$\mathcal{D}$-maximal sets

0.35 MB·

by  Peter Cholak

D-MAXIMAL SETS
4 PETERA.CHOLAK,PETERGERDES,ANDKARENLANGE
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0
2 Abstract. Soare [20] proved that the maximal sets form an orbit in E. We
consider here D-maximal sets, generalizations of maximal sets introduced by
c Herrmann and Kummer [12]. Some orbits of D-maximalsets are well under-
e
stood, e.g., hemimaximal sets [8], but many are not. The goal of this paper
D
istodefinenewinvariantsoncomputablyenumerablesetsandtousethemto
giveacompletenontrivialclassificationoftheD-maximalsets. Althoughthese
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invariants help us to better understand the D-maximal sets, we use them to
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showthatseveralclassesofD-maximalsetsbreakintoinfinitelymanyorbits.

]
O
L
.
h 1. Introduction
t
a
LetE denotethestructureofcomputablyenumerable(c.e.) setsundersetinclu-
m
sion. Understandingthe lattice-theoreticpropertiesofE andthe interplaybetween
[
computability and definability in E are longstanding areas of research in classical

2 computabilitytheory. Inparticular,researchershaveworkedtounderstandthe au-
v tomorphismgroupofE andtheorbitsofE. Theorbitofac.e.setAisthecollection
6 ∼
ofc.e.sets[A]={B ∈E |(∃ Ψ:E −→E) (Ψ(A)=B)}.Oneofthemajorquestions
6
in classical computability is the following.
2
1
Question 1.1. What are the (definable) orbits of E, and what degrees are realized
.
1
in these orbits? How can new orbits be constructed from old ones?
0
4
In seminal work [20], Soare proved that the maximal sets form an orbit using
1
his Extension Theorem. Martin [18] had previously shown that the maximal sets
:
v are exactly those c.e. sets of high degree, thus describing the definable property of
i
X being maximal in degree-theoretic terms. In addition, Harrington had shown that
the creative sets form an orbit (see [21], Chapter XV). In time, Soare’s Extension
r
a Theorem was generalized and applied widely to construct many more orbits of E.
Forexample,DowneyandStob[8]showedthatthe hemimaximal sets,i.e.,splits of
maximalsets,formanorbitandstudiedtheirdegrees. Inparticular,anymaximalor
hemimaximalsetis automorphictoa completeset. Onthe otherhand,Harrington
and Soare [9] defined a first order nontrivial property Q such that if A is a c.e. set
and Q(A) holds, then A is not automorphic to a complete set. These results are
the first partial answers to the following question related to Question 1.1.
Question 1.2. Which orbits of E contain complete sets?
Date:December18,2014.
This research was partially done while Cholak participated in the Buenos Aires Semester in
Computability,ComplexityandRandomness,2013. LangewaspartiallysupportedbyNSFDMS-
0802961andNSFDMS-1100604.
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2 P.CHOLAK,P.GERDES,ANDK.LANGE
It turns out that until recently all known definable orbits of E, besides the orbit
ofcreativesets, were orbits ofD-hhsimple sets, generalizationsof hhsimple (hyper-
hypersimple) sets (see [6]). (We give extensive background on all definitions and
ideas mentioned here in §2.)
The Slaman-Woodin Conjecture [19] asserts that the set
∼
{hi,ji|(∃ Ψ:E −→E)[Ψ(W )=W ]}
i j
is Σ1-complete. The conjecture was based on the belief that information could be
1
coded into the orbits of hhsimple sets. Cholak, Downey and Harrington proved a
stronger version of the Slaman-Woodin Conjecture.
Theorem 1.3 (Cholak, Downey and Harrington [6]). There is a computably enu-
merable set A such that the index set {i∈ω |Wi ∼=A} is Σ11-complete.
In a surprising twist (again see [6]), the sets A witnessing Theorem 1.3 cannot be
simple or hhsimple (showing that the original idea behind the conjecture fails).
It is still open, however, if the sets in Theorem 1.3 can be D-hhsimple. More-
over, the behavior of hhsimple sets under automorphisms is now completely un-
derstood. Specifically, two hhsimple sets are automorphic if and only if they are
∆0-automorphic [2, Theorem 1.3]. There is no similar characterization of when
3
D-hhsimple sets are automorphic.
Here we consider D-maximal sets, a special case of D-hhsimple sets but a gen-
eralization of maximal sets, to gain further insight into Questions 1.1 and 1.2. A
c.e. set A is D-maximal if for all W there is a c.e. set D disjoint from A such that
W ⊆∗ A⊔D or W ∪(A⊔D)=∗ ω. We can understand a given D-maximal set A
in terms of the collection D(A) of c.e. sets that are disjoint from A.
The goal of this paper is to provide a complete nontrivial classification of the
D-maximal sets in terms of how D(A) is generated. In Theorem 3.10, we describe
ten types of ways D(A) can be generated for any c.e. set A. We then show in
Theorem 4.1 that there is a complete and incomplete D-maximal set of each type.
The first six types of D-maximal sets were already well understood ([20], [8], [3],
[5]). Furthermore, Herrmann and Kummer [12] had constructed D-maximal sets
that were not of the first six types (in particular, as splits of hhsimple and atom-
less r-maximal sets). We, however, show that there are four types of examples of
D-maximal sets besides the first six and that each of these types breaks up into
infinitely many orbits. Moreover, we provide an overarching framework for under-
standing and constructing these examples. We discuss D-maximal sets of the first
six types and the type that arises as a split of an r-maximal set in §4. In §5, we
show how the remaining three types are very similar to the construction of splits
of hhsimple sets. For ease of reading, we discuss open questions as they arise. In
particular, open questions can be found in §3.5 and §5.8.
2. Background and definitions
All sets consideredin this paper are computably enumerable (c.e.), infinite, and
coinfinite unless explicitly specified. Let E∗ be the structure E modulo the ideal of
finite sets F. By Soare [20], it is equivalent to work with E∗ instead of E in the
sense that two sets A and B are in the same orbit in E if and only if they are in
D-MAXIMAL SETS 3
the same orbit in E∗. Given a c.e. set A, we define
L(A)=({W ∪A|W a c.e. set},⊆) and
E(A)=({W ∩A|W a c.e. set},⊆).
We let L∗(A) be the structure L(A) modulo F, and E∗(A) be the structure E(A)
modulo F. Recallthat A is maximal if for all B ∈L∗(A), if B 6=∗ A, then B =∗ ω.
IfweunderstandtheorbitofA,wecansometimesunderstandtheorbitsofsplits
of A.
Definition 2.1. (i) We call A ⊔A =A a splitting of A, and we call A and
0 1 0
A splits of A or halves of the splitting of A. We say that this splitting is
1
trivial if either of A or A are computable.
0 1
(ii) We call A ⊔A = A a Friedberg splitting of A if the following property
0 1
holds for any c.e. W: if W −A is not c.e. then neither of W −A are c.e.
i
as well.
(iii) Given a property P of c.e. sets, we say that a noncomputable c.e. set A
is hemi-P if there is a noncomputable c.e. set B disjoint from A such that
A⊔B satisfies P.
Note that if P is a definable property in E or E∗, then hemi-P is also definable
there.
2.1. D-hhsimple and D-maximal sets.
2.1.1. Motivation. Recall that a coinfinite set A is hhsimple if and only if L∗(A)
is a boolean algebra ([14], see also Soare [21]). Hence, A is maximal if and only if
L∗(A) is the two element boolean algebra.
Theorem 2.2 (Lachlan [14]). If a set H is hhsimple, then L∗(H) is a Σ0 boolean
3
algebra. Moreover, for every Σ0 boolean algebra B, there is a hhsimple set H such
3
that L∗(H) is isomorphic to B.
GivenTheorem2.2,wesaythatahhsimplesetH hasflavorBifL∗(H)isisomorphic
to the Σ0 boolean algebra B. Note that the ordering ≤ on a Σ0 boolean algebra is
3 3
0′′′-computable.
2.1.2. Working modulo D(A). Given a set A, we define
D(A)={B :B ∈L(A) & B−A is c.e.},
andletD∗(A) be the structureD(A)modulo F. SinceD∗(A)is anidealinthe lat-
tice L∗(A), we can take the quotient lattice L∗(A)/D∗(A). Theorem 2.2 motivates
the following definition.
Definition2.3. (HerrmannandKummer[12])AsetAisD-hhsimpleifL∗(A)/D∗(A)
isabooleanalgebra,andAisD-maximalifL∗(A)/D∗(A)isthetwoelementboolean
algebra.
By unraveling Definition 2.3, we have the following working definition of D-
maximality.
Definition 2.4. A set A is D-maximal if for all W there is a c.e. set D disjoint
from A such that W ⊆∗ A⊔D or W ∪(A⊔D)=∗ ω.
AnotherusefulcharacterizationoftheD-maximalsetsisgiveninthenextlemma.
4 P.CHOLAK,P.GERDES,ANDK.LANGE
Lemma 2.5 (Cholak et al. [5] Lemma 2.2). Let A be a c.e. noncomputable set.
The set A is D-maximal if and only if, for all c.e. W ⊇A, either W −A is c.e. or
there exists a computable R such that A⊆R⊆W.
Herrmannand Kummer [12] studied the D-hhsimple sets in the contextof diag-
onal sets. A set is diagonal if it has the form {e∈ω |ψ (e)} for some computable
e
enumeration{ψ } ofallpartialcomputable functions. In [12], they showedthat
i i∈ω
a set is not diagonal if and only if it is computable or D-hhsimple. Note that this
result implies that the property of being diagonal is elementary lattice-theoretic.
2.2. KnownexamplesofD-maximalsets. Maximalsetsandhemimaximalsets
(which form distinct orbits [20], [8]) are clearly D-maximal. Similarly, a set that
is maximal on a computable set is also D-maximal. In these three cases, for the
D-maximal set A there is a W such A∪W is maximal. As we will see, this does
not occur for other types of D-maximal sets. Others, however, have constructed
additional kinds of D-maximal sets, in particular, Herrmann and hemi-Herrmann
sets and sets with A-special lists, which we define now. It is easy to check that
these sets are D-maximal from their respective definitions.
Definition 2.6. (i) Wesaythatac.e.setAisr-separableif,forallc.e.setsB
disjoint from A, there is a computable set C such that B ⊆C and A⊆C.
We say that A is strongly r-separable if, additionally, we can choose C so
that C−B is infinite.
(ii) We say that a set A is Herrmann if A is both D-maximal and strongly
r-separable.
(iii) Given a set A, we call a list of c.e. sets F = {F : i ∈ ω} an A-special
i
list if F is a collection of pairwise disjoint noncomputable sets such that
F = A and for all c.e. sets W, there is an i such that W ⊆∗ F or
0 Fl≤i l
W ∪ F =∗ ω.
Fl≤i l
(iv) WesayasetAisr-maximal ifforeverycomputablesetR,eitherR∩A=∗ ∅
(so R ⊆∗ A) or R∩A =∗ ∅ (so A ⊆∗ R), i.e., no infinite computable set
splits A into two infinite sets.
(v) A c.e. set B is atomless if for every c.e. set C, if B ⊆ C 6=∗ ω, then there
is a c.e. set E such that C (∗ E (∗ ω, i.e., B does not have a maximal
superset.
Herrmann and hemi-Herrmann sets were defined by Hermann and further dis-
cussed in [5]. The main results in [5] for our purposes are that such sets exist
(Theorem 2.5) and that these sets form distinct (Theorem 6.9) definable (Defini-
tion 2.3) orbits (Theorems 4.1, 6.5) each containing a complete set (Theorems 7.2,
6.7(i)).
The notion of a set A with an A-special list was introduced in [3, §7.1]. There,
CholakandHarringtonshowedthatsuchsetsexistandformadefinable∆0 butnot
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∆0 orbit. This orbit remains the only concrete example of an orbit that is not ∆0.
3 3
Furthermore, as mentioned earlier Herrmann and Kummer [12] had constructed
D-maximal splits of hhsimple and atomless r-maximal sets in addition to the ones
mentioned above. We will discuss these examples later (see §5.1), but first we
explore the notion of a generating set for D(A) for an arbitrary (not necessarily
D-maximal) set A.
D-MAXIMAL SETS 5
3. Generating sets for D(A)
Inthissection,weonlyassumethatthesetsconsideredarecomputablyenumer-
able. In later sections, we will work explicitly with D-maximal sets. We will use
the framework of generating sets to understand and classify the different kinds of
D-maximal sets.
Definition 3.1. We say a (possibly finite or empty) collection of c.e. sets
G = {D ,D ,...} generates D(A) (equivalently G is a generating set for D(A)) if
0 1
each D is disjoint from A for all i ∈ ω and for all c.e. sets D that are disjoint
i
from A, there is a finite set F ⊂ ω such that D ⊆∗ D . In this case, we
Sj∈F j
say that {D | j ∈ F} covers D. If G generates D(A), we write D(A) = hGi.
j
We say {D ,D ,...} partially generates D(A) if there is some collection of sets G
0 1
containing {D ,D ,...} such that hGi=D(A).
0 1
We list a few basic observations.
Lemma 3.2. (i) Generating sets always exist for D(A). In particular, D(A) is
generated by the collection of all c.e. sets that are disjoint from A.
(ii) Let Φ be an automorphism of E∗. If for all c.e. W, we set Wˆ := Φ(W), then
{D ,D ,...} generates D(A) if and only if {Dˆ ,Dˆ ,......} generates D(Aˆ).
0 1 0 1
(iii) D(A)=h∅i iff A is simple.
3.1. Simplifyinggenerating sets. GeneratingsetsforD(A)arefarfromunique.
Here we develop some tools for finding less complex generating sets for D(A). We
use different tools based on whether or not D(A) has a finite generating set.
3.1.1. Finite generating sets.
Lemma 3.3. If a finite collection of sets G generates D(A), then D(A) = h∅i,
D(A)=hRi for some infinite computable set R, or D(A)=hWi for some noncom-
putable c.e. set W. Moreover, if {D} and {D˜} both generate D(A), then D =∗ D˜.
Proof. The union W of the finitely many sets in G is c.e. and disjoint from A and
clearly generates D(A). If W is finite, then D(A) = h∅i, and otherwise, we are in
the remaining two cases. For the last statement, D˜ ⊆∗ D and D ⊆∗ D˜ by the
definition of generating set. (cid:3)
The collection of all c.e. sets that have finite generating sets is definable.
Lemma 3.4. The statement “A single set generates D(A)” is an elementarily
definable statement in E∗ under inclusion.
3.1.2. Infinite generating sets. Infinite generating sets canbe much more complex,
dependingonwhetherall(ormanyof)theelementscanbechosentobecomputable
or pairwise disjoint.
Lemma 3.5. If {R ,R ,...}∪G generates D(A) where R is computable for all
0 1 i
i∈ω,thenthereexistsapairwisedisjoint collectionofcomputablesets{R˜ ,R˜ ,...}
0 1
so that {R˜ ,R˜ ,...}∪G generates D(A).
0 1
Proof. If {R ,R ,...}∪G generatesD(A), we inductively define {R˜ ,R˜ ,...}. Let
0 1 0 1
R˜ = R . Given the pairwise disjoint collection of computable sets {R˜ ,...,R˜ },
0 0 0 n
let m be the least index such that R − R˜ is infinite. If no such m exists,
m Fi≤n i
{R˜ ,...,R˜ }∪G generates D(A).
0 n
6 P.CHOLAK,P.GERDES,ANDK.LANGE
Otherwise, let R˜ = R − R˜ = R ∩ R˜ . The collection
n+1 m Fi≤n i m Fi≤n i
{R˜ ,R˜ ,...} satisfies the conclusion of the lemma since for each i ∈ ω there is an
0 1
m such that R ⊆∗ R˜ . (cid:3)
Sj≤i j Fj≤m j
Lemma 3.6. If D(A) is generated by an infinite collection of pairwise disjoint
sets, then D(A) is also generated by an infinite collection of pairwise disjoint sets
containing only computable sets, only noncomputable sets, or only computable sets
and one noncomputable set.
Proof. Suppose that the collection G = {D ,D ,...,R ,R ,...} generates D(A)
0 1 0 1
andconsistsofpairwisedisjointsetssuchthatD isnoncomputableandR iscom-
i i
putable for all i∈ω. Suppose that there are both computable and noncomputable
sets in G. We may assume all of these sets are infinite. If G contains only finitely
manyD ,the finite unionofthe D stogetherwith{R ,R ,...}generatesD(A). If
i i 0 1
G contains infinitely many D s, then {D˜ ,D˜ ,...}, where D˜ :=D ⊔R , generates
i 0 1 i i i
D(A). Note that D being noncomputable implies that D˜ is noncomputable. (cid:3)
i i
Lemma3.7. If{R ,R ,...}and{D ,D ,...}arepairwisedisjointgeneratingsets
0 1 0 1
for D(A) and all sets in {R ,R ,...} are computable, then all sets in {D ,D ,...}
0 1 0 1
are computable.
Proof. By definition of generating sets, there is a finite F ⊂ ω such that
D ⊆∗ R . It suffices to show that R − D is a c.e. set. There is
i Fj∈F j Fj∈F j i
a finite H ⊂ω such that R ⊆∗ D . Since the D are pairwise disjoint,
Fj∈F j Fj∈H j i
i ∈ H. Set H˜ := H −{i}. Then ( R −D ) ⊆∗ D . Since D and
Fj∈F j i Fj∈H˜ j i
D are disjoint,
Fj∈H˜ j
R −D =∗ R ∩ D , which is c.e.
G j i G j G j
j∈F j∈F j∈H˜
(cid:3)
We may assume that we have a generating set for D(A) whose union is A¯.
Lemma 3.8. If {D ,D ,...}∪G generates D(A) (and {D ,D ,...} is a collection
0 1 0 1
of pairwise disjoint sets), then there exists a (pairwise disjoint) collection of sets
{D˜ ,D˜ ,...} such that A= D˜ and {D˜ ,D˜ ,...}∪G generates D(A).
0 1 Fi∈ω i 0 1
Proof. IfX =A− D andX ={x <x <...},wecantakeD˜ :=D ⊔{x }.
Fi∈ω i 0 1 i i i
(cid:3)
We can also simplify partial generating sets that are not pairwise disjoint.
Lemma 3.9. Let {D ,D ,...} be a list of noncomputable c.e. sets whose union
0 1
with G generates D(A). Then, there is a collection of noncomputable c.e. sets
{D˜ ,D˜ ,...} whose union with G generates D(A) such that all the sets are either
0 1
pairwise disjoint or nested so that D˜ −D˜ is not c.e. for all n∈ω.
n+1 n
Proof. Inahighlynoneffectiveway,webuildalist{D˜ ,D˜ ,...},satisfyingourcon-
0 1
clusion. ToensurethatthislistpartiallygeneratesD(A)asdescribed,weconstruct
this list so that each D˜ is disjoint from A and every D is contained in the union
i i
of finitely many D˜ ’s.
i
D-MAXIMAL SETS 7
We attempt to inductively construct the list to consist of pairwise disjoint sets
based on an arbitrary starting point k ∈ ω. For each k ∈ ω, we inductively define
a function l :ω →ω. We set l (0):=k andD˜ = D . We let l (n+1)be
k k 0 Si≤lk(0) i k
theleastnumber(ifitexists)suchthat D − D isac.e.set. Let
Si≤lk(n+1) i Si≤lk(n) i
D˜ be this c.e. set. Then, D˜ = D . If for some initial choice of k,
n+1 Fi≤n i Si≤lk(n) i
the function l has domain ω, then the sets in {D˜ ,D˜ ,...} are pairwise disjoint.
k 0 1
Otherwise, the above procedure fails for all initial choices of k. Then, each l
k
is a strictly increasing function defined on some nonempty finite initial segment of
ω. Let m : ω → ω be defined so that m(k) is the maximum value of l . For all
k
k, m(k) ≥ k. Moreover, for all k and l > m(k), D − D is never a
Si≤l i Si≤m(k) i
c.e. set. We define a strictly increasing function m˜ : ω → ω inductively by setting
m˜(0) = m(0) and m˜(n+1) = m(m˜(n)+1). By construction, the list given by
D˜ = D has the desired nesting property. (cid:3)
n Si≤m˜(n) i
3.2. Standardized Types of generating sets. We use the results from §3.1 to
show that any c.e. set A has a generating set for D(A) of one of ten standardized
types. We canthen classify c.e.sets by the complexity of their generatingsets (see
Definition 3.11).
Theorem 3.10. For any c.e. set A, there exists a collection of c.e. sets, G, gener-
ating D(A) of one of the following types.
Type 1: G ={∅}.
Type 2: G ={R}, where R is an infinite computable set.
Type 3: G ={W}, where W is an infinite noncomputable set.
Type 4: G = {R ,R ,...}, where the R are infinite pairwise disjoint com-
0 1 i
putable sets.
Type 5: G = {D ,R ,R ,...}, where D is the only noncomputable set and
0 0 1 0
all the sets are infinite and pairwise disjoint.
Type 6: G = {D ,D ,...}, where the D are infinite pairwise disjoint non-
0 1 i
computable sets.
Type 7: G = {D ,R ,R ,...}, where D is the only noncomputable set, the
0 0 1 0
R are infinite pairwise disjoint computable sets, and D ∩ R 6= ∅ for
i 0 i
infinitely many i.
Type 8: G = {D ,D ,...,R ,R ,...}, where the D are pairwise disjoint
0 1 0 1 i
noncomputable sets and the R are infinite pairwise disjoint computable
i
sets.
Type 9: G = {D ,D ,...,R ,R ,...}, where the R are infinite pairwise
0 1 0 1 i
disjoint computable sets and the D are infinite nested noncomputable sets
i
such that, for all l ∈ω, D −D is not c.e. and there are infinitely many
l+1 l
j such that R −D is infinite.
j l
Type 10: G ={D ,D ,...}, where the D are infinite nested noncomputable
0 1 i
sets such that D −D is not c.e. for all l∈ω.
l+1 l
Proof. (1)If there is a finite generatingsetfor D(A), then there is a generatingset
of Type 1, 2 or 3 for D(A) by Lemma 3.3.
(2)IfD(A)hasaninfinitegeneratingsetconsistingofpairwisedisjointsets,then
D(A) has a generating set of Type 4, 5, or 6 by Lemma 3.6. We remark that, by
Lemma 3.7, if D(A) has a generating set of Type 4, then D(A) does not have a
generating set of Type 5 or Type 6. Note that if D(A) has an infinite generating
8 P.CHOLAK,P.GERDES,ANDK.LANGE
set consisting only of computable sets, we can assume these computable sets are
pairwise disjoint by Lemma 3.5 and hence (2) holds.
Assumetheantecedentsof(1)and(2)fail,andtakesomegeneratingsetforD(A).
Byrepeatedlytakingfiniteunionsofsomeofthesets,wecanassumethatthereare
zero, one, or infinitely many computable sets in this generating set. If there is one
computable set R, then we can assume that all other sets are noncomputable and
disjoint from R (by removing R from each of those sets). We can then take one of
thenoncomputablesets,W,andreplaceitwithR⊔W. So,wecanassumethatthe
generating set has either no computable sets or infinitely many pairwise disjoint
computable sets (again by Lemma 3.5). Similarly, by taking finite unions, we can
assume the generating set also has zero, one, or infinitely many noncomputable
sets. By the failure of the antecedent of (2) and Lemma 3.5, having zero is not a
possibility.
(3) If there is one noncomputable set, then D(A) has a generating set of Type
7 since the antecedents of (1) and (2) fail. Specifically, there must be infinitely
many disjoint computable sets in the generating set. If only finitely many of these
computable sets intersected with the one noncomputable set W, we could replace
W by its union with these finitely many sets to obtain a Type 5 generating set, a
contradiction.
Iftheantecedentof(3)fails,thegeneratingsetcontainsinfinitelymanynoncom-
putable sets {D ,D ,...}, and, by Lemma 3.9, these noncomputable sets can be
0 1
takentobe eitherpairwisedisjointornestedsothatD ⊂D andD −D is
n n+1 n+1 n
notc.e.foralln∈ω. Ifthegeneratingsetcontainsinfinitelymanycomputablesets
and the infinitely many noncomputable sets are pairwise disjoint, this generating
set is of Type 8.
(4) If D(A) has a generating set of Type 8 or Type 9, we are done.
Nowwewillarguethatthefailureofalltheantecedentsof(1)though(5)implies
that D(A) has a generating set of Type 10. Notice that if no computable sets
remain in our generating set, we are done. So, assume otherwise. Hence, the
noncomputable sets are nested, and almost all the computable sets are almost
contained in one of the noncomputable sets. For each noncomputable set D in
i
this generating set, we can take the union of D and the remaining finitely many
i
computable sets to obtain a new generating set {D˜ ,D˜ ,...} where the D˜ are
0 1 i
infinite nestedsets. Thisgeneratingsetcannotcontaininfinitely manycomputable
D˜ since the collection of the computable D˜ would be a generating set consisting
i i
of only computable sets by the note after (2). Hence, there are only finitely many
computable D˜ . The collectionofnoncomputable D˜ alsogeneratesD(A) since the
i i
D˜ are nested. If this generating set is not ofType 10, we can apply Lemma 3.9 to
i
obtain one of Type 10 since the antecedent of (2) fails. (cid:3)
NotethatD(A)mayhavegeneratingsetsofdifferentTypes. However,theTypes
are listed in order of increasing complexity. By following the procedure outlined in
the proof of Theorem 3.10, we will always find a generating set for D(A) of lowest
possible complexity. Hence, we can classify the c.e. sets by the Type complexity of
their generating set.
Definition 3.11. We say the c.e. set A is Type n if there is a generating set for
D(A) of Type n but no generating set for D(A) of Type m for all m<n.
D-MAXIMAL SETS 9
InTheorem4.1, the mainresultofthis paper, weshow thereis a D-maximalset of
each Type.
We more closely examine sets of a given Type in §3.4, but it is helpful to first
observe the behavior of generating sets under splitting.
3.3. Splits and generating sets for D(A).
Lemma 3.12. Suppose R is computable, A ⊔R = A, and G ⊆ D(A). Then G
0
generates D(A) iff G∪{R} generates D(A ).
0
Proof. LetD be c.e.anddisjointfromA . SinceD−A=D∩Risc.e.anddisjoint
0
from A, the set D is coveredby R and finitely many sets in G. (cid:3)
Corollary 3.13. A set A is half of a trivial splitting of a simple set iff D(A)=hRi
for some computable R.
Lemma3.14. SupposeA ⊔A =A. IfG∪{A }generatesD(A ),thenG generates
0 1 1 0
D(A).
Lemma 3.15. Suppose G generates D(A) and A ⊔ A = A. If A ⊔ A is a
0 1 0 1
Friedberg splitting of A, then G∪{A } generates D(A ).
1 0
Proof. Let D be a c.e. set. If D−A is not c.e., then D−A is not c.e. and D is
0
not disjoint from A . So, assume that D−A is a c.e. set. If D is disjoint from A ,
0 0
then D is covered by A and finitely many sets in G. (cid:3)
1
Corollary 3.16. If A is half of a Friedberg splitting of a simple set, then
D(A)=hWi where W is not computable.
Suppose that A ⊔A = A is a nontrivial splitting that is not Friedberg. We
0 1
would like a result describing a generating set for D(A ) similar to Lemmas 3.12
0
and 3.15, but such a result is not clear. For the splitting A ⊔A = A, there is a
0 1
set W such that W −A is not c.e. but W −A is a c.e. set. Since W −A may not
0
be contained in a finite union of generators for D(A) (for example, if A is simple),
the setA andthe generatorsfor D(A)maynotgenerateD(A ). Also,the workin
1 0
Section4.2showsthattheconverseofLemma3.15fails;forasplittingA ⊔A ,the
0 1
collectionG∪{A }generatingD(A )doesnotmeanthatthesplittingisFriedberg.
1 0
3.4. Understanding the Types. Sets of Types 1, 2, and 3 are particularly well
understood. By Lemma 3.2, S is simple iff S is of Type 1; there are no infinite c.e.
sets disjoint from S. By this fact and Lemma 3.12, A⊔R is simple iff A is Type
2. By Lemma 3.15, if A is half of a Friedberg splitting of a simple set, then A is of
Type 3. Moreover,sets of these Types are definable.
Lemma 3.17. The statement “A is Type 1 (respectively 2, 3)” is elementarily
definable in E∗ under inclusion.
Proof. ThesetAisType1iffAissimple,andAisType2iffthereisacomputable
set R disjoint from A such that A⊔R is simple. The set A is Type 3 iff there a
c.e. set D such that D is disjoint from A and for all c.e. sets W disjoint from A,
W ⊆∗ D. (cid:3)
In §4, we will show that there are D-maximal sets of all ten Types. Moreover,
we will show that D-maximal sets of Type 4, 5 and 6 are definable, somewhat
extending Lemma 3.17. However, the following question is open.
10 P.CHOLAK,P.GERDES,ANDK.LANGE
Question 3.18. Is there a result similar to Lemma 3.17 for the remaining Types
of sets in a general setting?
We finish this section with a remark on the behavior of sets of various Types
under trivial or Friedberg splitting.
Remark 3.19. SupposeA ⊔A =AisatrivialorFriedbergsplitting andA isnot
0 1 0
computable. If A is Type 1 or 2,then either A is Type 2 (A ⊔A =A is a trivial
0 0 1
splitting)orType 3(A ⊔A =Ais aFriedbergsplitting). IfAis Type 3,thenA
0 1 0
is Type 3. If A is Type 4, then A is Type 4 (A ⊔A =A is a trivial splitting) or
0 0 1
Type 5. If A is Type 5 (6, 7, or 8), then A is Type 5 (6, 7, or 8) (replace D with
0 0
the unionof D and A ). If A is Type 9 (10), then A is Type 9 (10)(replace each
0 1 0
D with the union of D and A ).
i i 1
We now examine the last four types more carefully. First, we explore the subtle
difference between Types 9 and 10, which is encoded in the last clauses of these
Types’ definitions.
3.4.1. Type 10 sets and r-maximality. Type10setscanariseassplitsofr-maximal
sets.
Lemma 3.20. If A is half of a splitting of an r-maximal set (so not of Type 1)
and A is not Type 2 or 3, then A is Type 10.
Proof. We will show that if A is Type 4, 5, 6, 7, 8, or 9 (and hence not of Type 1,
2, 3, or 10) then A is not half of a splitting of an r-maximal set. Fix some infinite
generating set G for D(A) of Type 4, 5, 6, 7, 8, or 9.
Let B be a c.e. set disjoint from A (such sets exist since A is not Type 1). We
showthatA⊔Bisnotr-maximal. SinceGisageneratingset,Biscontainedinsome
finite union of sets in G. Every c.e. superset of an r-maximal set is either almost
equal to ω or r-maximal itself. Since A does not have Type 2 or 3, A⊔B 6=∗ ω
and we can assume B is the union of these finitely many generators. We proceed
bycases. For G ofType 4, 5or7, anR notpartofthe unionwitnessesthat A⊔B
i
is not r-maximal. For G of Type 6, an infinite computable subset of some D not
i
part of the union demonstrates that A⊔B is not r-maximal. For G of Type 8 or
9, assume that B ⊆∗ R ∪ D . If G has Type 8, there is some l>i such
Sj≤i j Sj≤i j
that D ∩ R is infinite. An infinite computable subset of this intersection
l Sj≤i j
demonstrates that A⊔B is not r-maximal. Finally, suppose G is Type 9. By the
lastclauseofType9,thereisanr >isuchthatR −D isinfinite. Thecomputable
r i
set R witnesses that A⊔B is not r-maximal.
r
(cid:3)
Note that we cannot eliminate the assumption that A is not Type 2 or 3 in
Lemma3.20. IfA⊔Risatrivialsplittingofanr-maximalset,thenA⊔Rissimple.
By Corollary 3.13, D(A) = {R} and A is Type 2. Similarly, by Corollary 3.16, if
A⊔B is a Friedberg splitting of an r-maximal set, A is Type 3.
Question 3.21. If A is D-maximal and Type 10, then is A half of a splitting of
an r-maximal set?
We can, however, prove a stronger version of this statement with an additional
assumption.