\\\
\operatorname{non}(\mathcal{I})&=\min\\{|A|:A\notin\mathcal{I}\\}\\\
\operatorname{cof}(\mathcal{I})&=\min\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\land\forall
B\in\mathcal{I}\,\exists A\in\mathcal{A}\,(B\subseteq A)\\}\end{split}$

The above characteristics are useful in the case of ideals on an uncountable
set $X$ (for instance in the case of the $\sigma$-ideal $\mathcal{M}$ of all
meager sets and the $\sigma$-ideal $\mathcal{I}=\mathcal{N}$ of all Lebesgue
null sets). However, they are (but $\operatorname{cof}$) become trivial (and
so in some sense useless) in the case of ideals on $\omega$, since
$\operatorname{add}(\mathcal{I})=\operatorname{cov}(\mathcal{I})=\operatorname{non}(\mathcal{I})=\aleph_{0}$
for every ideal $\mathcal{I}$ on $\omega$. Fortunately, Hernández and Hrušák
introduced in [29] certain versions of these characteristics more suitable for
tall ideals on $\omega$.

######  Definition 6.2 ([29], see also [31]).

Let $\mathcal{I}$ be a tall ideal on $\omega$.

$\begin{split}\operatorname{\operatorname{add}^{*}}(\mathcal{I})&=\min\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\land\neg\exists
B\in\mathcal{I}\,\forall A\in\mathcal{A}\,(A\subseteq^{*}B)\\}\\\
\operatorname{\operatorname{cov}^{*}}(\mathcal{I})&=\min\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\land\neg\exists
B\in\mathcal{P}(\omega)\setminus\mathrm{Fin}^{*}\,\forall
A\in\mathcal{A}\,(A\subseteq^{*}B)\\}\\\
\operatorname{\operatorname{non}^{*}}(\mathcal{I})&=\min\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{P}(\omega)\setminus\mathrm{Fin}^{*}\land\forall
B\in\mathcal{I}\,\exists A\in\mathcal{A}\,(B\subseteq^{*}A)\\}\\\
\operatorname{\operatorname{cof}^{*}}(\mathcal{I})&=\min\\{|\mathcal{A}|:\mathcal{A}\subseteq\mathcal{I}\land\forall
B\in\mathcal{I}\,\exists A\in\mathcal{A}\,(B\subseteq^{*}A)\\}\end{split}$

If $\mathcal{U}$ is a free ultrafilter on $\omega$ and
$\mathcal{I}=\mathcal{U}^{*}$ is the dual ideal, the above mentioned
characteristics were earlier introduced by Brendle and Shelah in [10] where
the authors used the notations $\mathfrak{p}(\mathcal{I})$,
$\pi\mathfrak{p}(\mathcal{I})$, $\pi\chi(\mathcal{I})$ and $\chi(\mathcal{I})$
for $\operatorname{\operatorname{add}^{*}}(\mathcal{I})$,
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I})$,
$\operatorname{\operatorname{non}^{*}}(\mathcal{I})$ and
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I})$ respectively.

If an ideal $\mathcal{I}$ is not tall then
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I})$ is not well-defined (as
the $\min$ would be taken from the empty set) and
$\operatorname{\operatorname{non}^{*}}(\mathcal{I})=1$.

There is a close relation between “stared” characteristics and the classical
ones. Indeed, let us write $\widehat{A}=\\{B\in[\omega]^{\omega}:|A\cap
B|=\aleph_{0}\\}$ for $A\subseteq\omega$ and
$\widehat{\mathcal{I}}=\\{X\subseteq[\omega]^{\omega}:\exists
A\in\mathcal{I}\,(X\subseteq\widehat{A})\\}$ for an ideal $\mathcal{I}$ on
$\omega$. Then it is not difficult to see that

  1. (1)

$\widehat{\mathcal{I}}$ is an ideal on $[\omega]^{\omega}$,

  2. (2)

if $\mathcal{I}$ is tall, then $\widehat{\mathcal{I}}$ contains all
singletons,

  3. (3)

$\mathcal{I}$ is a P-ideal $\iff$ $\widehat{\mathcal{I}}$ is a $\sigma$-ideal,

and one can show (see [29, Proposition 1.2]) that
$\operatorname{\operatorname{add}^{*}}(\mathcal{I})=\operatorname{add}(\widehat{\mathcal{I}})$,
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I})=\operatorname{cov}(\widehat{\mathcal{I}})$,
$\operatorname{\operatorname{non}^{*}}(\mathcal{I})=\operatorname{non}(\widehat{\mathcal{I}})$
and
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I})=\operatorname{cof}(\widehat{\mathcal{I}})$.
Note also (see [11, p. 6]) that the set $[\omega]^{\omega}$ can be seen as a
Polish space because it is a $G_{\delta}$ subset of the Cantor space
$\\{0,1\\}^{\omega}$ (we identify a subset of $\omega$ with its characteristic
function), moreover, it is easy to see that $[\omega]^{\omega}$ is
homeomorphic to $\omega^{\uparrow\omega}=\\{x\in\omega^{\omega}:x\text{ is
strictly increasing}\\}$, a closed subset of $\omega^{\omega}$ which is
homeomorphic to $\omega^{\omega}$. Then, $\widehat{\mathcal{I}}$ is an ideal
with a Borel base (as $\widehat{A}$ is a $G_{\delta}$ subset of the Cantor
space for every $A$).

The inequalities holding among these characteristics (see e.g. [31, p. 578])
are summarized in the following diagram (where “$\kappa\to\lambda$” means
“$\kappa\leq\lambda$”):

$\aleph_{0}$$\operatorname{\operatorname{add}^{*}}(\mathcal{I})$$\operatorname{\operatorname{cov}^{*}}(\mathcal{I})$$\operatorname{\operatorname{non}^{*}}(\mathcal{I})$$\operatorname{\operatorname{cof}^{*}}(\mathcal{I})$$\mathfrak{c}$

###  6.1. Additivity

It is known that
$\operatorname{\operatorname{add}^{*}}(\operatorname{\mathcal{ED}})=\operatorname{\operatorname{add}^{*}}(\mathrm{Fin}\otimes\mathrm{Fin})=\aleph_{0}$
(see [alcantara-phd-thesis, Theorems 1.6.4 and 1.6.19]). Below we show that
the same holds for every ideal of the form $\mathcal{I}(\mathcal{F})$.

######  Proposition 6.3.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$. Then
$\operatorname{\operatorname{add}^{*}}(\mathcal{I}(\mathcal{F}))=\aleph_{0}$.

###### Proof.

By Proposition 3.6 the ideal $\mathcal{I}(\mathcal{F})$ is not a P-ideal, so
there is a countable family $\mathcal{A}\subseteq\mathcal{I}(\mathcal{F})$
such that there is no $B\in\mathcal{I}$ with $A\subseteq^{*}B$. Thus
$\operatorname{\operatorname{add}^{*}}(\mathcal{I}(\mathcal{F}))\leq\aleph_{0}$.
Since $\operatorname{\operatorname{add}^{*}}(\mathcal{I})\geq\aleph_{0}$ for
every ideal $\mathcal{I}$, the proof is finished. ∎

###  6.2. Uniformity

It is known that
$\operatorname{\operatorname{non}^{*}}(\operatorname{\mathcal{ED}})=\operatorname{\operatorname{non}^{*}}(\mathrm{Fin}\otimes\mathrm{Fin})=\aleph_{0}$
(see [alcantara-phd-thesis, Theorems 1.6.4 and 1.6.19]). Below we show that
the same holds for every tall ideal of the form $\mathcal{I}(\mathcal{F})$.

######  Proposition 6.4.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$ such that $\mathcal{I}(\mathcal{F})$ is tall. Then
$\operatorname{\operatorname{non}^{*}}(\mathcal{I}(\mathcal{F}))=\aleph_{0}$.

###### Proof.

Let $A_{n}=(\omega\setminus\\{n\\})\times\omega$ for each $n\in\omega$. Then
$\mathcal{A}=\\{A_{n}:n\in\omega\\}\subseteq\mathcal{P}(\omega)\setminus\mathrm{Fin}^{*}$.
Let $B\in\mathcal{I}(\mathcal{F})$. Then there are $n_{0}\in\omega$ and
$f\in\mathcal{F}$ such that $|\\{k:(n_{0},k)\in B\\}|<f(n_{0})$, so
$B\subseteq^{*}A_{n_{0}}$. Hence the inequality
$\operatorname{\operatorname{non}^{*}}(\mathcal{I}(\mathcal{F}))\leq\aleph_{0}$
is proved. Since
$\operatorname{\operatorname{non}^{*}}(\mathcal{I})\geq\aleph_{0}$ holds for
every tall ideal $\mathcal{I}$, the proof is finished. ∎

###  6.3. Cofinality

It is known that
$\operatorname{\operatorname{cof}^{*}}(\mathrm{Fin}\otimes\mathrm{Fin})=\mathfrak{d}$
(see [alcantara-phd-thesis, Theorem 1.6.19]). Below we show that
$\mathfrak{d}$ is a lower bound of the cardinal
$\operatorname{\operatorname{cof}^{*}}$ for every ideal of the form
$\mathcal{I}(\mathcal{F})$ but $\mathrm{Fin}\otimes\emptyset$.

######  Proposition 6.5.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$.
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I}(\mathcal{F}))\geq\mathfrak{d}$
$\iff$ $\mathcal{I}(\mathcal{F})\neq\mathrm{Fin}\otimes\emptyset$.

###### Proof.

($\implies$) Let $A_{n}=\\{0,1,\dots,n\\}\times\omega$ for every $n\in\omega$
and $\mathcal{A}=\\{A_{n}:n\in\omega\\}$. Since
$\mathcal{A}\subseteq\mathrm{Fin}\otimes\emptyset$ and for every
$B\in\mathrm{Fin}\otimes\emptyset$ there is $n\in\omega$ with $B\subseteq
A_{n}$,
$\operatorname{\operatorname{cof}^{*}}(\mathrm{Fin}\otimes\emptyset)\leq\aleph_{0}$.
On the other hand, it is easy to see that
$\operatorname{\operatorname{cof}^{*}}(\mathrm{Fin}\otimes\emptyset)\geq\aleph_{0}$.
Thus
$\operatorname{\operatorname{cof}^{*}}(\mathrm{Fin}\otimes\emptyset)=\aleph_{0}$.
Since $\mathfrak{d}>\aleph_{0}$, the proof is completed.

($\impliedby$) By Proposition 3.5(3) there is $f\in\mathcal{F}$ such that the
set $A=\\{n\in\omega:f(n)\neq 0\\}$ is infinite. Suppose, to the contrary,
that there is $\mathcal{B}\subseteq\mathcal{I}(\mathcal{F})$ such that
$|\mathcal{B}|<\mathfrak{d}$ and for each $A\in\mathcal{I}(\mathcal{F})$ there
is $B\in\mathcal{B}$ with $A\subseteq^{*}B$. Let
$\mathcal{D}\subseteq\omega^{\omega}$ be cofinal in $\omega^{\omega}$ and
$|\mathcal{D}|=\mathfrak{d}$.

For each $g\in\mathcal{D}$ we put $A^{g}=\\{(n,g(n)):n\in A\\}$. Since
$|(A^{g})_{n}|\leq f(n)$ for every $n\in\omega$,
$A^{g}\in\mathcal{I}(\mathcal{F})$. Hence there is $B^{g}\in\mathcal{B}$ with
$A^{g}\subseteq B^{g}$. Let $n_{g}\in\omega$ be such that $(B^{g})_{n}$ is
finite for all $n\geq n_{g}$. We define $h_{g}\in A^{\omega}$ by
$h_{g}(n)=\max((B^{g})_{n})$ for $n\geq n_{g}$ and $h_{g}(n)=0$ otherwise.

Let $\mathcal{H}=\\{h_{g}:g\in\mathcal{D}\\}$. Since $\mathcal{H}$ is cofinal
in $\mathcal{D}\restriction A=\\{g\restriction A:g\in\mathcal{D}\\}$ and
$\mathcal{D}\restriction A$ is cofinal in $A^{\omega}$, $\mathcal{H}$ is
cofinal in $A^{\omega}$. Thus, $|\mathcal{H}|\geq\mathfrak{d}$. On the other
hand, $|\mathcal{H}|\leq|\mathcal{B}|<\mathfrak{d}$, a contradiction. ∎

######  Remark.

It is known (see e.g. [31, Proposition 6.13]) that
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I})\geq\operatorname{cov}(\mathcal{M})$
for analytic ideals $\mathcal{I}$ that are not countably generated. Since
$\operatorname{cov}(\mathcal{M})\leq\mathfrak{d}$, Proposition 6.5 gives a
better bound for
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I}(\mathcal{F}))$ in the case
of analytic ideals of the form $\mathcal{I}(\mathcal{F})$.

It is known that
$\operatorname{\operatorname{cof}^{*}}(\operatorname{\mathcal{ED}})=\mathfrak{c}$
(see [alcantara-phd-thesis, Theorem 1.6.4]). Below we show that the same holds
for ideals “generated” by one essentially nonzero function.

######  Theorem 6.6.

Let $f\in\omega^{\omega}$ be such that $f(n)\neq 0$ for infinitely many
$n\in\omega$ and $\mathcal{F}=\\{k\cdot f:k\in\omega\\}$. Then
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I}(\mathcal{F}))=\mathfrak{c}$.

###### Proof.

Suppose, to the contrary, that
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I}(\mathcal{F}))<\mathfrak{c}$.
Then there is $\mathcal{B}\subseteq\mathcal{I}(\mathcal{F})$ such that
$|\mathcal{B}|<\mathfrak{c}$ and for every $A\in\mathcal{I}(\mathcal{F})$
there is $B\in\mathcal{B}$ with $A\subseteq B$.

We define $I_{n,0}=\\{k\in\omega:0\leq k<f(n)\\}$ and
$I_{n,i}=\\{k\in\omega:f(n)\cdot i\leq k<f(n)\cdot(i+1)\\}$ for every
$n,i\in\omega$, $i\geq 1$.

Let $\mathcal{A}=\\{A_{\alpha}:\alpha<\mathfrak{c}\\}$ be an almost disjoint
family on $\omega$ (i.e. $A_{\alpha}\cap A_{\beta}$ is finite for
$\alpha\neq\beta$ and $A_{\alpha}\in[\omega]^{\omega}$).

By $\phi_{\alpha}:\omega\to A_{\alpha}$ we denote the increasing enumeration
of the set $A_{\alpha}$.

For every $\alpha<\mathfrak{c}$, we define

$C_{\alpha}=\bigcup_{n\in\omega}\\{n\\}\times I_{n,\phi_{\alpha}(n)}$

and note that $C_{\alpha}\in\mathcal{I}(\mathcal{F})$ for every
$\alpha<\mathfrak{c}$.

Now for every $B\in\mathcal{B}$ let
$C_{B}=\\{\alpha<\mathfrak{c}:C_{\alpha}\subseteq B\\}$. Since
$C_{\alpha}\in\mathcal{I}(\mathcal{F})$,
$\bigcup_{B\in\mathcal{B}}C_{B}=\mathfrak{c}$.

If we show that $C_{B}$ is finite for every $B\in\mathcal{B}$, then
$|\bigcup_{B\in\mathcal{B}}C_{B}|\leq\omega\cdot|\mathcal{B}|<\mathfrak{c}$
and we obtain a contradiction that finishes the proof.

Let $B\in\mathcal{B}$ and $k\in\omega$ be such that $|B_{n}|\leq kf(n)$ for
all but finitely many $n\in\omega$. Then $|C_{B}|\leq k$. Indeed, if
$|C_{B}|>k$ then there is $D\in[\mathfrak{c}]^{k+1}$ such that
$C_{\alpha}\subseteq B$ for every $\alpha\in D$. Since
$\\{A_{\alpha}:\alpha\in D\\}$ is almost disjoint, the elements
$\phi_{\alpha}(n)$, where $\alpha\in D$, are pairwise distinct for all but
finitely many $n\in\omega$. Then the sets $(C_{\alpha})_{n}$ where $\alpha\in
D$ are pairwise disjoint for all but finitely many $n\in\omega$. Hence
$|B_{n}|\geq|D|\cdot|(C_{\alpha})_{n}|=(k+1)\cdot f(n)$ for all but finitely
many $n\in\omega$.

Since $f(n)\neq 0$ for infinitely many $n\in\omega$, there are infinitely many
$n\in\omega$ with $|B_{n}|>kf(n)$, a contradiction. ∎

######  Question 4.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty, cofinal in
$\widehat{\mathcal{F}}$ and
$\mathcal{I}(\mathcal{F})\neq\mathrm{Fin}\otimes\emptyset$. Does
$|\mathcal{F}|=\aleph_{0}$ imply
$\operatorname{\operatorname{cof}^{*}}(\mathcal{I}(\mathcal{F}))=\mathfrak{c}$?

###  6.4. Covering

For ideals $\mathcal{I},\mathcal{J}$ on $X$ and $Y$ respectively, we write
$\mathcal{I}\leq_{K}\mathcal{J}$ if there is a function $f:Y\to X$ such that
$f^{-1}[A]\in\mathcal{J}$ for every $A\in\mathcal{I}$. ($\leq_{K}$ is called
the _Katětov order_.)

######  Lemma 6.7.

Let $\mathcal{I},\mathcal{J}$ be tall ideals on $\omega$.

  1. (1)

If $\mathcal{I}\leq_{K}\mathcal{J}$, then
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I})\geq\operatorname{\operatorname{cov}^{*}}(\mathcal{J})$.

  2. (2)

If $\mathcal{I}\subseteq\mathcal{J}$, then
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I})\geq\operatorname{\operatorname{cov}^{*}}(\mathcal{J})$.

  3. (3)

If $X\notin\mathcal{I}$, then
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}\restriction
X)\geq\operatorname{\operatorname{cov}^{*}}(\mathcal{I})$.

###### Proof.

(1) It is proved in [29, Proposition 3.1], but we provide a short proof for
the completeness. Let $f:\omega\to\omega$ be such that
$f^{-1}[A]\in\mathcal{J}$ for every $A\in\mathcal{I}$. Let
$\mathcal{A}\subseteq\mathcal{I}$ be such that
$|\mathcal{A}|=\operatorname{\operatorname{cov}^{*}}(\mathcal{I})$ and for
every $B\in\mathcal{P}(\omega)\setminus\mathrm{Fin}^{*}$ there is
$A\in\mathcal{A}$ with $A\not\subseteq^{*}B$. Let
$\mathcal{B}=\\{f^{-1}[A]:A\in\mathcal{A}\\}\cup\\{f^{-1}[F]:F\in\mathrm{Fin}\\}$.
Then $\mathcal{B}\subseteq\mathcal{J}$ and
$|\mathcal{B}|\leq|\mathcal{A}|+\aleph_{0}=|\mathcal{A}|$. Moreover, it is not
difficult to see that for every
$B\in\mathcal{P}(\omega)\setminus\mathrm{Fin}^{*}$ there is $C\in\mathcal{B}$
such that $B\not\subseteq^{*}C$. Thus,
$\operatorname{\operatorname{cov}^{*}}(\mathcal{J})\leq|\mathcal{B}|\leq\operatorname{\operatorname{cov}^{*}}(\mathcal{I})$.

(2) It follows from (1) because the identity function shows
$\mathcal{I}\leq_{K}\mathcal{J}$.

(3) It follows from (1) because $\mathcal{I}\leq_{K}\mathcal{I}\restriction
X$. ∎

It is known that
$\operatorname{\operatorname{cov}^{*}}(\mathrm{Fin}\otimes\mathrm{Fin})=\mathfrak{b}$
and
$\operatorname{\operatorname{cov}^{*}}(\operatorname{\mathcal{ED}})=\operatorname{non}(\mathcal{M})$
(see [alcantara-phd-thesis, Theorems 1.6.19 and 1.6.4]). Below we observe that
$\mathfrak{b}$ is a lower bound and $\operatorname{non}(\mathcal{M})$ is an
upper bound of the cardinal $\operatorname{\operatorname{cov}^{*}}$ for every
tall ideal of the form $\mathcal{I}(\mathcal{F})$.

######  Proposition 6.8.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$ such that $\mathcal{I}(\mathcal{F})$ is tall. Then
$\mathfrak{b}\leq\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))\leq\operatorname{non}(\mathcal{M})$.

###### Proof.

By Proposition 3.5(1),
$\mathcal{I}(\mathcal{F})\subseteq\mathrm{Fin}\otimes\mathrm{Fin}$, so using
Proposition 6.7(2) we obtain
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))\geq\operatorname{\operatorname{cov}^{*}}(\mathrm{Fin}\otimes\mathrm{Fin})=\mathfrak{b}$.

Since $\mathcal{I}(\mathcal{F})$ is tall, there is $B\in[\omega]^{\omega}$ and
$f\in\mathcal{F}$ such that $f(n)\neq 0$ for every $n\in B$ (see Proposition
3.7). Then $X=B\times\omega\notin\mathcal{I}(\mathcal{F})$ and by Proposition
3.5(4) we have $\operatorname{\mathcal{ED}}\restriction
X\subseteq\mathcal{I}(\mathcal{F})\restriction X$.

Since $\operatorname{\mathcal{ED}}\restriction X$ and
$\operatorname{\mathcal{ED}}$ are isomorphic,
$\operatorname{\operatorname{cov}^{*}}(\operatorname{\mathcal{ED}}\restriction
X)=\operatorname{\operatorname{cov}^{*}}(\operatorname{\mathcal{ED}})=\operatorname{non}(\mathcal{M})$.

By Lemma 6.7(2) we have
$\operatorname{non}(\mathcal{M})=\operatorname{\operatorname{cov}^{*}}(\operatorname{\mathcal{ED}}\restriction
X)\geq\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F})\restriction
X)$ and by Lemma 6.7(3) we have
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F})\restriction
X)\geq\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))$. Thus,
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))\leq\operatorname{non}(\mathcal{M})$.
∎

######  Remark.

It is easy to see that
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I})\geq\mathfrak{p}$ for every
tall ideal $\mathcal{I}$, where $\mathfrak{p}$ is the pseudo-intersection
number. Since $\mathfrak{p}\leq\mathfrak{b}$ (see e.g. [5]), Proposition 6.8
gives a better lower bound for
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))$ in the case
of ideals of the form $\mathcal{I}(\mathcal{F})$.

In the sequel we will use the following characterization of the cardinal
$\operatorname{non}(\mathcal{M})$.

######  Lemma 6.9 (see e.g. [4, Lemma 2.4.8]).

$\operatorname{non}(\mathcal{M})=\min\\{|\mathcal{S}|:\mathcal{S}\subseteq\mathcal{C}\land\forall
f\in\omega^{\omega}\,\exists
S\in\mathcal{S}\,\exists^{\infty}n\in\omega\,(f(n)\in S(n))\\},$

where

$\mathcal{C}=\left\\{S\in\left([\omega]^{<\omega}\right)^{\omega}:\sum_{n\in\omega}\frac{|S(n)|}{(n+1)^{2}}<\infty\right\\}.$

######  Proposition 6.10.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$ such that $\mathcal{I}(\mathcal{F})$ is tall. If there
is an increasing sequence $(n_{i})$ such that
$\sum_{i\in\omega}f(i)/(n_{i}+1)^{2}<\infty$ for every $f\in\mathcal{F}$, then
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))=\operatorname{non}(\mathcal{M})$.

###### Proof.

($\leq$) It follows from Proposition 6.8.

($\geq$) Let $\mathcal{A}\subseteq\mathcal{I}(\mathcal{F})$ be a family of
cardinality less than $\operatorname{non}(\mathcal{M})$. For every
$A\in\mathcal{A}$ there is a function $f_{A}\in\mathcal{F}$ and
$n_{A}\in\omega$ such that $|A_{n}|\leq f_{A}(n)$ for every $n\geq n_{A}$. For
every $A\in\mathcal{A}$ we define the function
$S_{A}:\omega\to[\omega]^{<\omega}$ by

$S_{A}(n)=\begin{cases}A_{i}&\text{for }n=n_{i}\land i\geq n_{A},\\\
\emptyset&\text{otherwise}.\end{cases}$

Note that

$\sum_{n\in\omega}\frac{|S(n)|}{(n+1)^{2}}=\sum_{i\geq
n_{A}}\frac{|S(n_{i})|}{(n_{i}+1)^{2}}=\sum_{i\geq
n_{A}}\frac{|A_{i}|}{(n_{i}+1)^{2}}\leq\sum_{i\geq
n_{A}}\frac{f_{A}(i)}{(n_{i}+1)^{2}}<\infty$

and
$|\\{S_{A}:A\in\mathcal{A}\\}|\leq|\mathcal{A}|<\operatorname{non}(\mathcal{M})$.
By Lemma 6.9 there is $g\in\omega^{\omega}$ such that for every
$A\in\mathcal{A}$ and almost all $n\in\omega$ we have $g(n)\not\in S_{A}(n)$.

Let $h:\omega\to\omega$ be given by $h(k)=g(n_{k})$ for every $k\in\omega$.

Then $B=(\omega\times\omega)\setminus
h\in\mathcal{P}(\omega\times\omega)\setminus\mathrm{Fin}^{*}$ and for every
$A\in\mathcal{A}$ we have $A\subseteq^{*}B$. ∎

######  Corollary 6.11.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$ such that $\mathcal{I}(\mathcal{F})$ is tall. If
$\sum_{n\in\omega}f(n)/(n+1)^{2}<\infty$ for every $f\in\mathcal{F}$, then
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))=\operatorname{non}(\mathcal{M})$.

###### Proof.

Take $n_{i}=i$ and apply Proposition 6.10. ∎

######  Proposition 6.12.

Let $\mathcal{F}\subseteq\omega^{\omega}$ be nonempty and cofinal in
$\widehat{\mathcal{F}}$ such that $\mathcal{I}(\mathcal{F})$ is tall. If
$|\mathcal{F}|<\mathfrak{b}$, then
$\operatorname{\operatorname{cov}^{*}}(\mathcal{I}(\mathcal{F}))=\operatorname{non}(\mathcal{M})$.

###### Proof.

It is not difficult to see that for every $f\in\mathcal{F}$ there is an
increasing sequence $(n^{f}_{i})$ such that

$\sum_{i\in\omega}\frac{f(i)}{(n^{f}_{i}+1)^{2}}<\infty.$

Since $|\mathcal{F}|<\mathfrak{b}$, there is an increasing sequence $(n_{i})$
such that $(n^{f}_{i})\leq^{*}(n_{i})$ for every $f\in\mathcal{F}$. Let
$n^{f}\in\omega$ be such that $n^{f}_{i}\leq n_{i}$ for every $i\geq n^{f}$.
Then

$\sum_{i\in\omega}\frac{f(i)}{(n_{i}+1)^{2}}=\sum_{i<n^{f}}\frac{f(i)}{(n_{i}+1)^{2}}+\sum_{i\geq
n^{f}}\frac{f(i)}{(n_{i}+1)^{2}}\leq\sum_{i<n^{f}}\frac{f(i)}{(n_{i}+1)^{2}}+\sum_{i\geq
n^{f}}\frac{f(i)}{(n^{f}_{i}+1)^{2}}<\infty,$

so Proposition 6.10 finishes the proof. ∎

######  Acknowledgment.

The authors are thankful to Indian National Science Academy and Polish Academy
of Sciences for arranging the visit of the first author to University of
Gdańsk under INSA-PAS Bilateral Exchange Program during which this work was
done

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