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1 Introduction

Various notions of reducibilities stronger than Turing reducibility have been studied in computability theory, e.g. the so called classical strong reducibilities: one-one reducibility (1-reducibility), many-one reducibility (m-reducibility), truth-table reducibility (tt-reducibility), and weak truth-table reducibility (wtt-reducibility) (see e.g. Odifreddi [13]). More recently, one has started to look at the so called strongly bounded Turing reducibilities: identity bounded Turing reducibility (ibT-reducibility) and computable Lipschitz reducibility (cl-reducibility) which are defined in terms of Turing functionals where the use is bounded by the identity function and the identity function plus a constant and which were introduced by Soare [14] and Downey, Hirschfeldt, and LaForte [9, 10], respectively. cl-reducibility is not only a notion of relative complexity but can also be viewed as a notion of relative randomness and is hence important in the field of algorithmic randomness (see the monograph [8] by Downey and Hirschfeldt for more background). The degree structures of the strongly bounded Turing reducibilities on the c.e. sets have been studied intensively. Barmpalias [5] showed that the partial ordering \(\left( \mathbf {R}_{cl },\le \right) \) of the c.e. cl-degrees has no maximal elements; Fan and Lu [12] showed that there are maximal pairs hence the partial orderings of the ibT- and cl-degrees are not upper semilattices, and Barmpalias and Lewis [6] and Day [7] showed that these partial orderings are not dense. Ambos-Spies, Bodewig, Kräling, and Yu [3] embedded the nonmodular lattice N5 into the c.e. ibT- and cl-degrees thereby showing that these partial orderings are not distributive, and Ambos-Spies [1] proved some global results, e.g. showed that the first order theories of the partial orderings of the c.e. ibT- and cl-degrees are undecidable. Recently, Ambos-Spies [2] introduced a more general class of bounded Turing reducibilities, the uniformly bounded Turing reducibilities. A reducibility \(r\) is a (uniformly) bounded Turing reducibility ((u)bT-reducibility) if there is a family \(\mathcal {F}\) of (uniformly) computable functions such that, for all sets \(A\) and \(B\), \(A\) is \(r\)-reducible to \(B\) if and only if \(A\) is Turing reducible to \(B\) with use bounded by some function \(f\) in \(\mathcal {F}\). We call a (uniformly) bounded Turing reducibility admissible if it is reflexive and transitive and we call it monotone if it is induced by a family of strictly increasing functions. Examples of monotone admissible ubT-reducibilities are the strongly bounded Turing reducibilities ibT and cl as well as the linearly bounded and the primitive recursively bounded Turing reducibilities. An example of an admissible monotone bT-reducibility which is not uniformly bounded is wtt-reducibility. Here, we will only look at the monotone admissible bT-reducibilities.

If a reducibility \(r\) is stronger than a reducibility \(r'\), of course, every upper \(r\)-bound for some sets \(A\) and \(B\) is also an upper \(r'\)-bound for \(A\) and \(B\) and the same holds for lower bounds. But this does not necessarily imply that least upper \(r\)-bounds (joins) have to be a least upper \(r'\)-bounds, too. Again, the same holds for greatest lower bounds (meets). Here, we ask the question for which reducibilities \(r\) and \(r'\), joins and meets in the c.e. \(r\)-degrees are preserved in the c.e. \(r'\)-degrees. We say \(r\)-\(r'\) join (meet) preservation holds if, for all noncomputable c.e. sets \(A\), \(B\), and \(C\) such that the \(r\)-degree of \(C\) is the join (meet) of the \(r'\)-degrees of \(A\) and \(B\), it holds that the \(r'\)-degree of \(C\) is the join (meet) of the \(r'\)-degrees of \(A\) and \(B\), too.

For most of the classical reducibilities mentioned above, the structure of the c.e. degrees is an upper semilattice where the join of the degrees of two sets \(A\) and \(B\) is induced by the effective disjoint union \(A\oplus B\). So, for two such reducibilities where \(r\) is stronger than \(r'\), of course, \(r\)-\(r'\) join preservation holds. So, for example, \(m \)-\(tt \) join preservation, \(tt \)-\(wtt \) join preservation and \(wtt \)-\(T \) join preservation hold. For reducibilities \(r\) whose degree structures are not an upper semilattice with join induced by the effective disjoint union, the question of \(r\)-\(r'\) join preservation is less obvious. For the classical strong reducibilities, \(1\)-reducibility is an example of such a reducibility, but, as one can easily show (see Lemma 2 below), \(1\)-m join preservation holds. It easily follows that \(r\)-\(r'\) join preservation holds for all classical strong reducibilities where \(r\) is stronger than \(r'\). For the (uniformly) bounded Turing reducibilities, the question of join preservation is less straightforward. Ambos-Spies, Ding, Fan, and Merkle [4] showed that ibT-cl join preservation holds and Ambos-Spies, Bodewig, Kräling, and Yu (see [1]) showed that cl-wtt join preservation holds, too. This may lead one to conjecture that – just as in case of the classical strong reducibilities – \(r\)-\(r'\) join preservation holds for any monotone admissible (u)bT-reducibilities where \(r\) is stronger than \(r'\), too. As we will show here, however, this is not the case. In fact, for \(r\) = ibT,cl and for any monotone amissible ubT-reducibility \(r'\) which is strictly stronger than cl, \(r\)-\(r'\) join preservation fails (see Theorem 1 below).

We complement our main result by considering meet preservation in the monotone admissible bt-reducibilities, too. There we generalize the result in [4] that ibT-cl meet preservation holds by showing that indeed, \(r\)-\(r'\) meet preservation holds for all monotone admissible bT-reducibilities \(r\) and \(r'\) such that \(r\) is stronger than \(r'\) (see Lemma 5).

So, for the monotone admissible (uniformly) bounded Turing reducibilities, meet preservation holds in general while, in some instances, join preservation fails. For the classical reducibilities, i.e. the strong reducibilities together with Turing reducibility, the converse is true. There join preservation holds in general, whereas, as Downey and Stob [11] showed, wtt-T meet preservation fails.

2 Preliminaries

A reducibility \(r\) is admissible if it is reflexive and transitive. For two reducibilities \(r\) and \(r'\), we say that \(r\) is stronger than \(r'\) (denoted by \(r\preceq r'\)) if, for all sets \(A\) and \(B\), from \(A\le _r B\), it follows that \(A\le _{r'}B\), and \(r\) is strictly stronger than \(r'\) (\(r\prec r'\)) if \(r\preceq r'\) and \(r\ne r'\).

Definition 1

For two admissible reducibilities \(r\) and \(r'\), we say that \(r\)-\(r'\) join preservation holds (in the c.e. degrees) if, for any noncomputable c.e. sets \(A\), \(B\), and \(C\),

$$\begin{aligned} deg_r(A)\vee deg_r(B)=deg_r(C)\Rightarrow deg_{r'}(A)\vee deg_{r'}(B)=deg_{r'}(C) \end{aligned}$$

holds. Otherwise, we say that \(r\)-\(r'\) join preservation fails. Similarly, \(r\)-\(r'\) meet preservation holds (in the c.e. degrees) if, for any noncomputable c.e. sets \(A\), \(B\), and \(C\),

$$\begin{aligned} deg_r(A)\wedge deg_r(B)=deg_r(C)\Rightarrow deg_{r'}(A)\wedge deg_{r'}(B)=deg_{r'}(C) \end{aligned}$$

holds and \(r\)-\(r'\) meet preservation fails otherwise.

Let \(\lbrace \varPhi _e^X:e\ge 0\rbrace \) be a fixed enumeration of all Turing functionals obtained by Gödelization of the oracle Turing machines. Then, we obtain an enumeration \(\lbrace \varPhi ^{X,f}_e:e\ge 0\rbrace \) of all \(f\)-bounded Turing functionals by bounding the use of each \(\varPhi _e^X\) on input \(x\) by \(f(x)\) (by making the computation divergent in case of longer oracle queries). For any pair of sets \(A\) and \(B\), \(A\) is \(f\)-bounded Turing reducible to \(B\) (denoted by \(A\le _{f-T }B\)) if and only if there is an \(e\) such that \(A=\varPhi ^{B,f}_e\). By letting \(f=id\), we obtain an enumeration \(\lbrace \hat{\varPhi }_e^X\rbrace \) of all identity bounded Turing functionals.

We call a reducibility \(r\) a bounded Turing reducibility (bT-reducibility) if there is a family \(\mathcal {F}\) of computable functions such that \(A\le _r B\) if and only if \(A\le _{f-T }B\) for some function \(f\in \mathcal {F}\); in this case we say that \(r\) is induced by \(\mathcal {F}\). If \(\mathcal {F}\) is uniformly computable, \(r\) is called a uniformly bounded Turing reducibility (ubT-reducibility). We call a bounded Turing reducibility monotone if it is induced by a family \(\mathcal {F}\) which consists only of strictly increasing functions. Note that ibT and cl are ubT-reducibilities which are induced by \(\mathcal {F}_{ibT }=\lbrace id\rbrace \) and \(\mathcal {F}_{cl }=\lbrace id+e:e\ge 0\rbrace \), respectively.

Lemma 1

(Ambos-Spies [2]). Let \(r\) and \(r'\) be admissible ubT-reducibilities. Then, \(r\preceq r'\) if and only if there are uniformly computable families \(\mathcal {F}\) and \(\mathcal {F}'\) that induce \(r\) and \(r'\), respectively, such that \(\mathcal {F}\le ^*\mathcal {F}'\), i.e. for every function \(f\in \mathcal {F}\), there is a function \(f'\in \mathcal {F}'\) such that \(f(x)\le f'(x)\) for almost all \(x\in \omega \).

3 Join Preservation

It is a straightforward observation that \(r\)-\(r'\) join preservation holds for reducibilities \(r\) and \(r'\) such that \(r\) is stronger than \(r'\) and such that the structures of the c.e. \(r\)-degrees and of the c.e. \(r'\)-degrees form upper semilattices with join induced by the effective disjoint union. We will now observe (by giving an example) that \(r\)-\(r'\) join preservation may hold even if the structure of the c.e. \(r\)-degrees does not form an upper semilattice.

Lemma 2

\(1\)-m join preservation holds.

Proof

Given c.e. sets \(A_0\), \(A_1\), and \(B\) such that

$$\begin{aligned} deg_1(A_0)\vee deg_1(A_1)=deg_1(B) \end{aligned}$$
(1)

holds, we have to show that \(deg_{m }(A_0)\vee deg_{m }(A_1)=deg_{m }(B)\) holds, too. As we know that \(deg_{m }(A_0)\vee deg_{m }(A_1)=deg_{m }(A_0\oplus A_1)\), we only have to show that \(B=_{m } A_0\oplus A_1\). It is obvious that \(A_i\le _1 A_0\oplus A_1\) via \(f_i(x)=2x+i\) for \(i=0,1\), so, it follows from (1) that \(B\le _1 A_0\oplus A_1\), hence \(B\le _{m }A_0\oplus A_1\). On the other hand, if we fix \(g_i\) such that \(A_i\le _1 B\) via \(g_i\) for \(i=0,1\), it follows that \(A_0\oplus A_1\le _{m } B\) via \(g\) where \(g(2x+i)=g_i(x)\) for all \(x\ge 0\) and for \(i=0,1\).   \(\square \)

More examples of reducibilities \(r\) and \(r'\) where the structure of \(r\) does not form an upper semilattice but where \(r\)-\(r'\) join preservation still holds have been given in the bounded Turing degrees.

Lemma 3

(Ambos-Spies, Ding, Fan, and Merkle [4]; Ambos-Spies [1]). ibT-cl, ibT-wtt, and cl-wtt join preservation hold.

This result might lead to the assumption that cl-\(r\) join preservation holds for all reducibilities \(r\) with \(cl \preceq r \preceq wtt \), but this is not the case. We will now show that cl-\(r\) join preservation even fails for all admissible monotone ubT-reducibilities with \(cl \prec r\).

Theorem 1

Let \(r\) be a monotone admissible ubT-reducibility such that \(cl \prec r\). Then, for \(r'=ibT, cl ,\) \(r'\)-\(r\) join-preservation fails.

Proof

By Lemma 3, ibT-cl join preservation holds. So, it is enough to prove the theorem for \(r'\) = ibT. Since, by cl \(\prec r\), any upper ibT-bound for two sets \(A_0\) and \(A_1\) is also an upper \(r\)-bound for \(A_0\) and \(A_1\), it suffices to construct c.e. sets \(A_0\), \(A_1\), \(B\), and \(C\) such that \(deg_\mathrm{{ibT} }(A_0)\vee deg_\mathrm{{ibT} }(A_1)=deg_\mathrm{{ibT} }(B)\) and such that \(A_0,A_1\le _r C\) but \(B\not \le _r C\). Let \(\mathcal {F}\) be a uniformly computable admissible family of strictly increasing functions such that \(r\) is induced by \(\mathcal {F}\). As \(\mathcal {F}\) is uniformly computable, we can fix a computable function \(f\) such that \(f\ge ^*h\) for all \(h\in \mathcal {F}\). As \(cl \prec r\), hence \(r\not \preceq cl \), \(\mathcal {F}\not \le ^*\lbrace id+e:e\ge 0\rbrace \) holds, so, there is a function \(g\in \mathcal {F}\) such that \(\lbrace g\rbrace \not \le ^*\lbrace id+e:e\ge 0\rbrace \), i.e. for any \(e\ge 0\), \(g(x)>x+e\) for infinitely many \(x\). Since \(g\) is strictly increasing, this implies that for all \(e\ge 0\), \(g(x)>x+e\) for all but finitely many \(x\), so, \(id+e\le ^*g\) for all \(e\ge 0\). So, in order to complete the proof, it suffices to show that the following lemma holds.

Lemma 4

Let \(g\) be a strictly increasing computable function such that \(id+e\le ^* g\) for all \(e\) and let \(f\) be any computable function (in particular, \(f\) can be chosen as above). Then, there are c.e. sets \(A_0, A_1, B\) and \(C\) such that the following hold.

$$\begin{aligned}&deg_\mathrm{{ibT} }(A_0)\vee deg_\mathrm{{ibT} }(A_1)=deg_\mathrm{{ibT} }(B)\end{aligned}$$
(2)
$$\begin{aligned}&A_0, A_1 \le _{g-T } C \end{aligned}$$
(3)
$$\begin{aligned}&B\not \le _{f-T } C. \end{aligned}$$
(4)

Proof

We will enumerate c.e. sets \(A_0, A_1, B\), and \(C\) such that (2) to (4) hold using a tree argument. The construction will use ideas introduced in the proof that the nondistributive lattice N5 can be embedded into the partial orderings \(\left( \mathbf {R}_\mathrm{{ibT} },\le \right) \) and \(\left( \mathbf {R}_{cl },\le \right) \) in [3]. Our notation will be the same as in that proof. To guarantee that (3) holds and that \(B\) is an upper ibT-bound for \(A_0\) and \(A_1\), we will satisfy the following global permitting (or coding) requirement for \(i=0,1\).

$$ \begin{aligned} (x\searrow _{s+1} A_i \Rightarrow \exists y\le x(y\searrow _{s+1}B))\; \& \; (x\searrow _{s+1} A_i \Rightarrow \exists y\le g(x)(y\searrow _{s+1}C)) \end{aligned}$$
(5)

To guarantee that \(B\) is in fact the least upper ibT-bound for \(A_0\) and \(A_1\), i.e. that (2) holds, we will meet the following join requirements for \(e\ge 0\).

$$ \begin{aligned} \mathcal {Q}_e: A_0=\hat{\varPhi }_{e_1}^{W_{e_0}}\; \& \;A_1=\hat{\varPhi }_{e_2}^{W_{e_0}}\Rightarrow B\le _\mathrm{{ibT} }W_{e_0}\; (e=\langle e_0,e_1,e_2\rangle ). \end{aligned}$$

Finally, we will satisfy condition (4) by meeting the nonordering requirements

$$\begin{aligned} \mathcal {P}_e: B\ne \varPhi _e^{C,f} \end{aligned}$$

for \(e\ge 0\). Before we give the actual construction, we will explain the ideas underlying the strategies for meeting the individual requirements and how to combine them.

As the join requirements \(\mathcal {Q}_e\) are conditional requirements whose hypotheses are not decidable, we have to guess on the correctness of the hypotheses. We define the length of agreement between \(A_0\) and \(\hat{\varPhi }^{W_{e_0}}_{e_1}\) and between \(A_1\) and \(\hat{\varPhi }^{W_{e_0}}_{e_2}\) at stage \(s\) by letting

$$ \begin{aligned} l(e,s)=\max \lbrace x:\forall y<x(A_{0,s}(y)=\hat{\varPhi }^{W_{e_0,s}}_{e_1,s}(y)\; \& \;A_{1,s}(y)=\hat{\varPhi }^{W_{e_0,s}}_{e_2,s}(y))\rbrace . \end{aligned}$$

Since the \(\hat{\varPhi }\) are bounded functionals, \(\lim _{s\rightarrow \infty }l(e,s)\le \infty \) exists and the following holds.

$$ \begin{aligned} (A_0=\hat{\varPhi }_{e_1}^{W_{e_0}}\; \& \;A_1=\hat{\varPhi }_{e_2}^{W_{e_0}})\Leftrightarrow \lim _{s\rightarrow \infty }l(e,s)=\infty \Leftrightarrow \limsup _{s\rightarrow \infty }l(e,s)=\infty . \end{aligned}$$
(6)

In the following, we call a join requirement \(\mathcal {Q}_e\) infinitary if its hypothesis is true (i.e., if \(\lim _{s\rightarrow \infty }l(e,s)=\infty \)) and we call \(\mathcal {Q}_e\) finitary otherwise. The strategy for meeting the join requirements is the join strategy used by Ambos-Spies, Bodewig, Kräling, and Yu in [3]. For meeting an infinitary join requirement \(\mathcal {Q}_e\), we guarantee \(B\le _\mathrm{{ibT} }W_{e_0}\) by permitting (up to some computable subset of \(B\)). We work with a computable set \(S=\lbrace s_n:n\ge 0\rbrace \) of \(\mathcal {Q}_e\)-expansionary stages, i.e., \(s_0<s_1<s_2<\ldots \) and \(l(e,s_0)<l(e,s_1)<l(e,s_2)<\ldots \). We ensure that numbers put into \(B\) between stages \(s_n+1\) and \(s_{n+1}+1\) are greater than \(s_n+1\). So, it suffices to guarantee that if a number \(x\) enters \(B\) at a stage \(s+1\) where \(s\in S\) and \(x<l(e,s)\) then a number \(\le x\) will be enumerated into \(W_{e_0}\) after stage \(s\). This change in \(W_{e_0}\) is forced by putting a sufficiently small number into \(A_0\) or \(A_1\). As one can easily check, this is achieved by guaranteeing the following.

$$ \begin{aligned} \begin{aligned} x\searrow _{s+1} B\; \& \; x<l(e,s)\Rightarrow&\exists y<\min (x',l(e,s))(y\searrow _{s+1} A_0 \ \mathrm {or} \ y\searrow _{s+1} A_1)\\&\mathrm {where} \ x'=\mu z(z>x\; \& \; z\notin W_{e_0,s}) \end{aligned} \end{aligned}$$
(7)

For meeting the nonordering requirements \(\mathcal {P}_e\), we will use the Friedberg-Muchnik strategy. For a fixed unused number \(x\), we ensure \(B(x)\ne \varPhi ^{C,f}_e(x)\) by waiting for a stage \(s\) such that \(\varPhi ^{C_s,f}_{e,s}(x)=0\). Then, at stage \(s+1\), we put \(x\) into \(B\) and, in order to preserve the computation \(\varPhi ^{C_s,f}_{e,s}(x)\), we impose a restraint of length \(f(x)+1\) on \(C\), thereby ensuring

$$\begin{aligned} B(x)=1\ne 0=B_s(x)=\varPhi ^{C_s,f}_{e,s}(x)=\varPhi ^{C,f}_e(x). \end{aligned}$$
(8)

In the presence of the join requirements and the global permitting requirement, this strategy needs some amendments. To describe the potential conflicts, consider the situation in which we wish to meet requirement \(\mathcal {P}_e\) and simultaneously satisfy the global permitting requirement (5) and follow the join strategy (7) for a single infinitary join requirement \(\mathcal {Q}_{e'}\) of higher priority.

Now, when we put a number \(x\) into \(B\) at stage \(s+1\) in order to guarantee (8), then, according to (7), we have to put a number \(y<x'\) into \(A_0\) or \(A_1\) at stage \(s+1\) where

$$ x'=\mu z(z>x\; \& \; z\notin W_{e'_0,s}).$$

(In our case, we choose to put \(y\) into \(A_1\).) If we do so, then, as long as \(x \le y\), this is consistent with the first part of condition (5). But, for the second part of this condition, we have to put a number \(z \le g(y)\) into \(C\). In case that \(z \le f(x)\), however, this will injure the restraint imposed on \(C\) in order to preserve the computation \(\varPhi ^{C_s,f}_{e,s}(x)\). In order to overcome this problem, we will make sure that we can find a number \(y\) such that \(f(x) < y < x'\) where \(y\) is not yet in \(A_1\) and the interval \([y, g(y)]\) is not yet completely enumerated into \(C\). (Then putting \(y\) into \(A_1\) and some new number \(z\) with \(y \le z \le g(y)\) into \(C\) makes the enumeration of \(x\) into \(B\) compatible with (5) and (7).)

For that matter, we will assign a sufficiently long interval \(I_n\) of unused numbers to \(\mathcal {P}_e\). \(I_n\) will contain finitely many candidates \(x_{n,k}\) for a possible attack on \(\mathcal {P}_e\) where these numbers are chosen so that \(x_{n,k+1}> f(x_{n,k})\) and \(g(x_{n,k}) \ge x_{n,k}+k+2\) for all \(k\). (Note that the latter can be achieved since, by choice of \(g\), \(g(y) > y+k+2\) for all sufficiently large \(y\); also note that \(g(x_{n,k}) \ge x_{n,k}+k+2\) implies \(g(y) \ge y+k+2\) for all \(y \ge x_{n,k}\).) We will arrange that, for some \(k\) (and some stage \(s\)), \(\left( x_{n,k}, x_{n,k+1}\right] \subseteq W_{e'_0,s}\) where \(x_{n,k}\) is not in \(B_s\), \(x_{n,k+1}\) is not in \(A_{1,s}\) and the interval \([x_{n,k+1},g(x_{n,k+1})]\) is not completely contained in \(C_s\). (Hence, for \(x= x_{n,k}\) and \(y=x_{n,k+1}\), \(y<x'\) whence we can ensure (8) and simultaneously obey (5) and (7) by putting \(x_{n,k}\) into \(B\), \(x_{n,k+1}\) into \(A_1\), and some unused number from the interval \([x_{n,k+1},g(x_{n,k+1})]\) into \(C\) at stage \(s+1\).) In order to ensure \(\left( x_{n,k}, x_{n,k+1}\right] \subseteq W_{e'_0}\) for some \(k\), we will successively and in decreasing order put numbers \(w\) from \(I_n\) into \(A_0\) at stages \(s+1\) where \(l(e,s)\) is greater than the endpoint of \(I_n\). This forces \(W_{e'_0}\) to respond by enumerating more and more numbers from \(I_n\) (or smaller ones). As we will argue, this implies that, at some point \(s\), there will be an interval \(\left( x_{n,k},\ldots x_{n,k+1}\right] \subset I_n\) such that the enumeration of the numbers \(\ge x_{n,k}+1\) from \(I_n\) into \(A_0\) has forced all the numbers \(x_{n,k}+1,\ldots , x_{n,k+1}\) into \(W_{e'_0}\). (In the actual construction, all the numbers actually have to be forced simultaneously into all sets \(W_{e'_0}\) attached to the infinitary higher priority join requirements, but we will show that this can be achieved.) So we can use \(x_{n,k}\) for an attack on \(\mathcal {P}_e\) – provided that \(x_{n,k} \not \in B_s\), \(x_{n,k+1}\not \in A_{1,s}\) and \([x_{n,k+1},g(x_{n,k+1})] \not \subseteq C_s\).

The latter, however, is not trivially true, since to make the enumeration of \(w\) into \(A_0\) compatible with (5) simultaneously we have to put a trace \(w_B \le w\) into \(B\) and a trace \(w_C \le g(w)\) into \(C\). So whenever we put \(w\) into \(A_0\), then, simultaneously we put \(w\) into \(B\) (which is compatible with (7) since \(w\) goes simultaneously into \(A_0\)) and a number from the interval \([w,g(w))\) into \(C\). Since we put only numbers \(w > x_{n,k}\) into \(A_0\) this procedure also puts only numbers \(> x_{n,k}\) into \(B\) and no numbers into \(A_1\) hence guarantees \(x_{n,k} \not \in B_s\) and \(x_{n,k+1}\not \in A_{1,s}\). To ensure that \([x_{n,k+1},g(x_{n,k+1})] \not \subseteq C_s\), however, we have to choose the trace \(w_C \in [w,g(w))\) to be put into \(C\) carefully. Here we let \(w_C = w+k'+1\) for the unique \(k'\) such that \(w \in (x_{n,k'},x_{n,k'+1}]\). Note that, by choice of the numbers \(x_{n,k'}\) this ensures that \(w_C \le g(w)\). On the other hand, this ensures that \(x_{n,k+1}+k+2\) is not enumerated into \(C\) since, for \(w \le x_{n,k+1}\), \(w_C \le w+k+1\) while, for \(w > x_{n,k+1} < x_{n,k+1}+k+2\), \(w_C \ge w+(k+1)+1 > x_{n,k+1}+k+2\).

This completes the discussion of the basic conflicts among the different goals of the construction and how these conflicts can be resolved. We now turn to the actual construction.

We implement the guesses about which of the join requirements are infinitary on the full binary tree \(T=\lbrace 0,1 \rbrace ^{<\omega }\). A node \(\alpha \) codes a guess about the first \(n\) join requirements \(\mathcal {Q}_0,\ldots , \mathcal {Q}_{n-1}\) where, for \(e<n\), \(\alpha (e)=0\) codes the guess that \(\mathcal {Q}_e\) is infinitary and \(\alpha (e)=1\) codes the guess that \(\mathcal {Q}_e\) is finitary. So the true path \(f:\omega \rightarrow \lbrace 0,1 \rbrace \) of the construction is defined by

$$ f(e)={\left\{ \begin{array}{ll} 0&{}\mathrm {if} \ A_0=\hat{\varPhi }^{W_{e_0}}_{e_1}\; \& \;A_1=\hat{\varPhi }^{W_{e_0}}_{e_2}\\ 1&{}\mathrm {otherwise.} \end{array}\right. }$$

For each node \(\alpha \) of length \(e\) there is a strategy \(\mathcal {P}_\alpha \) for meeting requirement \(\mathcal {P}_e\) which is based on the guess \(\alpha \). We will show that the strategy \(\mathcal {P}_{f \upharpoonright e}\) on the true path will succeed in meeting \(\mathcal {P}_e\).

At any stage \(s\) of the construction we have an approximation \(\delta _s\) of \(f \upharpoonright s\), i.e., a guess which of the first \(s\) join requirements are infinitary. For the definition of \(\delta _s\), first we inductively define \(\alpha \)-stages for each node \(\alpha \) as follows. Each stage \(s\ge 0\) is a \(\lambda \)-stage. If \(s\) is an \(\alpha \)-stage, then we call \(s\) \(\alpha \)-expansionary if \(l(|\alpha |,s)>l(|\alpha |,t)\) for all \(\alpha \)-stages \(t<s\), and we call \(s\) an \(\alpha 0\)-stage if \(s\) is \(\alpha \)-expansionary and an \(\alpha 1\)-stage if \(s\) is an \(\alpha \)-stage but not \(\alpha \)-expansionary. Now, for each \(s\ge 0\), let \(\delta _s\in T\) be the unique \(\alpha \) of length \(s\) such that \(s\) is an \(\alpha \)-stage. So, the node \(\delta _s\) represents the guess at which of \(\mathcal {Q}_0,\ldots \mathcal {Q}_{s-1}\) are infinite which is made at the end of stage \(s\). It easily follows from (6) that the true path is the leftmost path visited infinitely often in the construction.

Claim 1 (True Path Lemma). \(f=\liminf _{s\rightarrow \infty }\delta _s\) , i.e., for any \(\alpha \), \(\alpha \sqsubset f\) if and only if \(\alpha \sqsubset \delta _s\) for infinitely many \(s\) and there are only finitely many \(s\) such that \(\delta _s<_L\alpha \).

The intervals \(I_n\) which might be assigned to the strategies for meeting the nonordering requirements are inductively defined as follows, where the \(n\)th interval \(I_n\) consists of \(n(x_{n,0}+1)\) subintervals \(I_{n,k}= (x_{n,k},x_{n,k+1}]\).

$$ \begin{aligned} x_{0,0}= & {} \mu x(g(x)\ge x+2)\\ x_{n,k}= & {} \mu x(x>f(x_{n,k-1})\; \& \; g(x)\ge x+k+2)\\&\mathrm {for} \ n\in \omega \, \mathrm {and} \ 1\le k\le n (x_{n,0}+1)\\ x_{n+1,0}= & {} \mu x (x>x_{n,n(x_{n,0}+1)}+n(x_{n,0}+1)+2\; \& \;g(x) \ge x+2) \,\mathrm {for} \ n\in \omega \\ I_{n,k}= & {} \left( x_{n,k},x_{n,k+1}\right] \ \mathrm {for} \ n\in \omega \ \mathrm {and} \ 0\le k\le n (x_{n,0}+1)-1\\ I_n= & {} \bigcup _{k=0}^{n(x_{n,0}+1)-1} I_{n,k} \end{aligned}$$

Note that this definition ensures that \(x_{n,k+1} > f(x_{n,k})\), \(g(w) \ge w+k+2\) for \(w \in I_{n,k}\) and \(g(w) < x_{n+1,0}\) for \(w \in I_n\).

For a node \(\alpha \) of length \(e\), we call a number \(x\in I_n\cup \lbrace x_{n,0}\rbrace \) \(\alpha \)-safe at stage \(s\) if

$$\begin{aligned} x=x_{n,k}\ \mathrm {for \ some} \,k \ \mathrm {with} \ 0\le k\le n(x_{n,0}+1)-1 \end{aligned}$$
(9)
$$\begin{aligned} x\notin B_s, x_{n,k+1}\notin A_{1,s}\ \mathrm {and} \ x_{n,k+1}+k+2\notin C_s, \ \mathrm {and} \end{aligned}$$
(10)
$$ \begin{aligned} \forall e' ([e'<e\; \& \; \alpha (e')=0]\Rightarrow I_{n,k}\subseteq W_{e_0',s}) \end{aligned}$$
(11)

hold where \(e'=\langle e_0',e_1',e_2'\rangle \).

Using the above definitions, the construction of the sets \(A_0\), \(A_1\), \(B\), and \(C\) is as follows where stage 0 is vacuous (i.e., \(A_{0,0}=A_{1,0}=B_0=C_0=\emptyset \)).

  • Stage s + 1. A strategy \(\mathcal {P}_\alpha \) with \(|\alpha |=e\) requires attention at stage \(s+1\) if \(\alpha \sqsubseteq \delta _s\), \(\mathcal {P}_\alpha \) is not satisfied at the end of stage \(s\), and one of the following cases applies.

    1. (i)

      No interval is assigned to \(\mathcal {P}_\alpha \) at the end of stage \(s\).

    2. (ii)

      Interval \(I_n=\left( x_{n,0},x_{n,n(x_{n,0}+1)}\right] \) is assigned to \(\mathcal {P}_\alpha \) at the end of stage \(s\),

      $$ \begin{aligned} \forall e' ([e'<e\; \& \;\alpha (e')=0]\Rightarrow l(e',s)> x_{n,n(x_{n,0}+1)}) \end{aligned}$$
      (12)

      holds, no number \(x\in I_n\cup \lbrace x_{n,0}\rbrace \) is \(\alpha \)-safe at stage \(s\), and \(I_n\not \subseteq A_{0,s}\).

    3. (iii)

      Interval \(I_n\) is assigned to \(\mathcal {P}_\alpha \) at the end of stage \(s\), (12) holds, and there is a number \(x\in I_n\cup \lbrace x_{n,0}\rbrace \) such that \(x\) is \(\alpha \)-safe at stage \(s\) and \(B_s(x)=\varPhi ^{C_s,f}_{e,s}(x)=0\).

    Fix \(\alpha \) minimal such that \(\mathcal {P}_\alpha \) requires attention (as \(\mathcal {P}_{\delta _s}\) requires attention, there is such an \(\alpha \)). Declare that \(\mathcal {P}_\alpha \) receives attention or becomes active, initialize all strategies \(\mathcal {P}_\beta \) with \(\alpha <\beta \) (i.e., if an interval is assigned to \(\mathcal {P}_\beta \) then cancel this assignment and if \(\mathcal {P}_\beta \) had been satisfied before, then declare \(\mathcal {P}_\beta \) to be unsatisfied), and perform the following action according to the case via which \(\mathcal {P}_\alpha \) requires attention.

    1. (i)

      For the least \(n>e,s\) such that the interval \(I_n\) has not been assigned to any strategy before, assign \(I_n\) to \(\mathcal {P}_\alpha \).

    2. (ii)

      Let \(y\) be the greatest number in \(I_n\setminus A_{0,s}\). Put \(y\) into \(A_0\) and \(B\) and, for the unique \(k\) such that \(y\in I_{n,k}\), put \(y+k+1\) into \(C\).

    3. (iii)

      Let \(x\) be the greatest \(\alpha \)-safe number in \(I_n\cup \lbrace x_{n,0}\rbrace \) such that \(B_s(x)=\varPhi ^{C_s,f}_{e,s}(x)=0\). Let \(k\) be the unique number such that \(x=x_{n,k}\). Put \(x\) into \(B\), \(x_{n,k+1}\) into \(A_1\), and \(x_{n,k+1}+k+2\) into \(C\). Then, declare \(\mathcal {P}_\alpha \) to be satisfied.

This completes the construction. We will prove a series of claims to show that the construction satisfies all of our requirements. The claims will essentially be the same as in the proof of Theorem 3.2 in [3]. The first of these claims is straightforward and we omit the proof.

Claim 2. Every strategy \(\mathcal {P}_\alpha \) on the true path (i.e., \(\alpha \sqsubset f\) ) is initialized only finitely often and requires attention only finitely often. Moreover, for any such strategy, there is an interval \(I_n\) which is permanently assigned to it.

Claim 3. The global permitting requirement ( 5 ) is satisfied.

Proof

It is crucial to note that numbers from \(I_n \cup \{x_{n,0}\} \cup \{g(x): x \in I_n\}\) can be enumerated into any of the sets under construction at stage \(s+1\) only by the strategy to which \(I_n\) is assigned at this stage. So, it follows by a straightforward induction that if a strategy \(\mathcal {P}_\alpha \) acts via (ii) at stage \(s+1\) then, for the number \(y\) there, neither \(y\) is in \(B_s\) nor \(y+k+1\) is in \(C_s\). And, similarly, if a strategy \(\mathcal {P}_\alpha \) acts via (iii) at stage \(s+1\) then neither \(x_{n,k}\) is in \(B_s\) nor \(x_{n,k+1}\) is in \(A_{1,s}\) nor \(x_{n,k+1}+k+2\) is in \(C_s\) where the latter follows from our observations preceding the construction. This easily implies the claim, since a number \(x\) is enumerated into \(A_0\) at some stage \(s+1\) only if some strategy \(\mathcal {P}_\alpha \) acts at stage \(s+1\) via (ii), hence \(x \in I_{n,k}\) for some \(k\) and, at stage \(s+1\), \(x\) is enumerated into \(B\) and \(x+k+1\) is enumerated into \(C\) where \(x+k+1 \le g(x)\) by choice of \(I_{n,k}\); and since a number \(x\) is enumerated into \(A_1\) at some stage \(s+1\) only if some strategy \(\mathcal {P}_\alpha \) acts at stage \(s+1\) via (iii), hence \(x=x_{n,k+1}\) for some \(n,k\) and, at stage \(s+1\), \(x_{n,k}<x_{n,k+1}\) is enumerated into \(B\) and \(x_{n,k+1}+k+2\) is enumerated into \(C\) where by choice of \(x_{n,k+1}\), \(x_{n,k+1}+k+2\le g(x)\).

Claim 4. The join requirements \(\mathcal {Q}_e\) are met.

Proof

The argumentation is very similar to the one in the proof of Claim 5 in the proof of Theorem 3.2 in [3]. We fix \(e=\langle e_0,e_1,e_2\rangle \) and assume w.l.o.g. that \(\mathcal {Q}_e\) is infinitary, so, \(\alpha 0\sqsubset f\) for \(\alpha =f\upharpoonright e\). Hence there are infinitely many \(\alpha 0\)-stages. By Claims 1 and 2, we can fix an \(\alpha 0\)-stage \(s_0>e\) such that no strategy \(\mathcal {P}_\beta \) with \(\beta \le \alpha 0\) becomes active after this stage. Let \(S=\lbrace s_n:n\ge 0\rbrace \) be the set of the \(\alpha 0\)-stages \(\ge s_0\). Then, \(S\) is computable, \(s_0<s_1<s_2<\ldots \), and \(l(e,s_0)<l(e,s_1)<l(e,s_2)<\ldots \). So, as explained in the discussion of the strategy for meeting the requirements \(\mathcal {Q}_e\), it suffices to show that (7) holds for \(s\in S\). But this is immediate by construction since at a stage \(s_m+1\) only a strategy \(\mathcal {P}_\beta \) with \(\alpha 0 \sqsubseteq \beta \) may act. Namely, if \(\mathcal {P}_\beta \) acts via (ii) then the number \(x\) enumerated into \(B\) is simultaneously enumerated into \(A_0\) and if \(\mathcal {P}_\beta \) acts via (iii) then the claim follows from the corresponding action by \(\beta \)-safeness of the number \(x\) put into \(B\).

Claim 5. The nonordering requirements \(\mathcal {P}_e\) are met.

Proof

For fixed \(e\), assume for a contradiction that \(\mathcal {P}_e\) is not met. Exactly as in [3], we can then argue that for \(\alpha =f\upharpoonright e\), an interval \(I_n\) becomes permanently assigned to \(\mathcal {P}_\alpha \) at some stage \(s_1+1\), that there is no number \(x\in I_n\cup \lbrace x_{n,0}\rbrace \) that is \(\alpha \)-safe at any stage \(s'>s_1\), and that all numbers in \(I_n\) are enumerated into \(A_0\) in decreasing order after stage \(s_1+1\) according to clause (ii) in the definition of requiring and receiving attention. As in [3], for \(x\in I_n\), let \(t_x>s_1\) be the \(\alpha \)-stage such that \(x\) is enumerated into \(A_0\) at stage \(t_x+1\). Then (12) holds for \(s=t_x\). So, for \(x \in I_n\) and for any infinitary higher priority join requirement \(\mathcal {Q}_{e'}\), \(W_{e'_0,t_x}\upharpoonright x+1 \ne W_{e'_0,t_{x-1}}\upharpoonright x+1\). So if we let \(J\) be the set of the numbers \(e'_0\), such that

$$ \begin{aligned} J=\lbrace e'_0:\exists e'_1, e'_2:(\langle e'_0,e'_1,e'_2\rangle < e \; \& \; \mathcal {Q}_{\langle e'_0, e'_1, e'_2\rangle } \ \mathrm {is} \ \mathrm {infinitary} \rbrace , \end{aligned}$$

then

$$\begin{aligned} \forall j\in J\;\forall x\in I_n (W_{j,t_x}\upharpoonright x+1 \subset W_{j,t_{x-1}}\upharpoonright x+1). \end{aligned}$$
(13)

Now, for \(x\in I_n\) and \(j\in J\), let

$$\begin{aligned} w_j(x)=|W_{j,t_x}\upharpoonright x+1| \ \mathrm {and} \ w_J(x)=\sum _{j\in J}w_j(x), \end{aligned}$$

and call \(x\) unsaturated if \(x\notin W_{j,t_x}\) for some \(j\in J\). By definition, \(|J|\le e\) and \(w_j(x)\le x+1\), hence

$$\begin{aligned} w_J(x_{n,0})\le e (x_{n,0}+1). \end{aligned}$$
(14)

As in [3], we will now argue that this bound is not compatible with (13) and the fact that there are no \(\alpha \)-safe numbers in \(I_n\cup \lbrace x_{n,0}\rbrace \). As shown in [3], it follows from (13) that

$$\begin{aligned} w_J(x_{n,0})\ge |\lbrace x\in I_n:x\,\mathrm {is}\,\mathrm {unsaturated} \rbrace |. \end{aligned}$$
(15)

Now, it suffices to give a lower bound on the number of unsaturated numbers in \(I_n\) that contradicts (14). For a number \(x_{n,k}\in I_n\cup \lbrace x_{n,0}\rbrace \) with \(0\le k\le n(x_{n,0}+1)-1\), (9) and (10) hold for \(t_x=s\). So, since there are no \(\alpha \)-safe numbers in \(I_n\cup \lbrace x_{n,0}\rbrace \) after stage \(s_1+1\), (11) must fail for \(t_x=s\). It follows that at least one number in \(I_{n,k}\) must be unsaturated for every \(k\). As there are \(n (x_{n,0}+1)\) many subintervals \(I_{n,k}\) in \(I_n\) each of which must contain at least one unsaturated number and as \(e<n\) by construction, it follows that there are at least \((e+1)(x_{n,0}+1)\) unsaturated numbers in \(I_n\), which, together with (15), leads to the desired contradiction.

This completes the proof of Lemma 4.   \(\square \)

4 Meet Preservation

In contrast to Theorem 1, meet preservation holds for the monotone admissible bounded Turing reducibilities in general. This is immediate by the following lemma which generalizes the observation in [4] that ibT-cl and cl-wtt meet preservation hold.

Lemma 5

Let \(r\) and \(r'\) be monotone admissible bounded Turing reducibilities induced by \(\mathcal {F}\) and \(\mathcal {F}'\), respectively, such that \(r\) is stronger than \(r'\). Then, \(r\)-\(r'\) meet preservation holds.

Proof

The proof is essentially the same as the one for the results in [4]. Let \(A_0\), \(A_1\), and \(B\) be c.e. sets such that

$$\begin{aligned} deg_r(A_0)\wedge deg_r(A_1)=deg_r(B) \end{aligned}$$
(16)

holds. As \(r\) is stronger than \(r'\), \(B\) is also an upper \(r'\)-bound for \(A_0\) and \(A_1\), so, it suffices to show that for a given c.e. set \(C\) such that \(C\le _{r'}A_0,A_1\), \(C\le _{r'}B\) holds. Fix functions \(f_i\in \mathcal {F}'\) such that \(C\le _{f_i-T } A_i\) for \(i=0,1\). Since \(r'\) is admissible, as shown in [2], we may assume that \(\mathcal {F}'\) is closed under composition, so, \(f_0\circ f_1=f\in \mathcal {F}'\). As \(r'\) is monotone, we may also assume that \(f_0\) and \(f_1\) are strictly increasing, so, \(\max (f_0,f_1)\le f\). It follows that \(C\le _{f-T } A_0, A_1\). Let \(C_f=\lbrace f(x):x\in C\rbrace \) be the \(f\)-shift of \(C\). Then, \(C_f\le _\mathrm{{ibT} }A_0, A_1\). As ibT is stronger than \(r\), \(C_f\le _r A_0, A_1\), so, by (16), \(C_f\le _r B\), hence \(C_f\le _{r'}B\). We know that \(C\le _{f-T }C_f\), hence by \(f\in \mathcal {F}'\), \(C\le _{r'}C_f\), so, by transitivity of \(r'\), \(C\le _{r'}B\).   \(\square \)

5 Open Problems

Contrasting previous positive results on join preservation in the bounded Turing degrees (see Lemma 3) we have shown that \(r\)-\(r'\) join preservation fails for the strongly bounded Turing reducibilities \(r\) = ibT,cl and any monotone admissible uniformy bounded Turing reducibility \(r'\) with \(cl\prec r'\). This naturally leads to the question of a classification of the monotone admissible bounded Turing reducibilities \(r\) and \(r'\) for which \(r\)-\(r'\) join preservation holds. Moreover, one may consider nonmonotone reducibilities, too. For the latter, a classification of the bT-reducibilities for which meet preservation holds is open, too.