# Extending Wadge Theory to *k*-Partitions

## Abstract

We extend some results about Wadge degrees of Borel subsets of Baire space to finite partitions of Baire space. A typical new result is the characterization up to isomorphism of the Wadge degrees of *k*-partitions with \(\mathbf {\Delta }^0_3\)-components.

### Keywords

Baire space Wadge reducibility Lipschitz reducibility Backtrack reducibility*k*-partition

*h*-preorder Well preorder Infinite game

## 1 Introduction

For subsets *A*, *B* of the Baire space \(\mathcal {N}=\omega ^\omega \), *A* is *Wadge reducible* to *B* (\(A\le _WB\)), if \(A=f^{-1}(B)\) for some continuous function *f* on \(\mathcal {N}\). The quotient-poset of the preorder \((P(\mathcal {N});\le _W)\) under the induced equivalence relation \(\equiv _W\) on the power-set of \(\mathcal {N}\) is called *the structure of Wadge degrees* in \(\mathcal {N}\). W. Wadge [15, 16] characterized the Wadge degrees of Borel sets up to isomorphism, in particular this poset is well-founded and has no 3 pairwise incomparable elements.

Let \({2}\le {k}<{\omega }\). By a *k**-partition of*\(\mathcal {N}\) we mean a function \(A:\mathcal {N}\rightarrow k=\{0,\ldots ,k-1\}\) often identified with the sequence \((A_0,\ldots ,A_{k-1})\) where \(A_i=A^{-1}(i)\) are the components of *A*. Obviously, 2-partitions of \(\mathcal {N}\) can be identified with the subsets of \(\mathcal {N}\) using the characteristic functions. The set of all *k*-partitions of \(\mathcal {N}\) is denoted \(k^\mathcal {N}\), thus \(2^\mathcal {N}=P(\mathcal {N})\). The Wadge reducibility on subsets of \(\mathcal {N}\) is naturally extended to *k*-partitions: for \(A,B\in k^\mathcal {N}\), \(A\le _W B\) means that \(A=B\circ f\) for some continuous function *f* on \(\mathcal {N}\). In this way, we obtain the preorder \((k^\mathcal {N};\le _W)\). For any pointclass \(\mathbf {\Gamma }\subseteq P(\mathcal {N})\), let \(\mathbf {\Gamma }(k^\mathcal {N})\) be the set of *k*-partitions of \(\mathcal {N}\) with components in \(\mathbf {\Gamma }\).

In contrast with the Wadge degrees of sets, the structure \(({\mathbf \Delta }^1_1(k^\mathcal {N});\le _W)\) for \(k>2\) has antichains of any finite size. Nevertheless, a basic property of the Wadge degrees of sets may be lifted to *k*-partitions, as the following very particular case of Theorem 3.2 in [4] shows:

### Proposition 1

For any \({2}\le {k}<{\omega }\), the structure \(({\mathbf \Delta }^{1}_{1}(k^\mathcal {N});\le _{W})\) is a well preorder, i.e. it has neither infinite descending chains nor infinite antichains.

Although this result gives an important information about the Wadge degrees of Borel *k*-partitions, it is far from a characterization. Our aim is to obtain such a characterization, continuing a series of earlier partial results (see e.g. [5, 11, 12, 14]). Our approach is to characterize the initial segments \(({\mathbf \Delta }^0_\alpha (k^\mathcal {N});\le _W)\) for bigger and bigger ordinals \(2\le \alpha <\omega _1\). In [11] we have done this for \(\alpha =2\), here we treat the case \(\alpha =3\) (a finitary version of this case was considered in [12]), a general case was outlined in Sect. 5 of [14]. Our original contribution is the discovery of useful properties of natural operations on the *k*-partitions and of the structures of labeled forests.

Let \(\bigoplus _iA_i\) be the disjoint union of a sequence of elements \(A_0,A_1,\ldots \) of \(k^\mathcal {N}\). Let \(\mathcal {N}^+:=\{1,2,\ldots \}^\omega \) and for \(x\in \mathcal {N}^+\) let \(x^-:=\lambda i.x(i)-1\), so \(x^-\in \mathcal {N}\). Define the binary operation \(+\) on \(k^\mathcal {N}\) as follows: \((A+B)(x):=A(x^-)\) if \(x\in \mathcal {N}^+\), otherwise \((A+B)(x):=B(y)\) where *y* is the unique element of \(\mathcal {N}\) such that \(x=\sigma 0y\) for a unique finite sequence \(\sigma \) of positive integers. For any \(i<k\), define a unary operation \(p_i\) on \(k^\mathcal {N}\) by \(p_i(A):=\mathbf {i}+A\) where \(\mathbf {i}:=\lambda x.i\) are the constant *k*-partitions (which are precisely the distinct minimal elements of \((k^\mathcal {N};\le _W)\)). For any \(i<k\), define a unary operation \(q_i\) on \(k^\mathcal {N}\) (for \(k=2\), \(q_0\) and \(q_1\) coincide with the Wadge’s operations \(\sharp \) and \(\flat \) from Sect. III.E of [16]) as follows: \(q_i(A)(x):=i\) if *x* has infinitely many zeroes, \(q_i(A)(x):=A(x^-)\) if *x* has no zeroes, and \(q_i(A)(x):=A(y^-)\) otherwise where *y* is the unique element of \(\mathcal {N}^+\) such that \(x=\sigma 0y\) for a string \(\sigma \) of non-negative integers. The introduced operations are correctly defined on Wadge degrees.

Our first result, which is proved with a heavy use of Proposition 1, characterizes some subalgebras of the Wadge degrees generated from the minimal degrees {\(\mathbf {0}\)}, ..., {\(\mathbf {k-1}\)}. The item (1) below follows from results in [11] but the proof here is slightly different and easier to generalize.

### Theorem 1

- (1)
The quotient-poset of \(({\mathbf \Delta }^0_2(k^\mathcal {N});\le _W)\) is generated from the degrees {\(\mathbf {0}\)}, ..., {\(\mathbf {k-1}\)} by the operations \(\bigoplus , p_0,\ldots ,p_{k-1}\).

- (2)
The quotient-poset of \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _W)\) is generated from {\(\mathbf {0}\)}, ..., {\(\mathbf {k-1}\)} by the operations \(\bigoplus ,+,q_0,\ldots ,q_{k-1}\).

Our second result characterizes the structures above in terms of the homomorphism preorder on labeled forests [11, 12, 14]. Let \((Q;\le )\) be a preorder. A *Q-poset* is a triple \((P,\le ,c)\) consisting of a countable nonempty poset \((P;\le )\) without infinite chains, and a labeling \(c:P\rightarrow Q\). A *morphism*\(f:(P,\le ,c)\rightarrow (P^\prime ,\le ^\prime ,c^\prime )\) of *Q*-posets is a monotone function \(f:(P;\le )\rightarrow (P^\prime ;\le ^\prime )\) satisfying \(\forall x\in P(c(x)\le c^\prime (f(x)))\). Let \(\widetilde{{\mathcal F}}_Q\) and \(\widetilde{{\mathcal T}}_Q\) denote the sets of all countable *Q*-forests and *Q*-trees without infinite chains, respectively. The *h-preorder*\(\le _h\) on \(\widetilde{{\mathcal F}}_Q\) is defined as follows: \(P\le _h P^\prime \), if there is a morphism from *P* to \(P^\prime \). If \(Q=\bar{k}\) of the antichain with *k* elements \(0,\ldots ,k-1\), we obtain the *Q*-preorders denoted by \(\widetilde{{\mathcal F}}_k\) and \(\widetilde{{\mathcal T}}_k\), respectively. We also need the preorder \(\widetilde{{\mathcal F}}_{\widetilde{{\mathcal T}}_k}\).

### Theorem 2

- (1)
The quotient-posets of \(({\mathbf \Delta }^0_2(k^\mathcal {N});\le _W)\) and of \((\widetilde{{\mathcal F}}_k;\le _h)\) are isomorphic.

- (2)
The quotient-posets of \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _W)\) and of \((\widetilde{{\mathcal F}}_{\widetilde{{\mathcal T}}_k};\le _h)\) are isomorphic.

Again, item (1) above was established in [11] and is now extended to (2) (a “finitary” version of (2) is described in [12]). The proof of Theorem 2 generalizes those in [11, 12]. It takes the *h*-quasiorders as natural naming systems for the subalgebras in Theorem 1, providing natural homomorphisms from the forest structures onto the corresponding degree structures. From the properties of the operations in Theorem 1 it follows by induction on the rank of the forests that these homomorphisms are in fact isomorphisms. Because of space bounds, we omit some proofs having published versions for the case of sets.

We do believe that the results above maybe extended to larger segments \(({\mathbf \Delta }^0_\alpha (k^\mathcal {N});\le _W)\), \(4\le \alpha <\omega _1\). Using the Kuratowski relativization technique [2, 8, 16], we can define for any \(1\le \beta <\omega _1\) the binary operation \(+_\beta \) on \(k^\mathcal {N}\) such that \(+_1\) coincides with \(+\) and, for any \(2\le \alpha <\omega _1\), the quotient-poset of \(({\mathbf \Delta }^0_\alpha (k^\mathcal {N});\le _W)\) is generated from {\(\mathbf {0}\)}, ..., {\(\mathbf {k-1}\)} by the operations \(\bigoplus \) and \(+_\beta \) for all \(1\le \beta <\alpha \). The extension of Theorem 2 is obtained by defining suitable iterated versions of the *h*-quasiorder in the spirit of [14]. Since \(\mathbf {\Delta }^1_1(k^\mathcal {N})=\bigcup _{\alpha <\omega _1}\mathbf {\Delta }^0_\alpha (k^\mathcal {N})\), we obtain the characterization of Wadge degrees of Borel *k*-partitions. Note that in [9] we considered a classification of hyperarithmetical *k*-partitions of \(\omega \) modulo *m*-reducibility which is in a precise sense the effective version of the Wadge degrees of *k*-partitions. Our “algebraic” approach to Wadge theory was motivated by the similar approach in [9].

## 2 Preliminaries

We use the standard set-theoretic notation. We identify the set of natural numbers with the first infinite ordinal \(\omega \). The first uncountable ordinal is denoted by \(\omega _1\). Let \(\mathcal {N}=\omega ^\omega \) be the set of all infinite sequences of natural numbers (i.e., of functions \(x :\omega \rightarrow \omega \)). Let \(\omega ^*\) be the set of finite sequences of elements of \(\omega \), including the empty sequence \(\varepsilon \). For \(\sigma ,\tau \in \omega ^*\) and \(x\in \mathcal {N}\), we write \(\sigma \sqsubseteq \tau \) (resp. \(\sigma \sqsubseteq x\)) to denote that \(\sigma \) is an initial segment of \(\tau \) (resp. of *x*). By \(\sigma x=\sigma \cdot x\) we denote the concatenation of \(\sigma \) and *x*, and by \(\sigma \cdot \mathcal {N}\) the set of all extensions of \(\sigma \) in \(\mathcal {N}\). For \(x\in \mathcal {N}\), we write \(x=x(0)x(1)\cdots \) where \(x(i)\in \omega \) for each \(i<\omega \). For \(x\in \mathcal {N}\) and \(n<\omega \), \(x[n]=x(0)\cdots x(n-1)\) is the initial segment of *x* of length *n*.

By endowing \(\mathcal {N}\) with the product of the discrete topologies on \(\omega \), we obtain the so-called *Baire space*. The product topology coincides with the topology generated by the collection of sets of the form \(\sigma \cdot \mathcal {N}\) for \(\sigma \in \omega ^*\). We recall the well-known (see e.g. [6]) relation of closed subsets of \(\mathcal {N}\) to trees. A *tree* is a non-empty set \(T\subseteq \omega ^*\) which is closed downwards under \(\sqsubseteq \). A *leaf* of *T* is a maximal element of \((T;\sqsubseteq )\). By \(\partial (T)\) we denote the set of minimal elements in \((\omega ^*\setminus T;\sqsubseteq )\). A *pruned tree* is a tree without leafs. A *path through* a tree *T* is an element \(x\in \mathcal {N}\) such that \(x[n]\in T\) for each \(n\in \omega \). For any tree *T*, the set [*T*] of paths through *T* is closed in \(\mathcal {N}\). For any non-empty closed set \(A\subseteq \mathcal {N}\) there is a unique pruned tree *T* with \(A=[T]\) and, moreover, there is a Lipschitz surjection \(t:\mathcal {N}\rightarrow A\) which is constant on *A* (such a surjection is called a retraction onto *A*). Therefore, there is a bijection between the pruned trees and the non-empty closed sets. Note that the well founded trees *T* (i.e., trees with \([T]=\emptyset \)) and non-empty well founded forests of the form \(F:=T\setminus \{\varepsilon \}\) are sufficient for defining the *h*-preorders in the Introduction.

We mention some reducibilities on the *k*-partitions of Baire space. Since these are many-one reducibilities, they are closely related to classes of functions \(\mathcal {F}\) on Baire space that are closed under composition and contain the identity function. Any such a class \(\mathcal {F}\) induces a “reducibility” (i.e., a preorder) \(\le _\mathcal {F}\) on \(k^\mathcal {N}\): \(A\le _\mathcal {F}B\), if \(A=B\circ f\) for some \(f\in \mathcal {F}\). If \(\mathcal {F}\) is the class of continuous functions we obtain Wadge reducibility. If \(\mathcal {F}\) is the class of Lipschitz functions we obtain Lipschitz reducibility which is denoted \(\le _L\) and plays an important role in the Wadge theory. Recall that a *Lipschitz function* (resp. a *strong Lipschitz function*) may be defined as a function *f* on \(\mathcal {N}\) satisfying \(f(x)(n)=\phi (x[n+1])\) (resp. \(f(x)(n)=\phi (x[n])\)), for some \(\phi :\omega ^*\rightarrow \omega \). Every strong Lipschitz function is Lipschitz and every Lipschitz function is continuous, so \(\le _L\) is contained in \(\le _W\) (but not vice versa). For a Lipschitz function *f* and a string \(\sigma \), \(f(\sigma )\) denotes the obvious initial segment of *f*(*x*) for each \(x\sqsupseteq \sigma \).

For any pointclass \(\mathbf {\Gamma }\subseteq P(\mathcal {N})\), by \(\mathbf {\Gamma }\)*-function* we mean a function *f* on \(\mathcal {N}\) such that \(f^{-1}(A)\in \mathbf {\Gamma }\) for each \(A\in \mathbf {\Gamma }\). Since the \(\mathbf {\Gamma }\)-functions are closed under composition and contain the identity function, we obtain the corresponding \(\mathbf {\Gamma }\)-reducibility \(\le _\mathbf {\Gamma }\). Among such reducibilities are \(\mathbf {\Delta }^0_\alpha \)-reducibilities, for each non-zero countable ordinal \(\alpha \). Note that \(\mathbf {\Delta }^0_\alpha \)-reducibility coincides with \(\mathbf {\Sigma }^0_\alpha \)-reducibility and \(\mathbf {\Delta }^0_1\)-reducibility coincides with Wadge reducibility. We usually shorten the notation \(\le _{\mathbf {\Delta }^0_\alpha }\) to \(\le _\alpha \), so \(\le _1\) coincides with \(\le _W\). Note that the relation \(\le _\alpha \) is contained in \(\le _\beta \) for all \(1\le \alpha<\beta <\omega _1\). The \(\mathbf {\Delta }^0_\alpha \)-functions and \( \mathbf {\Delta }^0_\alpha \)-reducibilities were investigated in [1, 2, 8]. By Jayne-Rogers theorem, the \(\mathbf {\Delta }^0_2 \)-functions coincide with the functions *f* on \(\mathcal {N}\) for which there is a partition \(\{A_n\}\) of \(\mathcal {N}\) to closed sets \(A_n\) such that \(f|_{A_n}\) is continuous for each \(n<\omega \).

Many reducibilities are closely related to infinite games. Relate to any \(A,B\in k^\mathcal {N}\) the *Lipschitz game*\(G_L(A,B)\) for players I and II as follows. Player I chooses a natural number *x*(0), Player II responses with his number *y*(0), I responses with *x*(1), and so on; every player knows all moves of the opponent. After \(\omega \) steps, I has produced some \(x\in \mathcal {N}\) while II has produced some \(y\in \mathcal {N}\); we say that II won this round if \(A(x)=B(y)\), otherwise I won the round. A *winning strategy* for II (resp. for I) in the game \(G_L(A,B)\) is identified with a Lipschitz function *f* (resp. a strong Lipschitz function *g*) such that \(A(x)=B(f(x))\) for each \(x\in \mathcal {N}\) (resp. \(A(g(y))\not =B(y)\) for each \(y\in \mathcal {N}\)). It is easy to see that II has a winning strategy in \(G_L(A,B)\) iff \(A\le _LB\). As it follows from Martin determinacy theorem, for any Borel *A*, *B* the game \(G_L(A,B)\) is determined, i.e. one of the players has a winning strategy. Versions of this fact are crucial for applications of infinite games to Wadge theory. In particular, define the *Wadge game*\(G_W(A,B)\) as the modification of \(G_L(A,B)\) in which Player II is allowed to pass (i.e., not choose a number) at any step. Player II wins a given round if he responses with infinitely many numbers during this round (thus producing some \(y\in \mathcal {N}\)) and \(A(x)=B(y)\). Then II has a winning strategy in \(G_W(A,B)\) iff \(A\le _WB\).

There are many other ingenious modifications of \(G_L(A,B)\) of which we mention one introduced by R. Van Wesep. The *backtrack game*\(G_{bt}(A,B)\) is the modification of \(G_W(A,B)\) by giving to Player II the additional ability to backtrack (i.e., delete all his previous moves and start the construction of *y* from scratch) at any step. Player II wins a given round if he makes only finitely many backtracks during this round (thus producing again some \(y\in \mathcal {N}\)) and \(A(x)=B(y)\). The winning strategies for II in the backtrack games are known as the *backtrack functions*. As shown in Theorem 21 of [1], the backtrack functions coincide with the \(\mathbf {\Delta }^0_2\)-functions, so II has a winning strategy in \(G_{bt}(A,B)\) iff \(A\le _2B\).

## 3 Operations on *k*-partitions

We use some obvious properties of the \(\omega \)-ary operation \(\bigoplus \) on \(S^\mathcal {N}\) (*S* is a non-empty set) defined by \(\bigoplus _nA_n(i\cdot x):=A_i(x)\). For any \(2\le m<\omega \), define the *m*-ary operation on \(S^\mathcal {N}\) by \(B_0\oplus \cdots \oplus B_{m-1}:=\bigoplus _nA_n\) where \(A_{mq+r}:=B_r\) for all \(q\ge 0, 0\le r<m\). For any \(\sigma \in \omega ^*,A\in S^\mathcal {N}\), define \(A^{\sigma }\in S^\mathcal {N}\) by \(A^{\sigma }(x):=A(\sigma x)\).

The next result is a straightforward extension to *k*-partitions of some properties of \(+\) established in Sect. III.C of [16] for sets.

### Proposition 2

- (1)
If \(A\le _1A'\) and \(B\le _1B'\) then \(A+B\le _1A'+B'\).

- (2)
\((A+B)+C\equiv _1A+(B+C)\).

- (3)
\((\bigoplus _nA_n)+B\equiv _1\bigoplus _n(A_n+B).\)

- (4)
\(A+B\equiv _2A\oplus B\).

- (5)
For any \(2\le \alpha <\omega _1\), the set \(\mathbf {\Delta }^0_\alpha (k^\mathcal {N})\) is closed under \(+\).

The next result is known (see Theorem 7.6 from [11] and references therein).

### Proposition 3

The structure \((k^\mathcal {N};\bigoplus ,\le _1,p_0,\ldots ,p_{k-1})\) is a \(\sigma \)-semilattice with discrete closures which by definition means that any \(p_i\) is a closure operation (i.e., \(A\le _1p_i(A)\), \(A\le _1B\) implies \(p_i(A)\le _1p_i(B)\), and \(p_i(p_i(A))\le _1p_i(A)\)), \(p_i(A)\le _1\bigoplus _nB_n\) implies that \(p_i(A)\le _1B_n\) for some \(n<\omega \), and \(p_i(A)\le _1p_j(B)\), \(i\not =j\) imply that \(p_i(A)\le _1B\).

Relate to any *k*-partition *A* the tree \(T(A):=\{\sigma \in \omega ^*\mid A\le _1A^{\sigma }\}\) which is known to be closely related to the Wadge theory, in particular the following relation to the operations \(p_0,\ldots ,p_{k-1}\) is straightforward (for related facts on sets see e.g. Sect. 2 of [1]):

### Proposition 4

- (1)
If \(x\in [T(A)]\) then \(p_{A(x)}(A)\le _1A\).

- (2)
For any \(i<k\), \(p_i(A)\le _1A\) iff \(i=A(x)\) for some \(x\in [T(A)]\).

A *k*-partition *A* is \(\alpha \)*-irreducible* (a more precise but more complicated term would be \(\sigma \)-join-irreducible w.r.t. \(\le _\alpha \), cf. [11]) if there do not exist *k*-partitions \(A_n<_\alpha A\), \(n<\omega \), with \(A\equiv _\alpha \bigoplus _nA_n\). If *A* is not \(\alpha \)-irreducible then we call it \(\alpha \text {-}reducible\). We collect some characterizations of 1-irreducible *k*-partitions. For the last one see e.g. [3, 7] (where the *k*-partitions *A* with the property from item (8) are called non-self-dual), the others are rather straightforward.

### Proposition 5

- (1)
The

*k*-partition*A*is 1-irreducible. - (2)
There is no \(\mathbf {\Delta }^0_1\)-partitions \(\{D_n\}\) of \(\mathcal {N}\) with \(A\circ d_n<_1A\) where \(d_n\) is a continuous retraction from \(\mathcal {N}\) onto \(D_n\).

- (3)
There are no 1-irreducible

*k*-partitions \(A_n<_1A\) with \(A\equiv _1\bigoplus _nA_n\). - (4)
If \(A\le _1\bigoplus _nB_n\) then \(A\le _1B_n\) for some \(n<\omega \).

- (5)
The tree

*T*(*A*) is not well founded. - (6)
The tree

*T*(*A*) is pruned. - (7)
There is \(i<k\) with \(p_i(A)\le _1A\).

- (8)
Any continuous function

*f*on \(\mathcal {N}\) has an*A*-fixed point (i.e., \(A(x)=A(f(x))\) for some \(x\in \mathcal {N}\)).

We state a corollary of (8) which extends to *k*-partitions a well known property of Wadge degrees (in the proof we use the idea of Lemma 29 in [7]).

### Proposition 6

Let \(A,B\in \mathbf {\Delta }^1_1(k^\mathcal {N})\) and *A* be 1-irreducible. Then \(B\le _WA\) implies \(B\le _LA\), \(A\le _WB\) implies \(A\le _LB\), and \(B\equiv _WA\) implies \(B\equiv _LA\).

### Proof

The third assertion follows from the first two. Let \(B\le _WA\) via *f*. Consider the game \(G_{diag}(A,B)\) where players I and II construct *x*, *y* as in the Lipschitz game and II wins iff \(A(x)\not =B(y)\). Player II does not have a winning strategy in this game since if *s* were such a strategy then \(A(x)\not =B(s(x))=Af(s(x))\) for all \(x\in \mathcal {N}\), contradicting Proposition 5(8). Thus, player I has a winning strategy *t* satisfying \(\forall y A(t(y))=B(y)\). Since *t* is Lipschitz, \(B\le _LA\).

Let now \(A\le _WB\) via *f*. Then player II does not have a winning strategy in \(G_{diag}(B,A)\) since if *s* were such a strategy then \(A(x)=B(f(x))\not =As(f(x))\) for all \(x\in \mathcal {N}\), contradicting Proposition 5(8). Thus, player I has a winning strategy *t* satisfying \(B(t(y))=A(y)\) for all \(y\in \mathcal {N}\). Since *t* is Lipschitz, \(A\le _LB\). \(\Box \)

Relate to any 1-irreducible \(A\in k^\mathcal {N}\) the *k*-partitions \(A'\in k^\mathcal {N}\) and \(\tilde{A}\in k^\mathcal {N}\cup \{\bot \}\) (here \(\bot \) is a new element strictly \(\le _1\)-below any *k*-partition) as follows: \(A'=A\circ t\) where \(t:\mathcal {N}\rightarrow [T(A)]\) is a Lipschitz retraction onto [*T*(*A*)] (modulo \(\equiv _1\), \(A'\) does not depend on the choice of *t*), and \(\tilde{A}=\bigoplus \{A^\sigma \mid \sigma \in \partial (T(A))\}\) (modulo \(\equiv _1\), \(\tilde{A}\) does not depend on the choice of a numbering of \(\partial (T(A))\) if the last set is non-empty, and if it is empty we set \(\tilde{A}=\bot \)).

### Proposition 7

- (1)
\(\tilde{A}=\bot \) iff \(T(A)=\omega ^*\);

- (2)
\(A'\le _1A\), \(\tilde{A}<_1A\) and \(A\equiv _1A'+\tilde{A}\);

- (3)
If \(A\equiv _1B\) then \(A'\equiv _1B'\) (but, in general, \(\tilde{A}\not \equiv _1\tilde{B}\));

- (4)
\(T(A')=\omega ^*\).

### Proof

Item (1) obvious. The first assertion in (2) is obvious, the second follows from 1-irreducibility of *A*, the third is straightforward.

(3) Let \(A\equiv _1B\). By symmetry, it suffices to show that \(A\circ t\le _1B\circ t_B\) where \(t_B\) is a continuous retraction onto [*T*(*B*)]. Let *f* witness \(A\le _1B\), then \(f(x)\in [T(B)]\) for each \(x\in [T(A)]\). (Suppose not, then \(\tau \sqsubseteq f(x)\) for a unique \(\tau \in \partial (T(B))\). By continuity of *f*, \(f(\sigma \cdot \mathcal {N})\subseteq \tau \cdot \mathcal {N}\) for some \(\sigma \sqsubseteq x\), hence \(A^\sigma \le _1B^\tau \). Since \(\sigma \in T(A)\), we have \(B\le _1A\le _1A^\sigma \le _1B^\tau \), so \(\tau \in T(B)\). A contradiction.) Since \(t_B\) is a retraction, \(f\circ t=t_B\circ f\circ t\), so \(A\circ t=B\circ f\circ t=(B\circ t_B)\circ (f\circ t)\), hence \(A\circ t\le _1B\circ t_B\) via \(f\circ t\).

(4) Let \(\sigma \in \omega ^*\) and \(B:=A^{t(\sigma )}\). Since \(t(\sigma )\in T(A)\), \(A\equiv _1B\). Note that \(T(B)=\{\tau \in \omega ^*\mid t(\sigma )\cdot \tau \in T(A)\}\). Since *t* is a Lipschitz retraction onto *T*(*A*), \(B'\equiv _1A'^\sigma \). Since \(A\equiv _1B\), by (3) we get \(A'\equiv _1B'\). Therefore, \(A'\equiv _1A'^\sigma \), \(\sigma \in T(A')\) and \(T(A')=\omega ^*\). \(\Box \)

The next properties of the operations \(+,q_0,\ldots ,q_{k-1}\) are straightforward:

### Proposition 8

- (1)
For \(3\le \alpha <\omega _1\), \(\mathbf {\Delta }^0_\alpha (k^\mathcal {N})\) is closed under \(q_0,\ldots ,q_{k-1}\).

- (2)
Structure \((k^\mathcal {N};\bigoplus ,\le _2,q_0,\ldots ,q_{k-1})\) is a \(\sigma \)-semilattice with discrete closures.

- (3)
For all \(A\in k^\mathcal {N}\) and \(i<k\), \(B\mapsto q_i(A)+B\) is a closure operator on \((k^\mathcal {N};\le _1)\).

- (4)
For all \(A,A_1,B,B_1\in k^\mathcal {N}\) and \(i,j<k\), if \(q_i(A)+B\le _1q_j(A_1)+B_1\) and \(q_i(A)\not \le _1q_j(A_1)\) then \(q_i(A)+B\le _1B_1\).

We proceed with some characterizations of 2-irreducible *k*-partitions.

### Proposition 9

- (1)
The

*k*-partition*A*is 2-irreducible. - (2)
There is no \(\mathbf {\Pi }^0_1\)-partition \(\{D_n\}\) of \(\mathcal {N}\) with \(A\circ d_n<_2A\) where \(d_n\) is a continuous retraction from \(\mathcal {N}\) onto \(D_n\).

- (3)
There are no 2-irreducible

*k*-partitions \(A_n<_2A\) with \(A\equiv _2\bigoplus _nA_n\). - (4)
If \(A\le _2\bigoplus _nB_n\) then \(A\le _2B_n\) for some \(n<\omega \).

- (5)
Any \(\mathbf {\Delta }^0_2\)-function

*f*on \(\mathcal {N}\) has an*A*-fixed point. - (6)
There is \(B\equiv _2A\) with \(T(B)=\omega ^*\).

- (7)
There is \(i<k\) with \(q_i(A)\le _1A\).

### Proof

The equivalence of (1)–(4) is rather straightforward.

(5)\(\rightarrow \)(1) Let *A* be 2 reducible, so \(A\equiv _2\bigoplus _nA_n\) for some \(A_n<_2A\). We have to find a \(\mathbf {\Delta }^0_2\)-function without *A*-fixed points. Clearly, it suffices to find such a function without \(\bigoplus _nA_n\)-fixed points. For any \(n<\omega \), choose \(n'<\omega \) with \(A_{n'}\not \le _2A_n\), then Player II does not win \(G_{bt}(A_{n'},A_n)\), hence Player I wins via a strategy \(s_n\), so \(A_{n'}(s_n(y))\not =A_n(y)\) for each \(y\in \mathcal {N}\). Define a Lipschitz function *f* by \(f(n\cdot y):=n'\cdot s_n(y)\). Then *f* has no \(\bigoplus _nA_n\)-fixed points.

(2)\(\rightarrow \)(5) Suppose for a contradiction that *A* satisfies (2) but some \(\mathbf {\Delta }^0_2\)-function has no *A*-fixed points. Then the following version of Claim 5.3.1 in [8] holds: for any closed set *D* and any continuous retraction *d* onto *D*, if \(A\equiv _2A\circ d\) then there is a \(\mathbf {\Delta }^0_2\)-function \(g:\mathcal {N}\rightarrow D\) without \((A\circ d)\)-fixed points. Indeed, since \(A\equiv _2A\circ d\), there is a \(\mathbf {\Delta }^0_2\)-function \(f:\mathcal {N}\rightarrow \mathcal {N}\) without \((A\circ d)\)-fixed points. Since *d* is a retraction, we can take \(g=d\circ f\). With this version at hand, we can repeat (with \(A\cap D\) replaced by \(A\circ d\)) the proof of Theorem 5.3 in [8] (which is based on the proof of Theorem 16 in [2]) and obtain a contradiction.

(5)\(\rightarrow \)(6) Consider the representation \(A\equiv _1A'+\tilde{A}\) from Proposition 7. By this proposition, \(A\equiv _2A'\oplus \tilde{A}\). Repeating the proof of Proposition 6 (with \(\le _W\) replaced by \(\le _2\)) we obtain that if any \(\mathbf {\Delta }^0_2\)-function has an *A*-fixed point then the \(\mathbf {\Delta }^0_2\)-degree of *A* coincides with the Lipshitz degree of *A*. Therefore, \(A\equiv _1A'\oplus \tilde{A}\). By (5), 2-irreducibility implies 1-irreducibility. Thus, *A* is 1-irreducible, hence \(A\le _1A'\) or \(A\le _1\tilde{A}\). But \(A\not \le _1\tilde{A}\), hence \(A\equiv _1A'\) and we can take \(B=A'\).

(6)\(\rightarrow \)(7). It suffices to show that if \(T(A)=\omega ^*\) then \(q_i(A)\le _1A\) for some \(i\in rng(A)\). By Proposition 6, for any \(\sigma \in \omega ^*\) there is a strong Lipschitz function \(f_\sigma \) with \(A=A^\sigma \circ f_\sigma \). To simplify notation a bit, we consider the particular case \(rng(A)=\{0,1,2\}\), it will be clear that the proof works for the general case. Towards a contradiction, suppose that \(q_i(A)\not \le _1A\) for each \(i<3\), then there are strong Lipschitz functions \(s_i\) such that \(q_i(A)(s_i(z))\not =A(z)\) for all \(i<3,z\in \mathcal {N}\). We construct \(y\in \mathcal {N}\) as follows.

Let \(x_0\) be the first number in the sequence \(s_0(y)\) (it does not depend on *y* because \(s_0\) is strong Lipschitz). If \(x_0>0\), we set \(y(0)=x_0-1\), find the second number \(x_1\) in the sequence \(s_0(y)\) and set \(y(1)=x_1-1\) if \(x_1>0\), and continue this process until we find the first zero in \(s_0(y)\). Such a zero exists because otherwise we would get \(y=s_0(y)^-\), hence \(A(y)=A(s_0(y)^-)=q_0(A)(s_0(y))\), contradicting the property of \(s_0\). Thus, we have \(\tau _00\sqsubseteq s_0(y)\) for a unique string \(\tau _0\) without zeroes. We proceed to construct *y* by concatenating the consecutive numbers of the sequence \(f_{\sigma _0}((x_0-1)(x_1-1)\cdots )\) to \(\sigma _0:=\tau _0^-\) where this time \(x_0,x_1,\ldots \) are the consecutive non-zero numbers in the sequence \(s_1(y)\); we continue until the first 0 in the last sequence is discovered. This 0 exists because otherwise we get \(y=\sigma _0f_{\sigma _0}(s_1(y)^-)\), hence \(A(y)=A^{\sigma _0}f_{\sigma _0}(s_1(y)^-)=A(s_1(y)^-)=q_1(A)(s_1(y))\), contradicting the property of \(s_1\). Thus, we have \(\tau _10\sqsubseteq s_1(y)\) for a unique string \(\tau _1\) without zeroes. We proceed to construct *y* by concatenating the consecutive numbers of the sequence \(f_{\sigma _1}((x_0-1)(x_1-1)\cdots )\) to \(\sigma _1:=\sigma _0f_{\sigma _0}(\tau _1^-)\) where this time \(x_0,x_1,\ldots \) are the consecutive non-zero numbers in the sequence \(s_2(y)\). Again we will find the first zero in \(s_2(y)\) and the corresponding \(\tau _2\) and \(\sigma _2\). Note that \(\sigma _0\sqsubset \sigma _1\sqsubset \sigma _2\) because we work with the strong Lipschitz functions.

At this point, we proceed to construct *y* by concatenating the consecutive numbers of the sequence \(f_{\sigma _2}((x_0-1)(x_1-1)\cdots )\) to \(\sigma _2\) where this time \(x_0,x_1,\ldots \) are the consecutive non-zero numbers in the unique sequence \(z_0\in \mathcal {N}\) satisfying \(s_0(y)=\tau _00z_0\); we continue until the first 0 in the sequence \(z_0\) is discovered. This 0 exists because otherwise we get \(y=\sigma _2f_{\sigma _2}(z_0^-)\), hence \(A(y)=A^{\sigma _2}f_{\sigma _2}(z_0^-)=A(z_0^-)=q_i(A)(s_0(y))\), contradicting the property of \(s_0\). Next we work in the same way with \(s_1(y)=\tau _10z_1\), \(s_2(y)=\tau _20z_2\), and so on.

By the construction, we obtain an infinite sequence \(\sigma _0\sqsubset \sigma _1\sqsubset \sigma _2\sqsubset \cdots \) with \(y=\bigcup _n\sigma _n\), and any of \(s_0(y),s_1(y),s_2(y)\) has infinitely many zeroes. Let \(i:=A(y),i<3\). By the definition of \(q_i(A)\) we have \(q_i(A)(s_i(y))=i=A(y)\), contradicting the property of \(s_i\).

(7)\(\rightarrow \)(6) Take \(B=q_i(A)\). By the definition of \(q_i(A)\), \(T(B)=\omega ^*\).

(6)\(\rightarrow \)(4) Choose \(B\equiv _2A\) with \(T(B)=\omega ^*\). First we show that \(B\le _2C\) implies \(B\le _1C\). Let \(B\le _2C\) via *f* such that for some \(\mathbf {\Pi }^0_1\)-partition \(\{D_n\}\) of \(\mathcal {N}\) the function *f* is continuous on \(D_n\) for each \(n<\omega \). By Baire category theorem, \(\sigma \cdot \mathcal {N}\subseteq D_n\) for some *n*. Thus, \(B^{\sigma }\le _1C\) via \(\lambda x.f(\sigma x)\). Since \(\sigma \in T(B)\), \(B\le _1B^\sigma \) and therefore \(B\le _1C\).

Let now \(A\le _2\bigoplus _nB_n\), hence \(B\le _2\bigoplus _nB_n\) and therefore \(B\le _1\bigoplus _nB_n\). Since \(T(B)=\omega ^*\), *B* is 1-irreducible by Proposition 5(6), hence \(B\le _1B_n\) for some \(n<\omega \). Therefore, \(A\le _2B_n\) and *A* is 2-irreducible. \(\Box \)

## 4 Generating Degree Structures

The next proposition coincides with item (1) of Theorem 1.

### Proposition 10

The quotient-poset of \(({\mathbf \Delta }^0_2(k^\mathcal {N});\le _W)\) is generated from the minimal degrees {\(\mathbf {0}\)}, ..., {\(\mathbf {k-1}\)} by the operations \(\bigoplus , p_0,\ldots ,p_{k-1}\) (and also by the operations \(\bigoplus ,+\)).

### Proof

Let \(\mathcal {S}\) (resp. \(\mathcal {S}_1\)) be the subalgebra of \((k^\mathcal {N};\bigoplus , p_0,\ldots ,p_{k-1})\) (resp. of \((k^\mathcal {N};\bigoplus , +)\)) generated by the set {\(\mathbf {0},\ldots ,\mathbf {k-1}\)}, then \(\mathcal {S}\subseteq \mathcal {S}_1\subseteq {\mathbf \Delta }^0_2(k^\mathcal {N})\) by Proposition 2(5). It remains to show that any \(A\in {\mathbf \Delta }^0_2(k^\mathcal {N})\) is Wadge equivalent to some \(B\in \mathcal {S}\). This is checked by the rank *rk*(*A*) of *A* in the well founded preorder \(({\mathbf \Delta }^0_2(k^\mathcal {N});\le _W)\). If \(rk(A)=0\) then \(A\in \){\(\mathbf {0},\ldots ,\mathbf {k-1}\)} and there is nothing to prove, so let *A* be non-constant. If *A* is 1-reducible then \(A\equiv _1\bigoplus _nA_n\) for some \(A_n<_1A\). By induction, \(A_n\equiv _1B_n\) for some \(B_n\in \mathcal {S}\), \(n\in \omega \). Then \(A\equiv _1\bigoplus _nB_n\in \mathcal {S}\), as desired.

Finally, let *A* be 1-irreducible. Consider the representation \(A\equiv _1A'+\tilde{A}\) from Proposition 7. Then \(A'\) is constant. (Otherwise, \(i,j\in rng(A')\) for some distinct *i*, *j*, hence \(p_i(A)\equiv _1p_j(A)\equiv _1A\) by Proposition 7(4). Since the Wadge degrees of \({\mathbf \Delta }^0_2\)-sets are generated by \(\bigoplus ,p_0,p_1\) from \(\{\emptyset \},\{\mathcal {N}\}\) (see Sect. III.C of [16]), any such degree is Wadge reducible to \(A_i\), hence \(A\not \in {\mathbf \Delta }^0_2(k^\mathcal {N})\). A contradiction.) If \(\tilde{A}=\bot \) then \(A\equiv _1A'\) is constant and we are done. Finally, let \(\tilde{A}\not =\bot \). Since \(\tilde{A}<_1A\), by induction \(\tilde{A}\equiv _1B\) for some \(B\in \mathcal {S}\). Then \(A\equiv _1p_i(B)\in \mathcal {S}\) for some \(i<k\), as desired. \(\Box \)

The next proposition is interesting in its own right.

### Proposition 11

The quotient-poset of \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _2)\) is generated from the minimal degrees {\(\mathbf {0}\)}, ..., {\(\mathbf {k-1}\)} by the operations \(\bigoplus , q_0,\ldots ,q_{k-1}\).

### Proof

Let \(\mathcal {S}\) be the subalgebra of \((k^\mathcal {N};\bigoplus , q_0,\ldots ,q_{k-1})\) generated by the set {\(\mathbf {0},\ldots ,\mathbf {k-1}\)}, then \(\mathcal {S}\subseteq {\mathbf \Delta }^0_3(k^\mathcal {N})\) by Proposition 8(1). It remains to show that any \(A\in {\mathbf \Delta }^0_3(k^\mathcal {N})\) is 2-equivalent to some \(B\in \mathcal {S}\). This is checked by the rank *rk*(*A*) of *A* in the well founded preorder \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _2)\). If \(rk(A)=0\) then \(A\in \){\(\mathbf {0},\ldots ,\mathbf {k-1}\)} and there is nothing to prove, so let *A* be non-constant. If *A* is 2-reducible then \(A\equiv _2\bigoplus _nA_n\) for some \(A_n<_2A\). By induction, \(A_n\equiv _2B_n\) for some \(B_n\in \mathcal {S}\), \(n\in \omega \). Then \(A\equiv _2\bigoplus _nB_n\in \mathcal {S}\), as desired.

Finally, let *A* be 2-irreducible. By Proposition 9(7), \(q_i(A)\equiv _2A\) for some \(i<k\). Such *i* is in fact unique. (Otherwise, \(q_i(A)\equiv _1q_j(A)\equiv _1A\) for some distinct *i*, *j*. Since any \({\mathbf \Delta }^0_3\)-set is Wadge reducible to a set generated by \(\bigoplus ,q_0=\sharp ,q_1=\flat \) from \(\{\emptyset \},\{\mathcal {N}\}\) by Sect. III.E of [16], any \({\mathbf \Delta }^0_3\)-set is Wadge reducible to \(A_i\), hence \(A\not \in {\mathbf \Delta }^0_3(k^\mathcal {N})\). A contradiction.) Moreover, \(A\equiv _2q_i(\tilde{A})\) for some \(\tilde{A}<_2A\) as it follows from the structure of the difference hierarchy of *k*-partitions over \(\mathbf {\Sigma }^0_2\) [14] (*A* will be Wadge complete in a non-self-dual level of this hierarchy similarly to Sect. 7 of [11], with \(p_i\) replaced by \(q_i\); in fact, the set of \({\mathbf \Delta }^0_2\)-degrees strictly below *A* is at most countable, hence we can take \(\tilde{A}\) as the disjoint union of all such degrees, then the \({\mathbf \Delta }^0_2\)-degree of \(\tilde{A}\) is the largest \(\mathbf {\Delta }^0_2\)-degree strictly below *A*). Since \(\tilde{A}<_2A\), by induction \(\tilde{A}\equiv _2B\) for some \(B\in \mathcal {S}\). Then \(A\equiv _2q_i(B)\in \mathcal {S}\), as desired. \(\Box \)

The next proposition coincides with item (2) of Theorem 1.

### Proposition 12

The quotient-poset of \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _W)\) is generated from the minimal degrees {\(\mathbf {0}\)},...,{\(\mathbf {k-1}\)} by the operations \(\bigoplus ,+,q_0,\ldots ,q_{k-1}\).

### Proof

Let \(\mathcal {S}\) be the subalgebra of \((k^\mathcal {N};\bigoplus ,+, q_0,\ldots ,q_{k-1})\) generated by the set {\(\mathbf {0},\ldots ,\mathbf {k-1}\)}, then \(\mathcal {S}\subseteq {\mathbf \Delta }^0_3(k^\mathcal {N})\) by Propositions 2(5) and 8. It remains to show that any \(A\in {\mathbf \Delta }^0_3(k^\mathcal {N})\) is Wadge equivalent to some \(B\in \mathcal {S}\). This is checked by the rank *rk*(*A*) of *A* in the well founded preorder \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _W)\). If \(rk(A)=0\) then \(A\in \){\(\mathbf {0},\ldots ,\mathbf {k-1}\)} and there is nothing to prove, so let *A* be non-constant. If *A* is 1-reducible then \(A\equiv _1\bigoplus _nA_n\) for some \(A_n<_1A\). By induction, \(A_n\equiv _1B_n\) for some \(B_n\in \mathcal {S}\), \(n\in \omega \). Then \(A\equiv _1\bigoplus _nB_n\in \mathcal {S}\), as desired.

Finally, let *A* be 1-irreducible. Consider the representation \(A\equiv _1A'+\tilde{A}\) from Proposition 7. Since \(T(A')=\omega ^*\), \(A'\) is 2-irreducible by Proposition 9(6). By Proposition 11, \(A'\equiv _2C\) for some \(C\in \mathcal {S}\), hence also \(A'\equiv _1C\) by the proof of (6)\(\rightarrow \)(4) in Proposition 9. If \(\tilde{A}=\bot \) then \(A\equiv _1C\in \mathcal {S}\) and we are done. Finally, let \(\tilde{A}\not =\bot \). Since \(\tilde{A}<_1A\), by induction \(\tilde{A}\equiv _1B\) for some \(B\in \mathcal {S}\). Then \(A\equiv _1C+B\in \mathcal {S}\), as desired. \(\Box \)

## 5 Operations on Labeled Forests

Let us briefly recall from [12, 13, 14] some operations of labeled forests and collect their properties used in the sequel (all these operations respect the *h*-equivalence). Recall that our labeled trees (*T*; *c*) in \(\widetilde{\mathcal {T}}_k\) (resp. in \(\widetilde{\mathcal {T}}_{\widetilde{\mathcal {T}}_k}\)) consist of a well founded tree *T* and a labeling \(c:T\rightarrow k\) (resp. \(c:T\rightarrow \widetilde{\mathcal {T}}_k\)). The non-empty labeled forests in \(\widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\) are obtained from such trees by removing the root \(\varepsilon \). The \(\omega \)-ary operation \(\bigoplus \) of disjoint union on \(\widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\) is defined in the obvious way.

For any \(i<k\) and \(F\in \widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\), let \(p_i(F)\) be the tree in \(\widetilde{\mathcal {T}}_{\widetilde{\mathcal {T}}_k}\) obtained from *F* by adjoining the empty string labeled by *i*. Note that the set \(\widetilde{\mathcal {F}}_k\) is closed under the operation \(p_i\) (we may think that \(\widetilde{\mathcal {F}}_k\subseteq \widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\) by identifying labels \(i<k\) with the singleton tree \(\mathbf {i}\) carrying the label *i*). Define the binary operation \(+\) on \(\widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\) as follows: \(F+G\) is obtained by adjoining a copy of *G* below any leaf of *F*. One easily checks that \(\mathbf{i}+F\equiv _hp_i(F)\), \(F\le _hF+G\), \(G\le _hF+G\), \(F\le _hF_1\rightarrow F+G\le _hF_1+G\), \(G\le _hG_1\rightarrow F+G\le _hF+G_1\), \((F+G)+H\equiv _hF+(G+H)\). Note that the set \(\widetilde{\mathcal {F}_k}\) is closed under the operation \(+\). Define the function \(s:\widetilde{\mathcal {T}}_k\rightarrow \widetilde{\mathcal {T}}_{\widetilde{\mathcal {T}}_k}\) as follows: *s*(*F*) is the singleton tree carrying the label *F*. Note that \(s(\mathbf i)=\mathbf {i}\) for each \(i<k\), and \(T\le _hS\) iff \(s(T)\le _hs(S)\), for all \(S,T\in \widetilde{\mathcal {T}}_k\). One easily checks the following properties:

### Proposition 13

- (1)
\((\widetilde{\mathcal {F}}_k;\bigoplus ,\le _h,p_0,\ldots ,p_{k-1})\) and \((\widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k};\bigoplus ,\le _h,p_0,\ldots ,p_{k-1})\) are \(\sigma \)-semilattices with discrete closures.

- (2)
For any \(T\in \widetilde{\mathcal {T}}_k\), \(F\mapsto s(T)+F\) is a closure operator on \(\widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\).

- (3)
For all \(T,T_1\in \widetilde{\mathcal {T}}_k\) and \(F,F_1\in \widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\), if \(s(T)+F\le _h s(T_1)+F_1\) and \(T\not \le _h T_1\) then \(s(T)+F\le _h F_1\).

## 6 Characterizing Degree Structures

The next proposition coincides with item (1) of Theorem 2.

### Proposition 14

The quotient-posets of \(({\mathbf \Delta }^0_2(k^\mathcal {N});\le _W)\) and of \((\widetilde{{\mathcal F}}_k;\le _h)\) are isomorphic.

### Proof

Let \((T;c)\in \widetilde{\mathcal {T}}_k\). Relate to any node \(\sigma \in T\) the *k*-partition \(\mu _T(\sigma )\) by induction on the rank \(rk(\sigma )\) of \(\sigma \) in \((T;\sqsupseteq )\) as follows: if \(rk(\sigma )=0\), i.e. \(\sigma \) is a leaf of *T* then \(\mu _T(\sigma ):=\mathbf {i}\) where \(i=c(\sigma )\); otherwise, \(\mu _T(\sigma ):=p_i(\bigoplus \{\mu _T(\sigma n)\mid n<\omega ,\sigma n\in T\})\).

Now we define a function \(\mu :\widetilde{\mathcal {T}}_k\rightarrow k^\mathcal {N}\) by \(\mu (T):=\mu _T(\varepsilon )\). Then \(T\le _hS\) iff \(\mu (T)\le _W\mu (S)\), for all \(T,S\in \widetilde{\mathcal {T}}_k\). This is checked using Propositions 13(1) and 3 by induction on \((rk_T(\varepsilon ),rk_S(\varepsilon ))\) in the lexicographic order of pairs of countable ordinals (for details see the proof of Theorem 5.1 in [10], although the induction there was not on the tree ranks of *T*, *S* but rather on their ranks in the well poset \((\widetilde{\mathcal {T}}_k;\le _h\))).

Next we extend \(\mu \) to \(\widetilde{\mathcal {F}}_k\) by \(\mu (F):=\bigoplus \{\mu _T(n)\mid n<\omega , (n)\in T\}\) where \(T:=\{\varepsilon \}\cup F\). Again, it is easy to see that \(F\le _hG\) iff \(\mu (F)\le _W\mu (G)\), for all \(F,G\in \widetilde{\mathcal {F}}_k\).

Repeating the proof of Proposition 10, we check by induction on *rk*(*A*) that for each \(A\in \mathbf { \Delta }^0_2(k^\mathcal {N})\) there is \(F\in \widetilde{\mathcal {F}}_k\) with \(\mu (F)\equiv _WA\). Thus, the function \(\mu \) induces the desired isomorphism of quotient-posets. \(\Box \)

The next proposition is interesting in its own right.

### Proposition 15

The quotient-posets of \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _2)\) and of \((\widetilde{{\mathcal F}}_k;\le _h)\) are isomorphic.

### Proof

The proof is a minor modification of the previous one. For any \((T;c)\in \widetilde{\mathcal {T}}_k\), define \(\nu _T:T\rightarrow k^\mathcal {N}\) as follows: if \(rk(\sigma )=0\) then \(\nu _T(\sigma ):=\mathbf {i}\) where \(i=c(\sigma )\), otherwise \(\nu _T(\sigma ):=q_i(\bigoplus \{\nu _T(\sigma n)\mid n<\omega ,\sigma n\in T\})\). Define \(\nu :\widetilde{\mathcal {T}}_k\rightarrow k^\mathcal {N}\) by \(\nu (T):=\nu _T(\varepsilon )\). Then \(T\le _hS\) iff \(\nu (T)\le _2\nu (S)\), for all \(T,S\in \widetilde{\mathcal {T}}_k\), because the operations \(q_i\) have the same algebraic properties as \(p_i\). Finally, extend \(\nu \) to \(\widetilde{\mathcal {F}}_k\) by \(\nu (F):=\bigoplus \{\nu _T(n)\mid n<\omega , (n)\in T\}\) where \(T:=\{\varepsilon \}\cup F\). Again, it is easy to see that \(F\le _hG\) iff \(\nu (F)\le _2\nu (G)\), for all \(F,G\in \widetilde{\mathcal {F}}_k\).

Repeating the proof of Proposition 11, we check by induction on *rk*(*A*) that for each \(A\in \mathbf { \Delta }^0_3(k^\mathcal {N})\) there is \(F\in \widetilde{\mathcal {F}}_k\) with \(\nu (F)\equiv _2A\). Thus, the function \(\nu \) induces the desired isomorphism of quotient-posets. \(\Box \)

The next proposition coincides with item (2) of Theorem 2.

### Proposition 16

The quotient-posets of \(({\mathbf \Delta }^0_3(k^\mathcal {N});\le _W)\) and of \((\widetilde{{\mathcal F}}_{\widetilde{{\mathcal T}}_k};\le _h)\) are isomorphic.

### Proof

Let \((T;c)\in \widetilde{\mathcal {T}}_{\widetilde{\mathcal {T}}_k}\). Relate to any node \(\sigma \in T\) the *k*-partition \(\rho _T(\sigma )\) by induction on the rank \(rk(\sigma )\) of \(\sigma \) in \((T;\sqsupseteq )\) as follows: if \(rk(\sigma )=0\) then \(\rho _T(\sigma ):=\nu (Q)\) where \(Q=c(\sigma )\in \widetilde{\mathcal {T}}_k\); otherwise, \(\rho _T(\sigma ):=\nu (Q)+(\bigoplus \{\rho _T(\sigma n)\mid n<\omega ,\sigma n\in T\})\). Now define a function \(\rho :\widetilde{\mathcal {T}}_{\widetilde{\mathcal {T}}_k}\rightarrow k^\mathcal {N}\) by \(\rho (T):=\rho _T(\varepsilon )\). Then \(T\le _hS\) iff \(\rho (T)\le _W\rho (S)\), for all \(T,S\in \widetilde{\mathcal {T}}_{\widetilde{\mathcal {T}}_k}\). This is checked using Propositions 13(2,3) and 2 by induction on \((rk_T(\varepsilon ),rk_S(\varepsilon ))\). Next we extend \(\rho \) to \(\widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\) by \(\rho (F):=\bigoplus \{\rho _T(n)\mid n<\omega , (n)\in T\}\) where \(T:=\{\varepsilon \}\cup F\). Again, it is easy to see that \(F\le _hG\) iff \(\rho (F)\le _W\rho (G)\), for all \(F,G\in \widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\).

Repeating the proof of Proposition 12, we check by induction on *rk*(*A*) that for each \(A\in \mathbf { \Delta }^0_3(k^\mathcal {N})\) there is \(F\in \widetilde{\mathcal {F}}_{\widetilde{\mathcal {T}}_k}\) with \(\rho (F)\equiv _WA\). Thus, the function \(\rho \) induces the desired isomorphism of quotient-posets. \(\Box \)

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