On Star operation and some ideals on the Baire space
May 31, 2026
Abstract
We investigate several \(\sigma\)-ideals on the Baire space \(\omega^\omega\) (\(\mathbb{Z}^\omega\)), introducing and studying the ideals \(\mathcal{G}\) and \(\mathcal{SMZ}^+\), alongside the classical ideals of meager sets, strong measure zero sets, the eventually different ideal and infinitely equal ideal We establish structural relationships and proper inclusions among these ideals. Also we compute the cardinal invariants of \(\mathcal{M}_-\), proving that they are same as invariants of \(\sigma\)-ideal of meager sets. We further analyze the operation \(^*\) on families of sets, establishing dual relationships such as \(\mathcal{ED}^* = \mathcal{IE}\), \(\mathcal{IE}^*=\mathcal{ED}\), \(\mathcal{M}_-^*=\mathcal{SMZ}^+\) and \(\mathcal{H}^* = \mathcal{G}\), and derive separations between ideals under additional set-theoretic assumptions. Finally, we prove a tree dichotomy theorem for the ideal \(\mathcal{M}_-\) and we study the associated forcing notion and its effects on cardinal invariants.
1 Introduction↩︎
We will be investigating subsets of the Cantor space \(2^\omega\) and the Baire space \(\omega^\omega\) and families of such subsets. The basis of the Cantor and Baire space is formed by clopen sets: \([\sigma]=\{x\in 2^\omega\text{ or } x\in\omega^\omega:\;\sigma\subseteq x\}\), where \(\sigma\in 2^{<\omega}\) or \(\sigma\in \omega^{<\omega}\).
Definition 1. We say that a subset \(\mathcal{I} \subseteq P(X)\) is an ideal if the following properties are satisfied:
\(\emptyset \in \mathcal{I}\),
When \(A \in \mathcal{I}\) and \(B \subseteq X\), then \(B \subseteq A \Longrightarrow B \in \mathcal{I}\),
If \(A,B \in \mathcal{I}\) then \(A \cup B \in \mathcal{I}\).
We say that a subset \(\mathcal{I} \subseteq P(X)\) is a \(\sigma\)-ideal if it is an ideal and it is closed under countable unions. We say that an ideal \(\mathcal{I}\subseteq\mathcal{P}(X)\) is proper if \(X\notin\mathcal{I}\). In this paper we will only consider proper ideals.
Other important families are the family of sets of Lebesgue’s measure zero, which we shall denote by \(\mathcal{N}\), and the family of meager sets denoted by \(\mathcal{M}\). By \(K_\sigma\) we denote the ideal of countable unions of compact sets in the Baire space \(\omega^\omega\) or \(\mathbb{Z}^\omega\).
For an ideal \(\mathcal{I}\) of polish space \(X\) we define standard cardinal coefficients:
\(add(\mathcal{I})=\min \{|\mathcal{A}|:\;\mathcal{A} \subseteq \mathcal{I} \wedge \bigcup \mathcal{A} \notin \mathcal{I} \},\)
\(cov(\mathcal{I})=\min \{|\mathcal{A}|: \mathcal{A} \subseteq \mathcal{I} \wedge \bigcup \mathcal{A} =X \},\)
\(non(\mathcal{I})=min\{|A|: A\notin \mathcal{I}\}\),
\(cof(\mathcal{I})=\min \{|\mathcal{B}|: \mathcal{B} \subseteq \mathcal{I} \wedge (\forall A \in \mathcal{I})(\exists B \in \mathcal{B})(A \subseteq B) \}.\)
These numbers are known as the additivity, the covering, the uniformity and the cofinality of the ideal \(\mathcal{I}\). For \(f,g\in \omega^\omega\) (or \(\mathbb{Z}^\omega\)) let \(f<^*g\) mean \(\forall^\infty n f(n)<g(n).\) We say that family \(F\subseteq\omega^\omega\) is unbounded if there is no function \(g\in\omega^\omega\) such that for ever \(f\in F\) we have \(f<^*g.\) We say that family \(F\subseteq\omega^\omega\) is dominating if for every function \(g\in\omega^\omega\) we can find \(f\in F\) and we have \(g<^*f.\) As \(\exists^\infty n\) we mean that something happens for infinitely many \(n\). As \(\forall^\infty n\) we mean that something happens for almost all \(n\).
Connected to those concepts we have next cardinal invariants:
\(\mathfrak{b}= min \{ |F| : F \subseteq \omega^\omega \;\wedge F \text{ is unbounded} \},\)
\(\mathfrak{d}=min \{ |F| : F \subseteq \omega^\omega \;\wedge F \text{ is dominating} \}.\)
Definition 2. We say that \(\bar{I}=(I_n)_{n<\omega}\) is an interval partition of \(\omega\) if \(\bigcup_{n<\omega} I_n=\omega\) and max \(I_{n}=\) min \(I_{n+1} -1\). By \(\mathbb{IP}\) we denote all interval partitions of \(\omega.\)
Definition 3. We say that for interval partitions \(\bar{I}=(I_n)_{n<\omega},\bar{J}=(J_n)_{n<\omega}\) we have \(\bar{I}\sqsubseteq^* \bar{J}\) if for almost every \(n\) there exists \(k\) such that \(I_k\subseteq J_n\).
We say that \(F\subseteq\mathbb{IP}\) is unbounded if there is no interval partition \(\bar{I}\in\mathbb{IP}\) such that for every partition \(\bar{J}\in F\) we have \(\bar{J}\sqsubseteq^* \bar{I}.\) We say that \(F\subseteq\mathbb{IP}\) is dominating if for every interval partition \(\bar{I}\in\mathbb{IP}\) we can find \(\bar{J}\in F\) such that \(\bar{I}\sqsubseteq^* \bar{J}.\) We will also use another useful representation of invariants \(\mathfrak{b}\) and \(\mathfrak{d}\) using interval partitions of \(\omega.\)
Theorem 4. (Blass, [1]) For interval partitions we have cardinal invariants:
\(\mathfrak{b}=min\{|F|: F\subseteq IP \wedge F\text{ is unbounded}\}\),
\(\mathfrak{d}=min\{|F|: F\subseteq IP\wedge F\text{ is dominating} \}\).
If \((X,+)\) is (an abelian) group then for \(A, B \in \mathcal{P}(X)\), we will write \[A+B= \{a+b: a \in A, b \in B\}.\] Next we define operation \(^*\) introduced by Seredyński in [2].
Definition 5. For a family \(\mathcal{F}\subseteq \mathcal{P}(X)\), where \(X\) is a Polish group let: \[\mathcal{F}^{*}=\{A\subseteq X:\;\forall F\in \mathcal{F} \;\; A+F \neq X \}.\]
The following diagram, called Cichoń’s diagram, summarizes all of the \(ZFC\)-provable inequalities between cardinal invariants of ideals \(\mathcal{M}\), \(\mathcal{N}\), \(\mathfrak{b}\), and \(\mathfrak{d}\).
Figure 1: 
.
2 Ideals on the Baire space↩︎
In this chapter, we will introduce some new ideals \(\mathcal{G}\) and \(\mathcal{SMZ}^+\) in the Baire space. We will investigate the structural relationships and interactions between these ideals and other known ideals in the Baire space.
We will begin by recalling the characterization of meager sets in the Cantor space and their fundamental properties.
Definition 6. For \(x\in 2^\omega\) and \(\bar{I}\in \mathbb{IP}\) let \(M(x,\bar{I})=\{y: \forall^{\infty} n \;\; x\upharpoonright I_n\neq y\upharpoonright I_n\}\).
Theorem 7. ([3] Talagrand)
For every \(x\in 2^\omega\) and \(\bar{I}\in \mathbb{IP}\) set \(M(x,\bar{I})\) is meager,
for every meager set \(M\) we have \(M\subseteq M(x,\bar{I})\) for some \(x\in 2^\omega\) and interval partition \(\bar{I}\),
if \(I\sqsubseteq^* J\) then \(M(x,\bar{I})\subseteq M(x,\bar{J}),\)
if \(M(y,\bar{I})\subseteq M(z,\bar{J})\), then \(\bar{I}\sqsubseteq^*\bar{J},\)
\(M(x,\bar{I})+y=M(x+y,\bar{I})\).
Strong measure zero sets were defined originally on the real line, by Borel, but we can translate the definition from \(\mathbb{R}\) to \(2^\omega\).
Definition 8. A set \(A \subseteq 2^\omega\) has strong zero measure \((A\in \mathcal{SMZ})\) if for every sequence \((k_n)_{n<\omega}\) of natural numbers there exists a sequence \((\sigma_n)_{n<\omega}\) such that \(A\subseteq \bigcup [\sigma_n]\) and \(|\sigma_n|=k_n\) for every \(n<\omega\).
Next, we move to the characterization of strong measure zero sets in the Cantor space.
Theorem 9. (Bartoszyński, Judah [3])Set \(X\subseteq2^\omega\) has strong measure zero if and only if for every interval partition \(\bar{I}\in \mathbb{IP}\) there exists \(z\in 2^\omega\) such that: \[\forall x\in X\;\; \exists^\infty n\;\; x\upharpoonright I_n=z\upharpoonright I_n.\]
The relationship between strong measure zero sets and meager sets on the real line was first shown in [4] by Galvin, Mycielski, and Solovay. Alternatively, this connection can be proven by using the characterizations of these two ideals.
Theorem 10. (Galvin-Myicelski-Solovay [4]) \(X\in\mathcal{SMZ}\) iff and only iff for all \(H\in \mathcal{M}\) we have \(X+H\neq 2^\omega\).
In [5], Wohofsky showed that the Galvin-Mycielski-Solovay Theorem doesn’t hold in the Baire space \((\mathbb{Z^\omega})\). He showed that \(\mathcal{M}^*\subseteq\mathcal{SMZ}\), and that assuming the Continuum Hypothesis it is possible to construct a set \(X\in \mathcal{SMZ}\backslash \mathcal{M}^*.\)
The characterization of meager sets was adapted to the Baire space in [6]. However, it turns out that in the Baire space the characterization given by 7.
Definition 11. Let \(\mathcal{M}_-\) be a \(\sigma\)-ideal generated by the sets \(M(x,\bar{I})\) for \(x\in \mathbb{Z}^\omega\) and \(\bar{I}\in \mathbb{IP}.\)
Now we will rewrite definition and characterizations of strong measure zero sets to the Baire space.
Definition 12. A set \(A \subseteq \mathbb{Z}^\omega\) has strong zero measure \((A\in \mathcal{SMZ})\) if for every sequence \((k_n)_{n<\omega}\) of natural numbers there exists a sequence \((\sigma_n)_{n<\omega}\) such that \(A\subseteq \bigcup [\sigma_n]\) and \(|\sigma_n|=k_n\) for every \(n<\omega\).
Proposition 13. A set \(X\subseteq \mathbb{Z}^\omega\) has strong measure zero if and only if for every increasing sequence \(k_n\in \omega^\omega\) there exists \((\sigma_n)_{n<\omega}\) such that \(\sigma_n\in \mathbb{Z}^{k_n}\) and: \[X\subseteq\{x:\exists ^\infty _n \;\; \sigma_n\subseteq x\}.\]
Proof. Proof is the same as in \(2^\omega.\) ◻
As it turns out, the characterization of strong measure zero sets provided in 9 does not hold in the Baire space, which we will demonstrate in Chapter 4. This motivates the following definition.
Definition 14. Set \(X\in \mathcal{SMZ}^+\) if and only if for every interval partition \(\bar{I}\in \mathbb{IP}\) there exists \(z\in\mathbb{Z}^\omega\) such that: \[\forall x\in X\;\; \exists^\infty n\;\; x\upharpoonright I_n=z\upharpoonright I_n.\]
Of course we have \(\mathcal{SMZ} \subseteq \mathcal{SMZ}^+\). In Chapter \(4\) we will show that under additional assumptions the reverse inclusion does not hold. We now present a version of the Galvin-Mycielski-Solovay theorem for the Baire space in terms of the ideals introduced above.
Theorem 15. For \(X\subseteq\mathbb{Z}^\omega\) we have \(X\in\mathcal{SMZ}^+\Longleftrightarrow\) for all \(H\in \mathcal{M}_-\) we have \(X+H\neq \mathbb{Z}^\omega\).
Proof. \((\Longleftarrow)\) Assume \(X+H\neq \mathbb{Z}^\omega\) for any \(H\in \mathcal{M_-}\). Take an interval partition \(\bar{I}\in \mathbb{IP}\) and take \(H\subseteq M(y,\bar{I})\), by translating \(H\) we may assume that \(y\) is constant with value \(0\) on every coordinate. Since \(X+H\neq \mathbb{Z}^\omega\) there exists \(z\notin X+H\subseteq\bigcup_{x\in X} M(x,\bar{I}).\) So for every \(x\in X\) there exists infinitely many \(n\) such that \(z\upharpoonright I_n = x\upharpoonright I_n\). So we get \(X \in\mathcal{SMZ}^+.\)
\((\Longrightarrow)\) Take any \(X\in \mathcal{SMZ}^+\) and any \(H\in \mathcal{M}_-\), where \(H\subseteq M(y,\bar{I})\) for some \(y\in \mathbb{Z}^\omega\) and \(\bar{I}\in \mathbb{IP}.\) Since \(X\in \mathcal{SMZ}^+\) we have \(X\subseteq\{x: \exists^{\infty} n \;\; x\upharpoonright I_n =z\upharpoonright I_n \}\). We have \(X+H\subseteq \bigcup_{x\in X} M(y+x,\bar{I})\). We have to show that there exists \(a\notin X+H\). Let \(a=y+z\). Assume that \(y+z\in X+H\), then for almost all \(n\) and some \(x\in X\) \(y+x \upharpoonright I_n \neq y+z\upharpoonright I_n\) , so \(x\upharpoonright I_n\neq z\upharpoonright I_n\). We get a contradiction, because \(X\in \mathcal{SMZ}^+\), so for infinitely many \(n\) we have \(x\upharpoonright I_n =z \upharpoonright I_n\). ◻
Corollary 16. In the Baire space we have \({\mathcal{SMZ}}^+=\mathcal{M}_-^*\).
The next ideal we consider is the eventually different ideal, which was studied in [7], where a tree dichotomy for this ideal was established.
Definition 17. Let \(\mathcal{ED}\) be \(\sigma\)-ideal \(\sigma\)-generated by \(E_f\) where \(f\in\mathbb{Z}^{\omega}\) and \[E_f=\{x\in\mathbb{Z}^{\omega}: \forall^{\infty}_n \;\; x(n)\neq f(n)\}.\]
The modern notion of porosity was formulated by J.Väisälä in [8], where he proved that that every porous set in \(\mathbb{R}^n\) has Hausdorff dimension strictly less than \(n\).
Definition 18. We say that a set \(X\subseteq \mathbb{Z}^{\omega}\) is porous \((X\in \mathcal{P})\) if there exist \(k\) such that for every \(\sigma \in \mathbb{Z}^{m}\) there exists \(\tau \in \mathbb{Z}^{m+k}\), \(\sigma \subseteq \tau\) and \([\tau]\cap X=\emptyset.\)
Theorem 19. There exists \(X\in\mathcal{ED}\) such that \(X\notin \mathcal{P}\).
Proof. Assume \(E_0\) is \(k\)-porous. Take any \(y\in \mathbb{Z}^n\). Take any \(\beta\in\mathbb{Z}^{n+k}\) such that \(y\subseteq \beta\). Take \(x\) such that for \(s\leq n+k+1\) we have \(x(s)=\beta (s)\). And for \(s>n+k+1\) let \(x(s)=1\). Of course \(x\in E_0\) and \(x\in [\beta]\), so \(E_0\notin\mathcal{P}.\) ◻
Proposition 20. \(\mathcal{ED}\subseteq\mathcal{M}_{-}\).
Proof. Take any set \(E_f\). Then for any partition \(\bar{I}\) we get \(E_f\subseteq M(f, \bar{I})\in \mathcal{M}_-.\) ◻
Proposition 21. There exists \(X\in \mathcal{M}_-\) such that \(X\notin \mathcal{ED}.\)
Proof. Let \(X=M(0,\bar{I})\) where \(\bar{I}\) is a partition into intervals such that \(|I_n|=n+2\), say \(I_n=\{i_0^n,_{\cdots}, i_{n+1}^n\}\). Assume there exists \((f_n)_{n<\omega}\subseteq \mathbb{Z}^\omega\) such that \(X\subseteq\bigcup_n E_{f_n}.\)
We will construct \(x\in X\) such that \(x\notin \bigcup_n E_{f_n}.\) Let:
\(x(i_j^n)=f_j(i_j^n)\) for \(j\leqslant n\),
\(x(i_{n+1}^n)=1\).
Of course we have \(x\in X\) because of \((2)\) and \(x\notin \bigcup_n E_{f_n}\) because of \((1).\) ◻
The next ideal was studied in [9].
Definition 22. Let \(\mathcal{IE}\) be \(\sigma\)-ideal \(\sigma\)-generated by \(I_g\) where \(g\in \mathbb{Z}^\omega\) and \[I_g=\{x\in \mathbb{Z}^\omega:\exists^\infty n \;\; x(n)=g(n)\}.\]
We will be also interested in the following ideal that was used by Bartoszyński and Judach in [3] to characterize \(add(\mathcal{M})\) and \(cof(\mathcal{M}).\)
Definition 23. Let \(\mathcal{H}\) be \(\sigma\)-ideal generated by \(H_{\bar{I},f}\) where \(f\in\mathbb{Z}^{\omega}\), \(\bar{I}\in \mathbb{IP}\) and \[H_{\bar{I},f}=\{x\in\mathbb{Z}^{\omega}: \forall^{\infty}n \exists m\in I_n \;\; x(m)= f(m)\}.\]
Note that \(\forall (\bar{I}^n)_{n<\omega}\in \mathbb{IP}\) and \(\forall(f_n)_{n<\omega}\in \mathbb{Z}^\omega\) there is \(\bar{J}\in \mathbb{IP}\) and \(g\in \mathbb{Z}^\omega\) such that \(\bigcup_{n} H_{\bar{I}^n, f_n}\subseteq H_{\bar{J},g}\).
Definition 24. Set \(X\in \mathcal{G}\) if and only if for every interval partition \(\bar{I}\in \mathbb{IP}\) there exists \(z\in\mathbb{Z}^\omega\) such that: \[\forall x\in X\;\; \exists^\infty n\;\; \forall m\in I_n \;\; x(m)\neq z(m).\]
Proposition 25. \(\mathcal{H}\subseteq\mathcal{M}_-\).
Proof. For any \(H_{\bar{I},f}\in \mathcal{H}\) we have \(H_{\bar{I},f}\subseteq M(f+1,\bar{I}).\) ◻
Proposition 26. There exists \(X\in \mathcal{M}_-\) such that \(X\notin \mathcal{H}\).
Proof. Let \(X=M(0,\bar{I})\) where \(\bar{I}\) is a partition into singletons. Assume there exists \(\bar{J}\in\mathbb{IP},f\in \mathbb{Z}^\omega\) such that \(M(0,\bar{I})\subseteq H_{\bar{J},f}\). We will construct \(x\in \mathbb{Z}^\omega\) as follows:
If \(f(n)\neq 1\), let \(x(n)=f(n)-1\),
if \(f(n)=1\) let \(x(n)=2\).
Of course \(x\in M(0,\bar{I})\) and \(x\notin H_{\bar{J},f}.\) ◻
Proposition 27. \(\mathcal{H}\subseteq\mathcal{IE}\).
Proof. Note that for any \(H_{\bar{I},f}\in \mathcal{H}\) we have \(H_{\bar{I},f}\subseteq I_f\). ◻
Proposition 28. There exists \(X\in \mathcal{IE}\) such that \(X\notin\mathcal{H}\).
Proof. Take any set \(I_f\in \mathcal{IE}.\) Assume that \(I_f\subseteq H_{\bar{I},f}\) for some interval partition \(\bar{I}.\) But there exists \(x\in I_f\) such that for every \(k\) and every \(m\in I_{2k}\) we have \(x(m)\neq f(m).\) ◻
Proposition 29. \(\mathcal{ED}\subseteq\mathcal{G}\).
Proof. Take any \(E_f\in \mathcal{ED}\). To show that \(E_f \in \mathcal{G}\) take any partition \(\bar{I}\). Take \(f\). Of course for every \(x\in E_f\), there exist infinitely many \(n\) such that for every \(m\in I_n\) we have \(f(m)\neq x(m),\) because for every \(x\in E_f\) we have \(x\neq^*f.\) ◻
All of these ideals can be reformulated in the Cantor space.
Proposition 30. In Cantor space \(\mathcal{H}=\mathcal{M}_-=\mathcal{M}\).
Proof. We only need to show that in Cantor space \(\mathcal{M}_- \subseteq\mathcal{H}\). Take any \(x\in\mathcal{M_-}\). We know that \(X\subseteq M(x,\bar{I})\) for some \(x\in 2^\omega\) and \(\bar{I}\in \mathbb{IP}.\) We have \(X\subseteq H_{\bar{I},x+1}\). ◻
Proposition 31. In Cantor space \(\mathcal{G}=\mathcal{SMZ}^+=\mathcal{SMZ}\).
Proof. We only need to show that in Cantor space \(\mathcal{G} \subseteq\mathcal{SMZ}^+\). Take any \(X\in\mathcal{G}\) and any partition \(\bar{I}\in \mathbb{IP}\). We will find \(z\in 2^\omega\) such that for every \(x\in X \;\;\exists^\infty n \;\; x\upharpoonright I_n=z\upharpoonright I_n\). Since \(X\in \mathcal{G}\) we know that there exists \(a\in2^\omega\) such that \(\forall x\in X\;\; \exists^\infty n\;\; \forall m\in I_n \;\; x(m)\neq a(m)\). Let \(z=a+1.\) ◻
Proposition 32. \(\mathcal{G}\) forms a \(\sigma\)-ideal in the Baire space.
Proof. Take \((X_l: l<\omega)\) where each \(X_l \in \mathcal{G}.\) We will show that \(\bigcup_{l<\omega } X_l\in\mathcal{G}\).
Take any interval partition \(\bar{I}.\) We will construct a bigger partition \(\bar{J}.\) Let \[J_0=I_0,\;J_1=I_1\cup I_2,\; J_n=I_{\frac{n\cdot(n-1)}{2}+n}\cup \cdots \cup I_{\frac{n\cdot(n-1)}{2}+2n}.\] Since every \(X_l\) is in \(\mathcal{G}\) there exists \(z_l\) for every \(X_l\) such that: \[\forall x\in X_l \;\; \exists^\infty n\; \forall m\in J_n \;\; x(m)\neq z_l(m).\] We want to construct \(z\) such that \(\forall l\;\forall x\in X_l \;\exists^\infty n\;\forall m\in J_n \;\; z(m)\neq x(m)\).
Let \(z\upharpoonright I_0 =z_1\upharpoonright I_0\). Next, if \(k=\frac{n(n-1)}{2}+n\) for some \(n\) and\)0<b<k+1\(define\)zI_k=z_1I_k\(and\)zI_k+b=z_b+1I_k+b\(. \end{proof} \begin{proposition}\)^+\(forms a\) \(-ideal in Baire space. \end{proposition} \begin{proof} Take\)(X_l: l<)\(where each\)X_l ^+.\(We will show that\)_l< X_l^+\(. Take any interval partition\)|I.\(We will construct a bigger partition\)|J.\(Let:\) \(J_0=I_0, \;J_1=I_1\cup I_2,\; J_n=I_{\frac{n\cdot(n-1)}{2}+n}\cup \cdots \cup I_{\frac{n\cdot(n-1)}{2}+2n}.\) \(Since every\)X_l\(is in\)^+\(there exists\)z_l\(for every\)X_l\(such that:\) \(\forall x\in X_l \;\; \exists^\infty n \;\; x\upharpoonright J_n= z_l\upharpoonright J_n.\) \(We want to construct\)z\(such that\)l xX_l ^_n zJ_n= xJ_n\(. Let\)zI_0 =z_1I_0\(. Next, if\)k=+n for some \(n\) and \(0<b<k+1\) define \(z\upharpoonright I_k=z_1\upharpoonright I_k\) and \(z\upharpoonright I_{k+b}=z_{b+1}\upharpoonright I_{k+b}\). ◻
The next ideal was studied in [10] where the authors proved that a Polish group \(G\) is locally compact if and only if \(\mathcal{M}(G) = \mathcal{UM}(G)\).
Definition 33. Set \(X\subseteq\mathbb{Z}^\omega\) is uniformly nowhere dense if for every \(k\in\omega\) there exists \(l\in \omega\) such that for every \(\alpha\in \mathbb{Z}^k\) there exists \(\beta\in \mathbb{Z}^{k+l}\) such that \(\alpha\subseteq\beta\) and \([\beta]\cap X=\emptyset\).
Definition 34. Set \(X\subseteq\mathbb{Z}^\omega\) \((X\in \mathcal{UM})\) is uniformly meager if it is a countable union of uniformly nowhere dense sets.
Theorem 35. Set \(X\subseteq\mathbb{Z}^\omega\) is uniformly meager if and only if there exists increasing sequence \(a\in \omega^\omega\) and a function \(\psi:\bigcup\mathbb{Z}^{a_{n}}\rightarrow\mathbb{Z}^{<\omega}\) such that \(|\psi(\sigma)|=a_{n+1}-a_n\), and \[X\subseteq M(a,\psi)=\{x: \forall^\infty n \;\; x\upharpoonright [a_n, a_{n+1}) \neq \psi (x\upharpoonright a_n) \}.\]
Proof. \((\Longrightarrow)\) First we will show that set \(Y=\{x: \forall\; n \;\; x\upharpoonright [a_n, a_{n+1}) \neq \psi (x\upharpoonright a_n) \}\) is uniformly nowhere dense. Take any \(k\in\omega\), \(x\in Y\) and \(x\upharpoonright k\). Of course \(k\in [a_n, a_{n+1})\) for some \(n.\) Let \(\alpha=(x\upharpoonright k)^\frown{{\underbrace{ 0^\frown 0^\frown\ldots^\frown 0}_{a_{n+1}-k}}}\). We know that there exists \(\psi(\alpha)\) such that \(Y\cap [\alpha]=\emptyset.\) Let \(\beta=(x\upharpoonright k)^\frown{{\underbrace{ 0^\frown 0^\frown\ldots^\frown 0}_{a_{n+1}-k}}}^\frown\psi(\alpha).\)
Now we will show that for every uniformly meager set \(X\) there exists \(M(a,\psi)\) such that \(X\subseteq M(a,\psi)\). Take \((X_k)_{n<\omega}\)- uniformly nowhere dense such that \(X=\bigcup_k X_k\), where \(X_k\subseteq X_{k+1}\). For every \(X_k\) we have \(X_k\subseteq \{x: \forall\; n \;\; x\upharpoonright [a^k_n, a^k_{n+1}) \neq \psi_k (x\upharpoonright a^k_n) \}\) and \(a_n^k\subseteq a_n^{k+1}.\) We will construct \(a\in \omega^\omega\), and \(\psi:\bigcup\mathbb{Z}^{a_{n}}\rightarrow\mathbb{Z}^{<\omega}\). Let \(a_n=a_n^n\) and \(\psi(y\upharpoonright a_n)= \psi_n(y\upharpoonright a_n)^\frown{{\underbrace{ 0^\frown 0^\frown\ldots^\frown 0}_{a_{n+1- a^n_{n+1}}}}}.\) Of course we have \(X\subseteq M(a,\psi)\)
\((\Longleftarrow)\) For every \(M(a,\psi)\) we have \(M(a,\psi)=\bigcup_m \{x: \forall\; n\geqslant m \;\; x\upharpoonright [a_n, a_{n+1}) \neq \psi (x\upharpoonright a_n) \}\). Of course \(\{x: \forall\; n\geqslant m \;\; x\upharpoonright [a_n, a_{n+1}) \neq \psi (x\upharpoonright a_n) \}\) is a uniformly nowhere dense set. ◻
Proposition 36. \(\mathcal{M}_-\subseteq\mathcal{UM}\).
Proof. Take any \(M(x,\bar{I})\in \mathcal{M}_-\) where \(\bar{I}=\{[a_n,a_{n+1}):n\in\omega\}\). For \(\sigma\in \mathbb{Z}^{a_{n}}\) and \(n\in \omega\) set \(\psi (\sigma)=x\upharpoonright [a_n,a_{n+1}).\) Of course \(M(x,\bar{I})\subseteq M(a,\psi)\). ◻
Theorem 37. There exists \(X\in \mathcal{UM}\) such that \(X\notin \mathcal{M}_-.\)
Proof. Let \(\psi\) be a \(1-1\) and onto function and let \(a(n)=n\). Take \(X=M(a,\psi)=\{x:\forall^\infty n \;\; x(n)\neq \psi(x\upharpoonright n)\}\). We will show that \(X\nsubseteq M(y,\bar{I})\) for any \(x\in \mathbb{Z}^\omega\) and \(I\in \mathbb{IP}.\) We will construct \(z\in X\backslash M(y,\bar{I})\).
Let \(z\in \mathbb{Z}^{\bigcup_{i<n}I_{2i}\cup I_{2i+1}}\) be such that:
\(z_n(j)\neq\psi(z_n\upharpoonright j)\) for \(j\in I_{2n}\cup I_{2n+1}\),
\(z\upharpoonright I_{2n+1}=y\upharpoonright I_{2n+1}.\)
We can construct such \(z\), because for every \(i\in I_{2n}\cup I_{2n+1}\) there exist finitely many \(\tau_i\in \mathbb{Z}^{\max I_{2n+1}}\) such that:
\(z \upharpoonright min I_{2n}\subseteq \tau_i,\)
\(\tau_i (j)=\psi (y\upharpoonright j)\) for \(j\in I_{2n+1}\).
◻
The inclusions discussed above are illustrated in the diagram below. All inclusions are proper, with proof of \(\mathcal{ED}\neq \mathcal{G}\) provided in chapter \(4\) under assumption of \(add(\mathcal{H})=\mathfrak{c}.\)
Figure 2: 
.
3 Cardinal invariants of \(\mathcal{M}_-\)↩︎
In this chapter, we will show that cardinal invariants of \(\mathcal{M}_-\) are equal to the corresponding cardinal invariants of a \(\sigma\)-ideal of meager sets. We say that family \(F\subseteq \omega^\omega\) is infinitely equal if for every \(g\in\omega^\omega\) we can find \(f\in F\) such that \(\exists^\infty n\) such that \(f(n)=g(n).\) We say that family \(F\subseteq \omega^\omega\) is eventually different if for any \(g\in \omega^\omega\) we can find \(f\in F\) such that \(\forall^\infty n\) we have \(f(n)\neq g(n).\) We recall the combinatorial characterizations of the cardinal coefficients \(non(\mathcal{M})\) and \(cov(\mathcal{M})\) in terms of functions in the Baire space.
Theorem 38. (Bartoszyński, Miller [3])
\(non(\mathcal{M})=min\{|F|: F\subseteq \omega^\omega \wedge F \text{ is infinitely equal}\}\),
\(cov(\mathcal{M})=min\{|F|: F\subseteq \omega^\omega \wedge F \text{ is eventually different}\}\).
Proposition 39. \(cov(\mathcal{M}_-)= cov(\mathcal{M})\) and \(non(\mathcal{M}_-)=non(\mathcal{M})\).
Proof. Since \(\mathcal{M}_- \subseteq\mathcal{M}\) we clearly have \(cov(\mathcal{M}_-)\geqslant cov(\mathcal{M})\) and \(non(\mathcal{M}_-)\leqslant non (\mathcal{M})\).
To prove the other inequality we will use characterization of \(cov(\mathcal{M)}\) from 38. Let family \(F=\{x_\alpha\in \omega^\omega : \alpha<cov(\mathcal{M})\}\) be an eventually different family.
Take any interval partition \(\bar{I}\) and let \(M_\alpha=M(x_\alpha,\bar{I})\). We will show that \(\bigcup M_\alpha =\omega^\omega\). Take any \(y\in\omega^\omega.\) Since family \(F\) is of size \(cov(\mathcal{M})\) we can find \(x_\alpha \in F\) such that \(x_\alpha\neq^*y\). So \(y\in M(x_\alpha,I).\)
To prove the other inequality take set \(\{x_\alpha: \alpha < non(\mathcal{M}_-)\}\notin M(y, \bar{I})\) for every \(y\in \omega^\omega\) and any interval partition \(\bar{I}.\) So for every \(y\in\omega^\omega\) there exists \(x_\alpha\) such that \(x_\alpha=^*y\). So the family \(\{x_\alpha: \alpha<non(\mathcal{M}_-)\}\) is exactly the family from 38. ◻
Theorem 40. \(add(\mathcal{M}_-)= add(\mathcal{M})\).
Proof. Recall that \(\mathrm{add}(\mathcal{M})=min\{\mathfrak{b},cov(\mathcal{M})\}\). We first show that \(add(\mathcal{M}_-)\leq \mathfrak{b}\). We will use characterization of \(\mathfrak{b}\) from 4.
We want to construct family \(\{M_\alpha : \alpha < \mathfrak{b}\}\subseteq \mathcal{M}_-\) such that \(\bigcup_{\alpha} M_{\alpha}\notin \mathcal{M}_-.\)
Take \(\bar{I}^\alpha\), unbounded family of interval partitions. Define \(M_\alpha=M(0,\bar{I}^\alpha)\).
Assume that there exist \(X\in \mathcal{M}_-\) such that \(\bigcup_\alpha M_\alpha =X\) and \(X=M(y,\bar{J})\). Then for every \(\alpha\) we have \(M(0,\bar{I}^\alpha) \subseteq M(y, \bar{J})\). Using 7 we get that for all \(\alpha\) \(\bar{I}^\alpha\sqsubseteq^*\bar{J}\). That is contradiction, because \(\bar{I}^\alpha\) was unbounded.
Next we will show \(add(\mathcal{M}_-) \leq cov(\mathcal{M})\). Since \(cov(\mathcal{M}_-)= cov(\mathcal{M})\) we have that \(add(\mathcal{M}_-)\leq cov(\mathcal{M}_-)\leq cov(\mathcal{M}).\)
To prove the other inequality assume \(add(\mathcal{M}_-)< add(\mathcal{M})=min\{\mathfrak{b}, cov(\mathcal{M})\}\).
Take \((M_\alpha:\alpha<\kappa<add(\mathcal{M}))\) and \(M_\alpha=M(x_\alpha,\bar{I}^\alpha).\) We will show that \(\bigcup_{\alpha<\kappa} M_\alpha\in\mathcal{M}_-\).
Since \(\kappa<cov(\mathcal{M})\) there exists \(z\notin\bigcup M_\alpha\). Hence, \(\forall\alpha<\kappa\;\; \exists^\infty n \;\; z\upharpoonright I_n^\alpha=x_\alpha \upharpoonright I_n^\alpha\).
Take \((\bar{J}^\alpha:\alpha<\kappa)\) such that \(\forall n\; \exists k\;(I_k^\alpha \subseteq J_n^\alpha \wedge z\upharpoonright I_k^\alpha= x_\alpha\upharpoonright I_n^\alpha)\). Since \(\kappa<\mathfrak{b}\) there is \(\bar{R}\in\mathbb{IP}\) such that \(\forall\alpha \;\; \bar{J}^\alpha \sqsubseteq^* \bar{R}\). So, \(\bigcup_{\alpha<\kappa} M_\alpha \subseteq M(z,\bar{R})\). ◻
To show \(cof(\mathcal{M}_-)=cof(\mathcal{M})\) we will need Tukey reduction. We recall basic definitions below.
Definition 41. Let \(R_0=(X_0, \sqsubseteq_0, Y_0), \;\; R_1=(X_1,\subseteq_1, Y_1)\), where \(X_0, X_1, Y_0, Y_1\) are sets and \(\sqsubseteq_o, \subseteq_1\) relations. We say that \(R_0\leq_T R_1\) if there exist:
\(\varphi: X_0\rightarrow X_1\)
\(\psi: Y_1 \rightarrow Y_0\)
such that for all \(y\in Y_0, z\subseteq X_0\) if \(\phi[z]\sqsubseteq_1 y\) then \(z\subseteq_1 \psi (y)\).
Definition 42. \(R_0 \otimes R_1 =(X_0 \times X_1 ^{Y_0}, \sqsubseteq_\otimes, Y_0\times Y_1)\), where for \(x\in X_0, f\in X_1^{Y_0}, y\in Y_0, z\in Y_1\) we have \((x,f)\sqsubseteq_\otimes (y,z)\) if and only if \(x\sqsubseteq_0y \wedge f(y)\subseteq_1 z.\)
Theorem 43. If for ideals \(\mathcal{I},\mathcal{J}\) we get \(\mathcal{I}\leq_T \mathcal{J}\), then:
\(add(\mathcal{I})\geq \mathrm{add}(\mathcal{J})\)
\(cof(\mathcal{I})\leq cof(\mathcal{J})\)
Theorem 44. \((\mathcal{M}_-,\notin, \omega^\omega)\otimes(\mathbb{IP},\sqsubseteq^*,\mathbb{IP})\geq_T(\mathcal{M}_-,\subseteq, \mathcal{M}_-)\)
Proof. We will need two functions:
\(\varphi:\mathcal{M}_-\rightarrow \mathcal{M}_- \times \mathbb{IP}^{\omega^\omega}\)
\(\psi:\omega^\omega\times\mathbb{IP} \rightarrow \mathcal{M}_-\)
Let \(\psi(x,\bar{I})=M(x,\bar{I})\in \mathcal{M}_-\). Let \(\varphi(N)=(\tilde{N},f)\in \mathcal{M}_- \times \mathbb{IP}^{\omega^\omega}\), where:
\(N\subseteq\tilde{N}=M(y, \bar{J})\)
\(f(x)=\bar{R}\) such that:
if \(x\in \tilde{N}\) we can take anything
if \(x\notin \tilde{N}\) take \(\bar{R}\) such that \(\bar{J}\sqsubseteq^* \bar{R}\) and \(\forall n \exists k \left( J_k \subseteq R_n \right)\) and \(x\upharpoonright J_k= y \upharpoonright J_k\)
We have to show that \(\varphi (N) \sqsubseteq_\otimes (x, \bar{I}) \Longrightarrow N\subseteq\psi(x,\bar{I})=M(x,\bar{I})\)
Since \(\varphi (N) \sqsubseteq_\otimes (x, \bar{I})\) we know that:
\(x\notin \tilde{N}\)
\(f(x) \sqsubseteq^{*} \bar{I}\)
So we have that \(\exists^\infty n \;\; x\upharpoonright J_n =y\upharpoonright J_n\), and \(\forall n \exists k (J_k \subseteq R_n \wedge x\upharpoonright J_k =y \upharpoonright J_k)\).
So \(\forall^{\infty} n \exists k (R_k\subseteq I_n \wedge x\upharpoonright R_k=y\upharpoonright R_k).\)
Take any \(w\in N\subseteq \tilde{N} =M(y,\bar{J})\). So \(\forall ^{\infty} n \;\; w\upharpoonright J_n \neq y \upharpoonright J_n\). So we get \(w\in M(x, \bar{I})=\psi(x, \bar{I}).\) ◻
Corollary 45. \(add(\mathcal{M}_-)\geq add(\mathcal{M})\), \(cof(\mathcal{M_-})\leq cof(\mathcal{M})\).
Proposition 46. \(cof(\mathcal{M_-})\geq cof(\mathcal{M})\).
Proof. We will use the fact that \(cof(\mathcal{M})=max\{\mathfrak{d},non(\mathcal{M})\}\)
First we will show \(cof(\mathcal{M}_-)\geq non(\mathcal{M})\).
Since \(non(\mathcal{M_-})=non(\mathcal{M})\) we have \(cof(\mathcal{M}_-)\geq non(\mathcal{M}_-)=non(\mathcal{M}).\)
Next we will show \(cof(\mathcal{M}_-)\geq \mathfrak{d}\). Again we will use characterization from 4
Take family \(\mathcal{B}\subseteq\mathcal{M}_-\) such that \(\forall A \in \mathcal{M}_- \exists B_\alpha\in \mathcal{B}\) and \(A\subseteq B_\alpha.\) Let \(B_\alpha\subseteq M(x_\alpha, \bar{I}^\alpha).\) Take any partition \(\bar{J}\) and \(A\subseteq M(y,\bar{J})\). There exists \(B_\alpha\) such that \(A\subseteq B_\alpha\); so \(M(y,\bar{J})\subseteq M(x_\alpha, \bar{I}^\alpha)\) and so \(\bar{J}\sqsubseteq^*\bar{I}^\alpha.\) And we get that \(\{\bar{I}^\alpha:\alpha < cof(\mathcal{M}_-)\}\) is a dominating family. ◻
Definition 47. For \(\sigma\)-ideal \(\mathcal{I}\) on a Polish space \(X\) we define transitive covering of \(\mathcal{I}\) as: \[cov_t(\mathcal{I})=min\{|\mathcal{A}|: \exists Y \in \mathcal{I} \;\; Y+\mathcal{A}=X\}.\]
Of course for any \(\sigma\)-ideal \(\mathcal{I}\) we have \(cov_{t}(\mathcal{I})\geq cov(\mathcal{I})\) \(cov_t(\mathcal{I})=non(\mathcal{I})^*.\)
In [11] authors showed that \(cov_t(\mathcal{M})\) on \(\mathbb{Z}^\omega= cov(\mathcal{M})\). We can prove the same with \(\mathcal{M}_-.\)
Proposition 48. \(cov_t(\mathcal{M}_-)=cov(\mathcal{M}).\)
Proof. We only need to show the inequality \(cov_t(\mathcal{M}_-)<cov(\mathcal{M})\).
Take \(F=\{x_\alpha: \alpha< cov(\mathcal{M})\}\) and \(N\subseteq M(0,\bar{I})\) for some \(\bar{I}\in \mathbb{IP}.\) So \(N+F=\bigcup_{\alpha<cov(\mathcal{M})} (x_\alpha, \bar{I})\). Assume that there exists \(y\notin N+F\). So, \(\forall \alpha\exists^\infty _n \;\; y\upharpoonright I_n =x_\alpha \upharpoonright I_n\), so \(\forall \alpha \exists ^\infty_n \;\; y(n)=x_\alpha(n)\). Contradiction, because \(F\) is of size \(cov(\mathcal{M}).\) ◻
4 Operation \(^*\)↩︎
The operation \(^*\) was first defined and investigated in [2] by Seredyński. It was later studied by Pawlikowski and Sabok who showed in [12] that \(Count = Count^{**}\). Moreover, Solecki in [13] constructed a \(\sigma\)-ideal \(\mathcal{I}\) satisfying \(\mathcal{I}^* = Count\).
We begin with some basic facts about operation \(^*\).
Fact 49. For any \(\mathcal{F,G} \subseteq \mathcal{P}(X)\) we have:
\(\mathcal{G}\subseteq\mathcal{F}^{*} \Longrightarrow \mathcal{F}\subseteq\mathcal{G}^{*}\),
\(\mathcal{F}\subseteq\mathcal{F}^{**}\),
\(\mathcal{G}\subseteq\mathcal{F} \Longrightarrow \mathcal{F}^*\subseteq\mathcal{G}^{*}\),
\(\mathcal{F}^*=\mathcal{F}^{***}\),
\(\mathcal{F}^{*}\) is closed under taking subsets and translation invariant,
\(\mathcal{F}=\mathcal{F}^{**}\Longleftrightarrow \exists \mathcal{A}\; \mathcal{F}=\mathcal{A}^{*}.\)
Now, we will demonstrate some relation between the ideals defined in Chapter \(2\) with respect to operation \(^*\).
Proposition 50. \(\mathcal{ED}^*=\mathcal{IE}\) and \(\mathcal{IE}^*=\mathcal{ED}\).
Proof. It will be enough to check it for generators.
First we will how that \(\mathcal{ED}^*\subseteq\mathcal{IE}\). Take any \(X\in \mathcal{ED}^*\). For every \(E_f \in \mathcal{ED}\) there exists \(z\notin \bigcup_{x\in X} E_f+x\). So for every \(x\in X\) we have \(x\notin -E_f+z=E_g\) for some \(g\in \mathbb{Z}^\omega\). So \(x\in\{y:\exists^\infty n\; y(n)=g(n)\}\) and so \(X\subseteq I_g\).
Next we will show \(\mathcal{IE}\subseteq\mathcal{ED}^*\). Take any \(I_g\in\mathcal{IE}\), and any \(E_f\in \mathcal{ED}.\) We have to find \(z\notin E_f+I_g\). Let \(z=f+g.\)
Next we will show \(\mathcal{IE}^*\subseteq\mathcal{ED}\). Take any \(X\in \mathcal{IE}^*\). For every \(I_f \in \mathcal{IE}\) there exists \(z\notin \bigcup_{x\in X} I_f+x\). So for every \(x\in X\) we have \(x\notin -I_f+z=I_g\) for some \(g\in \mathbb{Z}^\omega\). So \(x\in\{y:\forall^\infty n \;\; g(n)\neq y(n)\}\) and so \(X\subseteq E_g\).
Next we will show \(\mathcal{ED}\subseteq\mathcal{IE}^*\). Take any \(E_f\in \mathcal{ED}\) and any \(I_g\in\mathcal{IE}\). We have to find \(z\notin E_f+I_g\). Let \(z=f+g.\) ◻
Corollary 51. \(\mathcal{ED}=\mathcal{ED^{**}}\), \(\mathcal{IE}=\mathcal{IE}^{**}\), \(\mathcal{SMZ}^+\subseteq\mathcal{IE}\).
Theorem 52. \(X\in \mathcal{G}\Longleftrightarrow \forall H\in\mathcal{H} \;\; H +X\neq \mathbb{Z}^\omega\).
Proof. \(\Longrightarrow\) Take any \(X \in \mathcal{G}\) and \(H_{\bar{I},f}\). We want to find \(z \notin X + H_{\bar{I},f}\). Since \(X \in \mathcal{G}\) we can find \(g\) such that \(\exists^\infty n\; \forall m \in I_n \;x(m) \neq g(m)\). Let \(z(m) = f(m) + g(m)\).
\(\Longleftarrow\) Assume \(X \notin \mathcal{G}\). We will show that there exists \(\bar{I}\in \mathbb{IP}\) and \(g\in\mathbb{Z}^\omega\) such that \(X + H_{\bar{I},g} = \mathbb{Z}^\omega\).
Since \(X \notin E\), the negation of the definition holds: \[\exists \bar{I} \in \mathbb{IP} \;\; \forall f \in \mathbb{Z}^\omega \;\; \exists x \in X \;\; \forall^\infty n \;\; \exists m \in I_n \;\; x(m) = f(m).\] Take this partition \(\bar{I}\). Take any \(g \in \mathbb{Z}^\omega\).
Take any \(a \in \mathbb{Z}^\omega\). We want to find \(x \in X\) and \(y \in H_{\bar{I},g}\) such that \(a = x + y\).
We know that \(a=f+g\) for some \(f\). Take \(x_f\in X\) such that \(\forall^\infty n \;\;\exists m\in I_n \;\; x(m)=f(m).\) Let \(x=x_f\).
Now we will construct \(y\in H_{\bar{I},g}\) such that \(a=x_f+y.\)
If \(f(n)=x_f(n)\) let \(y(n)=g(n)\),
else let \(y(n)=a(n)-x_f(n).\)
Of course \(y\in H_{\bar{I},g}\), because \(y(n)=g(n)\) if and only if \(x_f(n)=f(n)\). ◻
Corollary 53. \(\mathcal{H}^*=\mathcal{G}\), \(\mathcal{SMZ}^+\subseteq\mathcal{G}\).
The diagram below presents the relationships between the ideals, including \(\mathcal{SMZ}\) and \(\mathcal{SMZ}^+.\)
Figure 3: 
.
We will need Luzin set to show that \(\mathcal{G}\nsubseteq\mathcal{M}_-\) and \(\mathcal{IE}\nsubseteq \mathcal{M}_-\).
Definition 54. A subset \(L\) of \(\omega^\omega\) is called a Luzin set if \(L\) is uncountable, but for every nowhere dense subset \(K\) of \(\omega^\omega\) the intersection \(K \cap L\) is countable.
It is known that Luzin set has strong measure zero, but is not meager.
Corollary 55. Assuming the existence of a Luzin set, we have \(\mathcal{G}\nsubseteq\mathcal{M}_-\) and \(\mathcal{IE}\nsubseteq \mathcal{M}_-\).
Theorem 56. There exists \(M\in\mathcal{M}_-\) and \(K\in K_\sigma\) such that \(M+K=\mathbb{Z}^\omega.\)
Proof. Let \(M=M(0,\bar{I})\), where \(\bar{I}\) is a partition into singletons. Let \(K=\{x\in \mathbb{Z}^\omega:\forall^\infty_n \;\;|x(n)|<2\}\). Take any \(z\in \mathbb{Z}^\omega.\) We have to find \(x\in M\) and \(y\in K\) such that \(x+y=z.\)
If \(\forall^\infty_n z(n)\neq0\), then \(z\in M\), so \(z=z+0\).
If \(\exists^\infty_n \;z(n)=0\) take \(x,y\) as follows:
if \(z(n)\neq0\) let \(x(n)=z(n)\) and \(y(n)=0.\)
if \(z(n)=0\) let \(x(n)=1\) and \(y(n)=-1.\)
◻
Theorem 57. \(K_\sigma^*\) is not an ideal.
Proof. Let \(\mathcal{U}\) be an ultrafilter on \(\omega\). Let \(A=\{x\in\mathbb{Z}^\omega: \{n:x(n)\geqslant0\}\in \mathcal{U}\}\). Let \(B=\mathbb{Z}^\omega\backslash A=\{x\in\mathbb{Z}^\omega: \{n:x(n)<0\}\in \mathcal{U}\}\) Note that \(A,B\in K_\sigma^*\) and \(A\cup B=\mathbb{Z}^\omega\) it follows that \(K_\sigma^*\) is not an ideal. ◻
Next, we will use the operation \(^*\) to show that some of the inclusions the diagram are proper.
Proposition 58. There exists \(X\in \mathcal{H}\) such that \(X\notin \mathcal{SMZ}^+.\)
Proof. Take \(\bar{I}\)- interval partition into intervals of length two. Take \(H_{\bar{I},1}\) and \(M(0,\bar{I})\). We will show that \(H_{\bar{I},1}+M(0,\bar{I})=\mathbb{Z}^\omega\). Take any \(a\in \mathbb{Z}^\omega\), and construct \(y\in M(0,\bar{I})\) and \(x\in H_{\bar{I},1}\) such that:
If \(n\) is odd, let \(x(n)=1\) and \(y(n)=a(n)-1\),
if \(n\) is even, let \(x(n)=a(n)-1\) and \(y(n)=1.\)
Of course \(y\in M(0, \bar{I})\), because in every interval we can find \(m\) such that \(y(m)=1\). Similarly \(x\in H_{\bar{I},f}\) because in every interval we can find \(m\) such that \(x(m)=1.\) ◻
Corollary 59. There exists \(X\in \mathcal{IE}\) such that \(X\notin \mathcal{SMZ}^+.\)
Theorem 60. There exists \(Y\in \mathcal{ED}\) such that \(Y\notin \mathcal{SMZ}^+.\)
Proof. Let \(Y=\{x:\forall^\infty n \;\; x(n)\neq 0\}\in \mathcal{ED}.\) We will show that \(Y\notin \mathcal{M}_-^*=\mathcal{SMZ}^+.\) Take \(M(0,\bar{I})\), where \(\bar{I}\) is partition into singletons. Take any \(a\in \mathbb{Z}^\omega.\) We have to find \(f\in Y\) and \(g\in M(0,\bar{I})\) such that \(f+g=a\).
If \(a(n)\neq 1\) let \(f(n)=z-1\) and \(g(n)=1\)
If \(a(n)=1\) let \(f(n)=2\) and \(g(n)=-1.\)
Of course \(g\in M(0,\bar{I})\) because \(g(n)\neq 0\) for every \(n\). Also \(f\in Y\) because \(f(n)\neq 0\) for every \(n.\) ◻
Corollary 61. There exists \(Y\in \mathcal{G}\) such that \(Y\notin\mathcal{SMZ}^+.\)
Next, we will need a Theorem from [14] to show that, under additional assumptions, we can separate some of these ideals.
Theorem 62. If \(\mathcal{J} \subseteq \mathcal{P}(\mathbb{Z}^{\omega})\) is a translation and reflection invariant proper \(\sigma\)-ideal such that \(cof(\mathcal{J})\leqslant \mathfrak{c}\), \(add(\mathcal{J})= \mathfrak{c}\) and for any \(A \notin \mathcal{J}\), we have: \[(\mathcal{J} \cup \{A\})^* \neq \mathcal{J}^*.\]
Corollary 63. Assuming \(add(\mathcal{H})=\mathfrak{c}\) there exists \(X\in \mathcal{G}\) such that \(X\notin\mathcal{ED}\).
Proof. Since assuming \(add(\mathcal{H})=\mathfrak{c}\) from Theorem 62 we get \(\mathcal{ED}\neq \mathcal{G}.\) ◻
More general Theorem was showed in [10], but we present an easier proof.
Theorem 64. \(\mathcal{SMZ}\subseteq \mathcal{UM}^*\).
Proof. Take any \(X\in\mathcal{SMZ}\) and \(H\in \mathcal{UM}\) by 35 there are \(\bar{a}=(a_n)_{n<\omega}\) and \(\psi:\bigcup\omega^{a_{n}}\rightarrow\mathbb{Z}^{<\omega}\) such that \(H\subseteq M(a,\psi)\). Take \(a_0=0.\) Let \((\sigma_n)_{n<\omega}\) be such that \(|\sigma_n|=a_{n+1}-a_n\) and \(X\subseteq\{x:\exists ^\infty _n \;\; \sigma_n\subseteq x\}.\)
We will construct \(\tau\notin X+H\). Let:
\(\tau_0=\psi(\emptyset)+\sigma_0\)
\(\tau_n= \tau_{n-1}^\frown (\psi (\tau_{n-1} -\sigma_n \upharpoonright [a_0,a_n))+\sigma_n\upharpoonright [a_n, a_{n+1}))\)
Let \(\tau=\bigcup_{n} \tau_n\). ◻
Corollary 65. \(add(\mathcal{M})=\mathfrak{c}\) implies that there is \(X\in \mathcal{SMZ}^+\) such that \(X\notin\mathcal{SMZ}\).
Proof. We know that \(\mathcal{SMZ}\subseteq\mathcal{UM}^*\). By 62 we get that \(\mathcal{SMZ}\neq\mathcal{SMZ}^+\). ◻
Next we will show corollary related to cardinal invariants of ideals in \(\mathbb{Z}^\omega\).
Corollary 66. \[cov(\mathcal{M})=cov_t(\mathcal{M}_-)=non(\mathcal{SMZ}^+)= non(\mathcal{SMZ})= non(\mathcal{M}^*)=cov_t(\mathcal{M})= cov(\mathcal{M})\]
Proof. From 48 we know that \(cov(\mathcal{M})=cov_t(\mathcal{M}_-)\). Since \(\mathcal{M}_-^*=\mathcal{SMZ}^+\) we get the next equality. Because \(\mathcal{SMZ}\subseteq \mathcal{SMZ}^+\) we get \(non(\mathcal{SMZ})\leqslant non(\mathcal{SMZ}^+)\). Since \(\mathcal{M}^*\subseteq\mathcal{SMZ}\) we have \(non(\mathcal{M}^*)\leqslant non (\mathcal{SMZ})\). Of course \(non(\mathcal{M}^*)=cov_t(\mathcal{M})\). Also we have \(cov(\mathcal{M})= cov_t(\mathcal{M})\). So we get that all the above cardinal invariants are equal to \(cov(\mathcal{M}).\) ◻
5 Tree dichotomy for \(\mathcal{M}_-\)↩︎
In [7] Khomskii and Laguzzi created a tree dichotomy among others for the ideal \(\mathcal{ED}\). Now we will do the same for \(\mathcal{M}_-\) ideal.
Definition 67. Let \(\mathbb{T}_{\mathcal{M}_-}\) be the collection of all trees \(T\subseteq\omega^\omega\) such that: \[\forall \sigma \in T \;\; \exists M >|\sigma|\;\; \forall I\subseteq \omega \backslash M\;\;\forall \rho \in \omega ^I \;\;\exists \tau \in T\upharpoonright\sigma \;\; \tau \upharpoonright I=\rho,\] where \(I\) is an interval.
Note that for every \(T\in \mathbb{T_{\mathcal{M}_-}}\) we have that \([T]\notin\mathcal{M}_-.\)
Theorem 68. For any analytic \(A\subseteq\omega^{\omega}\) we have that
either \(A\in \mathcal{M}_-\) or \(\exists T \in \mathbb{T}_{\mathcal{M}_-}\) such that \([T]\subseteq A.\)
In particular, the assignment \(T\in \mathbb{T}_{\mathcal{M}_-}\longrightarrow [T]\in Bor /{\mathcal{M}_-}\) is a dense embedding and the forcing notions \((\mathbb{T}_{\mathcal{M}_-}, \subseteq)\) and \((Bor /{\mathcal{M}_-}, \subseteq)\) are forcing equivalent.
Proof. Let \(T\) be a tree on \(\omega \times \omega\) such that \(A=\pi[T]\) (here \(\pi\) denote the projection on to the first coordinate). For any tree \(S\) on \(\omega \times \omega\) define the derivative: \[S'=\{(\sigma,\tau)\in S:(\exists M>|\sigma|) (\forall I\subseteq \omega\backslash M)(\forall \rho \in \omega^I)(\exists (\sigma',\tau')\in T\upharpoonright_{(\sigma,\tau)}) (\sigma'\upharpoonright I =\rho)\}.\]
Next define inductively:
\(T^{(0)}:=T,\)
\(T^{(\alpha+1)}:=\left(T^{(\alpha)}\right)^{\prime}\),
\(T^{(\lambda)}=\bigcap_{\alpha<\lambda} T^{(\alpha)}\).
Let \(\alpha_0<\omega_1\) be the least such that \(T^{(\alpha_0)}=T^{(\alpha_0+1)}\). Consider the following two cases:
Case \(1)\) \(T^{(\alpha_0)}=\emptyset\).
Note that \([\pi (T)]\subseteq \bigcup_{\beta < \alpha_0} [\pi [T^\beta \backslash T^{\beta +1}]].\) It will be enough to show that \(\forall \beta<\alpha_0\) we have \([\pi[T^\beta\setminus T^{\beta+1}]]\in \mathcal{ M}_-.\) Fix \(\beta\) and \((\sigma, \tau)\in T^\beta \backslash T^{\beta+1}.\) We will show that \([\pi [T^\beta\upharpoonright_{(\sigma, \tau)}]]\in \mathcal{M}_-\).
Since \((\sigma, \tau )\notin T^{\beta+1}\) we get: \[\forall M >|\sigma|=|\tau|\;\; \exists I\subseteq \omega\backslash M \;\; \exists \rho \in \omega^I \;\; \forall (\sigma',\tau')\in T^\beta \upharpoonright_{(\sigma, \tau)} \;\; \sigma' \upharpoonright_I \neq \rho .\;\; \;\; (*)\]
We will inductively build \(x\in \omega^\omega\) and \(J\in \mathbb{IP}\) such that \([\pi[T^\beta \upharpoonright_{(\sigma ,\tau )}]]\subseteq M(x,J).\) Assume that \(J_n\) and \(x\upharpoonright J_n\) are constructed. By \((*)\) there is \(I_n\subseteq \omega \backslash max(J_n)\) and \(\rho_n \in \omega^{I_n}\) such that: \[\forall (\sigma',\tau') \in T^\beta \upharpoonright _{(\sigma,\tau)} \;\; \sigma'\upharpoonright I_n \neq \rho_n\]
Let \(J_{n+1}=[max(J_n)+1, max (I_n)]\) and let \(x\upharpoonright J_{n=1}\) be such that \(x\upharpoonright I_n =\rho_n\).
Claim 69. \([\pi [T^\beta \upharpoonright _{(\sigma, \tau)}]] \subseteq M(x,\bar{J})\).
Proof. Take any \(y\in [\pi [T^\beta \upharpoonright _{(\sigma, \tau)}]]\). There exists \((y,z)\) such that \((y,z)\in T^\beta \upharpoonright_{(\sigma,\tau)}\). So by \((**)\) for every \(n\) we have \(y\upharpoonright I_n \neq \rho\). Since for every \(J_n\) \(I_n\subseteq J_n\) we have \(\forall^\infty \;\; y\upharpoonright J_n \neq x\upharpoonright J_n\), because \(x\upharpoonright I_n =\rho_n.\) Thus we have \([\pi [T^\beta \upharpoonright _{(\sigma, \tau)}]] \subseteq M(x,\bar{J})\). ◻
Case \(2)\) \(T^{(\alpha_0)}\neq \emptyset\).
We will show that \(\pi [T^{\alpha_0}] \in \mathbb{T}_{\mathcal{M}_-}\). Take any \(\sigma \in \pi [T^{\alpha_0}]\). So there exists \(\tau\) such that \((\sigma, \tau) \in T^{\alpha_0}.\) Then we can find \(M> |\sigma |=|\tau|\) such that for every interval \(I\subseteq \omega \backslash M\) and for every \(\rho \in \omega ^I\) we can find \((\sigma',\tau ')\in T^{\alpha_0}\) such that \((\sigma,\tau )\subseteq (\sigma', \tau ')\) and \(\sigma' \upharpoonright I=\rho.\) It follows that \(\sigma\subseteq\sigma' \in \pi [T^{\alpha_0}]\) and \(\sigma' \upharpoonright I =\rho.\) ◻
It follows from [15] that \(\mathbb{T}_{\mathcal{M}_-}\) forcing is proper as \(\mathcal{M}_-\) has an \(F_{\sigma}\)- basis.
Now we move to some corollaries about forcing with \(\mathbb{T}_{\mathcal{M}_-}\).
Definition 70. A real \(x\) is called infinitely equal over a model \(M\), if and only if \(\forall y\in \omega^\omega \cap M\;\; \exists^\infty n \; \; x(n)=y(n)\)
Theorem 71. \(\mathbb{T}_{\mathcal{M}_-}\) adds infinitely equal real.
Proof. For any \(f\in \omega^\omega\), by prop.. we have that \(E_f\subseteq\mathcal{M}_-\). So there is \(S\leqslant T\) such that \([S]\cap E_f=\emptyset\). In particular \(S\Vdash{"} \exists^{\infty}_{n}\) \(\dot{r}_{gen}(n)=f(n)"\) ◻
Fact 72. (Folklore) If \(\mathbb{P}\) adds an infinitely equal real and \(\mathbb{P} \Vdash^" \dot{\mathbb{Q}}\) adds infinitely equal real\(^"\) then \(\mathbb{P}*\dot{\mathbb{Q}}\) adds a Cohen real.
Corollary 73. If \(\mathbb{P}^{\omega_2}\) is a countable support iteration of length \(\omega_2\) over a model of \(CH\) with \(\mathbb{T}_{\mathcal{M}_-}\) as each iterand, then \(\mathbb{V}^{\mathbb{P}_{\omega_2}} \Vdash(cov(\mathcal{M})=\omega_2=\mathfrak{c})\).
Question 74. Does for any \(T\in \mathcal{M}_-\), \(\mathbb{T}_{\mathcal{M}_-} \upharpoonright T\) adds a Cohen real?
Theorem 75. \(\mathbb{T}_{\mathcal{M}_-}\) does not have \(c.c.c\)
Proof. For every \(x\in 2^\omega\) we create \(T_x\in\mathbb{T}_{\mathcal{M}_-}.\) Enumerate \(\omega^{<\omega}\) as \(\{\sigma_n:n<\omega\}\). Let \(T_x=\bigcup_n F_n^x\) where \((F_n^x:n<\omega)\) is build inductively as follows:
\(F_n^x\subseteq F^x_{n+1}\),
\(F_n\)- well founded.
If \(F_n^x\) is done let: \[F^x_{n+1}=\bigcup_{\tau\in term(F_n^x)}\{\tau^\frown n^\frown x(n+1):n<\omega \}\] Of course \(\forall x\neq y\) we have \([T_x]\cap [T_y]=\emptyset\) ◻
Fact 76. If \(\mathcal{I}\) has an \(F_\sigma\)- base, then the idealized forcing notion \(Bor/{\mathcal{I}}\) preserves Baire Category examined in [15].
Fact 77. ([16]) If we have countable support iteration \((\mathbb{P}_\alpha, \dot{\mathbb{Q}}_\alpha; \alpha<\omega_2)\) and for all \(\alpha<\omega_2\) we have \(\mathbb{P}_\alpha \Vdash(\dot{\mathbb{Q}}_\alpha =Bor/{\dot{\mathcal{I}}_\alpha} \text{ and } \dot{\mathcal{I}}_\alpha\) has an \(F_\sigma\)-basis \()\) then \(\mathbb{P}_{\omega_2}\) preserves Baire Category Theorem.
It follows that countable support iteration of \(\mathbb{T}_{\mathcal{M}_-}\) forces the same constelation of Cichoń’s diagram as finite support iteration of Cohen forcing.
Corollary 78. If \(\mathbb{P}^{\omega_2}\) is countable support iteration of \(\mathbb{T}_{\mathcal{M}_-}\) of length \(\omega_2\) over a model of \(CH\) then \(\mathbb{V}^{\mathbb{P}_{\omega_2}} \Vdash(non(\mathcal{M})=\omega_1)\).