Jump degrees of torsion free abelian groups

33 Pages·2011·0.45 MB·English

JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS
BROOKEM.ANDERSEN,ASHERM.KACH,ALEXANDERG.MELNIKOV,
ANDREEDSOLOMON
Abstract. Weshow,foreachcomputableordinalαanddegreea>0(α),the
existenceofatorsion-freeabeliangroupwithproperαth jumpdegreea.
1. Introduction
An important topic of effective algebra is concerned with determining the com-
putational strength of an algebraic structure. For an isomorphism type , it is
A
natural to define the computational strength of (the degree of ) to be the least
A A
degree (if it exists) in its degree spectrum
DegSpec( )= d: is d-computable .
A { A }
We recall an infinite algebraic structure is d-computable if its universe can be
A
identified with the natural numbers ω in such a way that the atomic diagram of
A
becomes d-computable.
Often, the degree spectrum fails to capture the computational strength of as
A
in many cases it has no least element (for example, see [18]). As a result, there has
been a line of study into the jump degrees of isomorphism types.
Definition 1.1. If is any countable structure, α is any computable ordinal,
A
and a 0(α) is any degree, then has αth jump degree a if the set
≥ A
d(α) :d DegSpec( )
{ ∈ A }
has a as its least element. In this case, the structure is said to have αth jump
A
degree.
A structure has proper αth jump degree a if has αth jump degree a but
A A
not βth jump degree for any β < α. In this case, the structure is said to have
A
proper αth jump degree.
For any computable ordinal α, it is well-known that an arbitrary structure may
nothaveαth jumpdegree(forexample, see[6]). Theexistenceornonexistenceofa
structure with proper αth jump degree a for a 0(α) depends heavily on the class
≥
of algebraic structures considered. Within the context of linear orders, if an order
typehasadegree,itmustbe0;ifanordertypehasfirstjumpdegree,itmustbe0;
(cid:48)
and yet for each computable ordinal α 2 and degree a 0(α), there is a linear
≥ ≥
order having proper αth jump degree a (see [6], culminating results in [1], [13], [15]
and [18]). Within the context of Boolean algebras, if a Boolean algebra has nth
Date:June30,2011.
Key words and phrases. torsion-freeabeliangroup,jumpdegree.
ThesecondauthorwaspartiallysupportedbytheMarsdenFundofNewZealandviaaPost-
DoctoralFellowship.
1
2 ANDERSEN,KACH,MELNIKOV,ANDSOLOMON
jump degree (for any n ω), it must be 0(n); yet for each a 0(ω), there is a
∈ ≥
Boolean algebra with proper ωth jump degree a (see [14]).
The subject of this paper is the existence of jump degrees of torsion-free abelian
groups. For α 0,1,2 , it is known that every possible proper αth jump degree
∈ { }
is realized.
Theorem 1.2 (Downey [5]; Downey and Jockusch [5]). For every degree a 0,
≥
there is a (rank one) torsion-free abelian group having degree a.
For every degree b 0, there is a (rank one) torsion-free abelian group having
(cid:48)
≥
proper first jump degree b.
Indeed, every finite rank torsion-free abelian group (see Definition 2.2) has first
jump degree as a consequence of a difficult computability theoretic theorem of
Coles, Downey, and Slaman [3]. (See Melnikov [17] for a complete discussion.)
In contrast, not every infinite rank torsion-free abelian group has first jump
degreeasaconsequenceofthefollowingtheoremofMelnikov. Recallthatanonzero
degree a is low if a =0 and nonlow otherwise.
(cid:48) (cid:48)
Theorem 1.3 (Melnikov [17]). There is a torsion-free abelian group such that
G
DegSpec( )= a:a is nonlow . Consequently, thereisatorsion-freeabeliangroup
G { }
with proper second jump degree 0 .
(cid:48)(cid:48)
As a consequence of work by Ash, Jockusch, and Knight (see [1]), two obser-
vations of Melnikov (see Theorem 3 and Proposition 10 of [17]) have implications
about proper second jump degrees and proper third jump degrees.
Theorem1.4. Foreverydegreea>0 ,thereisatorsion-freeabeliangrouphaving
(cid:48)(cid:48)
proper second jump degree a. For every degree b > 0 , there is a torsion-free
(cid:48)(cid:48)(cid:48)
abelian group having proper third jump degree b.
The main purpose of this paper is to generalize Theorem 1.4 to an arbitrary
computable ordinal α.
Theorem 1.5. For every computable ordinal α and degree a > 0(α), there is a
torsion-free abelian group having proper αth jump degree a.
Unfortunately,ourtechniquesdonotallowustoproduceexampleswitha=0(α).
Question 1.6. For each computable ordinal α 3, is there a torsion-free abelian
≥
group having proper αth jump degree 0(α)?
Fixing α, we prove Theorem 1.5 by coding sets S ω into groups in such
S
⊆ G
a way that is X-computable if and only if S Σ0(X). The coding method
GS ∈ α
is based on techniques in Fuchs (see Section XIII, Chapter 88 and Chapter 89,
of [11]) and Hjorth (see [12]). In particular, given torsion-free abelian groups
A
and of a certain type and elements a A and b B, Fuchs adds elements
B ∈ ∈
of the form p n(a + b) for n ω to to build an indecomposable group
−
∈ A ⊕ B
containing . Hjorth uses this idea to encode certain labeled graphs into
A ⊕ B
countable torsion-free abelian groups. Hjorth’s ideas were then used by Downey
and Montalb´an to show the isomorphism problem for torsion-free abelian groups is
analyticcomplete(see[7])andfurtherstudiedbyFokina,Knight,Maher,Melnikov,
andQuinn(see[8]). Wereferthereadertothesepapersforadditionalbackground,
thoughwedonotassumeknowledgeoftheminourpresentation. Ournotationand
terminology will follow Fuchs (see [10] and [11]) rather than Hjorth.
JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS 3
Section 2 discusses background, notation, and conventions, though we also refer
the reader to [10] and [11] for classical background on torsion-free abelian groups
and to [2] for background on effective algebra. Section 3 describes the encoding of
sets S ω into groups (this encoding depends on α). Theorem 1.5 is demon-
S
⊆ G
strated in Section 4.
2. Background, Notation, and Conventions
In this section, we review basic terminology and results relevant to torsion-
free abelian groups. We also introduce some classical notation and adopt some
conventions that will simplify the exposition.
Definition 2.1. An abelian group =(G:+,0) is torsion-free if
G
( x G)( n ω)[x=0 n=0 = nx=0]
∀ ∈ ∀ ∈ (cid:54) ∧ (cid:54) ⇒ (cid:54)
where nx denotes x+ +x.
(cid:124) ·(cid:123)·(cid:122)· (cid:125)
ntimes
Definition 2.2. If is a torsion-free abelian group, a (finite or infinite) set of
G
nonzero elements g G is linearly independent if α g + +α g =0 has
no solution with α{ i}iZ∈Ifo⊂r each i, i ,...,i I, and α1 i=1 0·fo·r· somkeiik. A basis
i 1 k i
∈ { }⊆ (cid:54)
for is a maximal linearly independent set and the rank of is the cardinality of
G G
a basis.
Thegroups constructedforTheorem1.5willhavecountablyinfiniterank. The
G
key coding mechanism will be the existence or nonexistence of elements divisible
by arbitrarily high powers of a prime.
Definition 2.3. If is a torsion-free abelian group and x G, we write p x
∞
G (cid:2) ∈(cid:3) |
and say p infinitely divides x if ( k ω)( y G) x=pky , i.e., if p divides x
∀ ∈ ∃ ∈
arbitrarily many times. We say p finitely divides x and write p x otherwise.
∞
(cid:54)|
Definition 2.4. A subgroup of an abelian group is pure if for every n Z
G H ∈
and every b G:
∈
If =n b, then =n b.
H| | G | |
In other words, if an integer n divides an element b G within , then n divides b
∈ H
within .
G
We note that if is torsion-free, then x=pky can have at most one solution y
G
for any integer k.
Remark 2.5. Within any presentation of , the set x G : p x of elements
∞
G { ∈ | }
infinitely divisible by p is Πc( ). Indeed, this set is a subgroup of under the
2 G G
group operation (which we use without further mention).
Definition 2.6. A torsion-free abelian group is divisible if it is the additive group
of a Q-vector space (or equivalently, if x=ny has a solution y G for every x G
∈ ∈
and n 1).
≥
If is a torsion-free abelian group, its divisible closure (denoted D( )) is the
G G
divisible torsion-free abelian group of the least rank containing . (This least rank
G
is the rank of .)
G
4 ANDERSEN,KACH,MELNIKOV,ANDSOLOMON
Thus, the countable divisible torsion-free abelian groups are the groups Qn
(for n ω) and Qω, and the divisible closure of Z is Q. Classically, the divisible
∈
closure D( ) exists, is unique, and contains as a subgroup. In terms of effective
G G
algebra, Smith (see [20]) proved that every computable torsion-free abelian group
has a computable divisible closure and that there is a uniform procedure for pass-
ing from to D( ).1 However, in general the divisible closure is not effectively
G G
unique (i.e., unique up to computable isomorphism) and the canonical image of
G
inD( )iscomputablyenumerablebutnotnecessarilycomputable(see[9]and[21]
G
foracompletediscussionoftheseissues). Therefore,whenweconsideraparticular
copy ofatorsion-freeabeliangroup,weuseD( )todenotethecanonicaldivisible
G G
closure as in [20]. Thus, we have a uniform way to pass from any given copy of
G
to a copy of D( ).
G
Inourconstruction,wewilluseamorelimitednotionofclosureunderdivisibility
by certain primes.
Definition 2.7. If p ω is prime and is a torsion-free abelian group, define the
∈ G
p-closure of (denoted [ ] ) to be the smallest subgroup of D( ) containing
p
G G H G G
having the property ( g G)[p g].
∞
∀ ∈ |
More generally, if P is a set of prime numbers and is a torsion-free abelian
G
group, define the P-closure of (denoted [ ] ) to be the smallest subgroup of
P
G G H
D( ) containing having the property ( g G)( p P)[p g]. We often write
∞
G G ∀ ∈ ∀ ∈ |
[ ] for [ ] with P = p ,p and [ ] for [ ] .
G p0,p1 G P { 0 1} G P,q G P∪{q}
If is any torsion-free abelian group and P is any set of prime numbers, we say
G
that is P-closed if =[ ] .
G G ∼ G P
Thefollowinglemmasaysthatif isP-closed,thentheresultofclosing under
G G
additionalprimeswillstillbeP-closed. Inparticular,wecanviewtheprimeclosure
[ ] astheresultofclosing undereachoftheindividualprimesinP inanyorder.
P
G G
Lemma 2.8. If is a torsion free abelian group, P is a set of primes and q is
G
prime not in P, then [[ ] ] =[ ] .
G P q ∼ G P,q
Proof. Since [[ ] ] is clearly a subgroup of [ ] and since element of [[ ] ] is
P q P,q P q
G G G
infinitely divisible by q, it suffices to show that each element of [[ ] ] is infinitely
P q
G
divisible by each prime p P. Fix p P and g [[ ] ] . We need to find
P q
∈ ∈ ∈ G
h [[ ] ] such that ph = g. By the definition of [[ ] ] , there is a k 0 such
P q P q
∈ G G ≥
that qkg =(cid:98)h [ ]P. Let g(cid:98) [ ]P be such that pg(cid:98)=(cid:98)h and let h [[ ]P]q be such
∈ G ∈ G ∈ G
that qkh=g. Then
(cid:98)
qk(ph)=p(qkh)=pg(cid:98)=(cid:98)h=qkg
Since is torsion free, qk(ph)=qkg implies ph=g as required. (cid:3)
G
By an obvious variation of the construction in [20], there is an effective way to
pass from to a copy of [ ] which is uniform in both and P. As above, the
P
G G G
closure operation sending to [ ] is not necessarily effectively unique so we fix
P
G G
this uniform procedure to define a particular copy of [ ] given a particular copy
P
G
of .
G
Convention 2.9. We will write statements such as ([Z] [Z] ) [Z] = .
Such statements are intended to apply within a fixed (onρe1,-Pd\imenPsio∩nal)ρc2,oρp3y,Pof Q∅,
1OneformsD(G)frompairs(cid:104)g,n(cid:105)withg∈G andn≥1modulothecomputableequivalence
relation(cid:104)g,n(cid:105)∼(cid:104)h,m(cid:105)ifandonlyifmg=nh.
JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS 5
whereZ(cid:53)Qisfixedaswell. Inparticular,theindicatedprimeclosuresofZshould
all be seen as being taken within a fixed copy of Q.
Often,wewillwriteelementsoftheform x+y as x+y eventhough x and y may
p p p p p
notexistwithinthegroup. Wejustifythisbypassingtothedivisibleclosureofthe
groupandconsideringthecanonicalimageofthegroupwithinitsdivisibleclosure.
Thus, x+y = x + y in D( ) even though x and y may not be in the image of .
p p p G p p G
However,whenwespeakofanelementz beinginfinitelydivisiblebyaprimep,
∈G
we mean its image in the divisible closure is infinitely divisible by the prime p and
this divisibility is witnessed by elements that are the images of elements of .
G
Definition 2.10. A rooted torsion-free abelian group is a torsion-free abelian
G
group with a distinguished element (termed the root of ).
G
We use rooted torsion-free abelian groups to help build our groups inductively.
When we consider isomorphisms, we always consider group isomorphisms with no
assumption that roots are preserved. That is, the root is only used as a tool in the
inductive definitions and is not a formal part of the algebraic structure.
Definition 2.11. Let be a torsion-free abelian group and d D( ) be a
i i I
subset of its divisible clGosure. We define the extension of by{ d}∈ ⊆, denGoted
i i I
G { }∈
; d :i I ,
i
(cid:104)G ∈ (cid:105)
to be the smallest subgroup of D( ) containing and d for i I.
i
G G ∈
Notethatif iscomputableand d isacomputablesetofelementsofD( )
i i I
G { }∈ G
(indeed, computably enumerable suffices), then the subgroup ;d :i I is com-
i
(cid:104)G ∈ (cid:105)
putably enumerable in D( ). Since there is a uniform procedure to produce a
G
computable copy of any computably enumerable subgroup of D( ) (by letting n
G
denote the n-th element enumerated into the subgroup and defining the group op-
erationsaccordingly)wehaveauniformproceduretopassfrom to ;d :i I .
i
G (cid:104)G ∈ (cid:105)
We continue by introducing some (important) conventions that will be used
throughout the paper without further mention.
Convention 2.12. If β is any nonzero ordinal, when we write β = δ+i or β =
δ+2(cid:96)+i for some i ω, we require δ to be either zero or a limit ordinal (allowing
∈
zero only if β <ω) and (cid:96) to be a nonnegative integer.
If i is even, we say the ordinal β is even; if i is odd, we say the ordinal β is odd.
When at limit ordinals, it will be necessary to approximate the ordinal effec-
tively from below. We therefore fix a computable ordinal λ and increasing cofinal
sequences for ordinals less than λ.
Definition 2.13. Fix a computable ordinal λ.
Fix a computable function f : λ ω λ such that f(α+1,n) = α for all
× →
successor ordinals α+1 λ and n ω, and such that f(α,n) is a sequence
n ω
∈ ∈ { } ∈
of increasing odd ordinals (greater than one) with α = f(α,n) for all limit
n ω
∪ ∈
ordinals α λ.
∈
We denote f(α,n) by f (n).
α
3. The Group (For Successor Ordinals α)
S
G
Fixing a computable successor ordinal α below λ, the group will be a direct
S
G
sumofrootedtorsion-freeabeliangroups (n)codingwhethernisorisnotinS. It
S
G
6 ANDERSEN,KACH,MELNIKOV,ANDSOLOMON
willbeusefultohaveaplethoraofdisjointsetsofprimes. Wethereforepartitionthe
prime numbers into uniformly computable sets P = p , Q = q ,
β β α+1 β β α+1
{ } ∈ { } ∈
U = u , V = v , D = d , and E = e ,.
β,k β α+1,k ω β,k β α+1,k ω n n ω n n ω
Mo{re sp}ec∈ifically∈, the iso{morp}hi∈sm ty∈pe of {(n)}wi∈ll be either (cid:2){ (Σ}0∈)(cid:3) or
(cid:2) (cid:3) (cid:2) (cid:3) (cid:2) (cid:3) GS G α dn
(Π0) ,or (Σ0) or (Π0) (alldescribedlater)dependingonwhetherα
G α dn H α dn H α dn
is even or odd (deciding versus ) and whether n is in S (deciding Σ versus Π).
G H
Thegroup (n)willbeX-computable(uniformlyinn)ifS Σ0(X). Conversely,
GS ∈ α
there will be an effective enumeration Υ of computable infinitary Σc sen-
tences such that = Υ if and only{ ifn}nn∈ω S.2 Thus, the group wαill be
S n S
G | ∈ G
X-computable if and only if S Σ0(X).
∈ α
The definition of the rooted torsion-free abelian groups (Σ0), (Π0), (Σ0),
G α G α H α
and (Π0)isdonebyrecursion. Unfortunately,therecursionisnotstraightforward
H α
for technical reasons within the algebra (discussed in Remark 3.2). Indeed, we
introduce additional rooted torsion-free abelian groups (Σ0(m)) for m ω if α is
G α ∈
an even ordinal.
We define some of these groups pictorially in Section 3.1. The hope is these
examplesprovideenough intuitiontothereadersothattheformaldefinitionof
S
G
(and all the auxiliary groups) is not (too) painful.
3.1. Defining (Σ0), (Σ0(m)), and (Π0) Pictorially. For each successor or-
G β G β G β
dinal β 3, we give a pictorial description of the groups (Σ0), (Σ0(m)), and
≥ G β G β
(Π0). The recursion starts with (Σ0(m)) as Z with root r = pm and (Π0)
G β G 2 1 G 2
as [Z] with root r = 1. The recursion continues as illustrated in Figure 1 and
p1
Figure 2.
Within these figures, the recursively defined rooted torsion-free abelian groups
aredenotedwithtriangles(thetextinsidespecifieswhichrecursivelydefinedgroup),
with the root denoted by a circle at the top. A line segment connecting two roots
andwithalabelpdenotesthesumoftherootsisinfinitelydivisiblebyp. Brackets
aroundarecursivelydefinedrootedgroupwithalabelpdenotesthep-closureofthe
recursively defined rooted group. A prime p next to a root r denotes r is infinitely
divisible by p.
3.2. Defining Formally. Havingpictoriallydescribedthegroups (Σ0), (Σ0(m)),
GS G β G β
and (Π0), we formalize the definition of . Of course, doing so requires formal-
G β GS
izing the definition of all the auxiliary groups.
Definition 3.1. For each ordinal β with 1 < β α, define rooted torsion-free
≤
abelian groups (Σ0) and (Π0) (if β is odd) or (Σ0(m)) for m ω and (Π0)
G β G β G β ∈ G β
(if β is even) by recursion as follows.
For β =2, define (Σ0(m)) to be the group Z with root r =pm and define
• G β 1
(Π0) to be the group [Z] with root r =1.
G β p1
For odd β =δ+2(cid:96)+1 3, define (Σ0) to be the group
• ≥ G β
(cid:42) (cid:43)
(cid:77)(cid:77) (cid:77)
[Z]pβ ⊕ G(Σ0β−1(m))⊕ G(Π0β−1); qβ−t(r+rk),qβ−t(r+rk,m):k,m,t∈ω ,
k ωm ω k ω
∈ ∈ ∈
2Wereferthereaderto[2],forexample,fordefinitionsandbackgroundoncomputableinfinitary
formulas.
JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS 7
r
pβ
qβ qβ qβ qβ qβ qβ
r0,0 r1,0 r0,1 r1,1 r0 r1
(Σ0 (0)) (Σ0 (0)) (Σ0 (1)) (Σ0 (1)) (Π0 ) (Π0 )
G β−1 G β−1 G β−1 G β−1 G β−1 G β−1
r
pβ
qβ qβ qβ qβ
r0,0 r1,0 r0,1 r1,1
(Σ0 (0)) (Σ0 (0)) (Σ0 (1)) (Σ0 (1))
G β−1 G β−1 G β−1 G β−1
Figure 1. (Σ0) (Top) and (Π0) (Bottom) if β =2δ+2(cid:96)+1>1
G β G β
r
pβ vβ,0 r1
vβ,1
(Σ0 )
G β−1 uβ,0 G(Σ0β−1) uβ,1 rm vβ,m rm+1 vβ,m+1
(Σ0 )
r G β−1 uβ,m G(Π0β−1) uβ,m+1
pβ vβ,0 r1
vβ,1
(Σ0 )
G β−1 uβ,0 G(Σ0β−1) uβ,1 rm vβ,m rm+1 vβ,m+1
(Σ0 )
G β−1 uβ,m G(Σ0β−1) uβ,m+1
Figure 2. (Σ0(m)) (Top) and (Π0) (Bottom) if β =δ+2(cid:96)+2>2
G β G β
with root r = 1 in [Z] , where r is the root of the kth copy of (Π0 )
and r is the root ofpβkth copy okf (Σ0 (m)). G β−1
Fork,omdd β =δ+2(cid:96)+1 3, definGe β(Π−10) to be the group
≥ G β
(cid:42) (cid:43)
(cid:77)(cid:77)
[Z]pβ ⊕ G(Σ0β−1(m)); qβ−t(r+rk,m):k,m,t∈ω ,
k ωm ω
∈ ∈
withrootr =1in[Z] ,wherer istherootofkthcopyof (Σ0 (m)).
These are illustratepdβ in Figurek,m1. G β−1
8 ANDERSEN,KACH,MELNIKOV,ANDSOLOMON
r
pβ
qβ qβ qβ qβ
r0 r1 r00 r10
(Σ0 ) (Σ0 ) (Π0 ) (Π0 )
G β−1 G β−1 G β−1 G β−1
r
pβ
qβ qβ
r0 r1
(Σ0 ) (Σ0 )
G β−1 G β−1
Figure 3. (Σ0) (Top) and (Π0) (Bottom) if β =δ+2(cid:96)+2>2
H β H β
For even β =δ+2(cid:96)+2>2, define (Σ0(m)) to be the group
• G β
(cid:42) (cid:43)
(cid:77) (cid:2)G(Σ0β−1)(cid:3)uβ,k ⊕ (cid:77)(cid:2)G(Π0β−1)(cid:3)uβ,k; p−βtr0,vβ−,tk(rk+rk+1):k,t∈ω ,
0 k m k>m
≤ ≤
with root r = r , where r is the root of the kth copy of (Σ0 ) or of
(Π0 ) depend0ing on whekther k m or k >m. G β−1
G Foβr−1even β =δ+2(cid:96)+2>2, de≤fine (Π0) to be the group
G β
(cid:42) (cid:43)
(cid:77)(cid:2)G(Σ0β−1)(cid:3)uβ,k; p−βtr0,vβ−,tk(rk+rk+1):k,t∈ω ,
k ω
∈
with root r =r , where r is the root of the kth copy of (Σ0 ).
These are illu0strated inkFigure 2. G β−1
For limit β =δ >0, define the group (Σ0(m)) to be
• G β
(cid:42) (cid:43)
(cid:77) (cid:104) (cid:105) (cid:77)(cid:104) (cid:105)
0 k m G(Σ0fβ(k)) uβ,k ⊕k>m G(Π0fβ(k)) uβ,k; p−βtr0,vβ−,tk(rk+rk+1):k,t∈ω ,
≤ ≤
with root r = r , where r is the root of the kth copy of (Σ0 ) or of
0 k G fβ(k)
(Π0 ) depending on whether k m or k >m.
G fβ(k) ≤
Define (Π0) to be the group
G β
(cid:42) (cid:43)
(cid:77)(cid:104) (cid:105)
k ω G(Σ0fβ(k)) uβ,k; p−βtr0,vβ−,tk(rk+rk+1):k,t∈ω ,
∈
with root r =r , where r is the root of the kth copy of (Σ0 ).3
0 k G fβ(k)
3We emphasize that the definition of G(Σ0(m)) and G(Π0) for β =δ+2(cid:96)+2 is identical to
β β
thecaseofβ=δ asbydefinitionf (k)=δ+2(cid:96)+1forallk. Weseparatethem,hereand
δ+2(cid:96)+2
insomelaterproofs,inhopesofnotobfuscatingtheintuition.
JUMP DEGREES OF TORSION-FREE ABELIAN GROUPS 9
This completes the formal descriptions of these groups.
For odd β 3, the group (Σ0) has the form
≥ G β
(cid:42) (cid:43)
(cid:77)(cid:77) (cid:77)
[Z]pβ ⊕ G(Σ0β−1(m))⊕ G(Π0β−1); qβ−t(r+rk),qβ−t(r+rk,m):k,m,t∈ω .
k ωm ω k ω
∈ ∈ ∈
We refer to the subgroups (indexed by k and m) of the form (Σ0 (m)) with
roots r as the (Σ0 (m)) components of (Σ0). SimilarlyG, weβ−r1efer to the
(Π0 k),msubgroupGs (inβd−e1xed by k) with root rG aβs the (Π0 ) components of
G(Σ0β−).1We use similar language in the case of ekven β anGd limβ−it1β, as well as for
G β
other groups defined inductively within the paper.
We emphasize the components of a group do not detach as direct summands.
For clarity, we always refer to direct components (when the directness is an issue)
without omitting the word “direct”.
When we speak of components, we mean the components which are used in the
inductive definition of these groups (or their prime closures), and we do not care
if there are alternate ways to present the group. More formally, every such group
will be considered as an image of one canonical copy given by the definition, and
a subgroup is a component if and only if it is an image of a component which was
used in the definition of this canonical copy. The isomorphism is chosen once and
forever.
The important relationship between (Σ0) and (Π0) for odd β and (Σ0(m))
G β G β G β
and (Π0) for even β is whether each embeds within the other. For small β, one
G β
canseethatthegroupsdefinedabovesatisfythefollowingembeddabilityrelations:
if β > 1 is odd, then (Σ0) (cid:53) (Π0) and (Π0) (cid:53) (Σ0); if β > 0 is even, then
G β (cid:54) G β G β G β
(Π0) (cid:53) (Σ0(m)) and (Σ0(m)) (cid:53) (Π0) for all m ω. For larger ordinals β,
G β (cid:54) G β G β G β ∈
the formal proof of these properties is less straightforward. Moreover, stronger
properties of such groups are needed to run a successful induction. We avoid these
formal difficulties by not using these embedability relations in later proofs, stating
them only in order to aid intuition. Though they will not be formally used, the
reader may find it useful to keep in mind which groups are “bigger”.
The embeddability relations discussed reflect the utility of the coding. Infor-
mally, we will ask
Is there a large subgroup attached to x?
aboutanelementxthatisinfinitelydivisiblebyanappropriateprime. Theanswer
will allow us to extract whether the Σ0 outcome or the Π0 outcome was the case.
β β
We(informally)justifynotusingasimplerrecursiveschemetodefinethegroups
(Σ0) and (Π0) in the following remark.
G β G β
Remark 3.2. It would of course be simpler if Definition 3.1 used only the odd
recursion schema (for all successor ordinals). Unfortunately, the embeddability
relations would not be satisfied in this case, e.g., when β =4 it would be the case
that (Σ0) (cid:53) (Π0) and (Π0) (cid:53) (Σ0). The reason is (Σ0) would contain
G β G β G β G β G 4
infinitely many copies of (Π0) and infinitely many copies of (Σ0) whereas (Π0)
only would contain infiniGtely3many copies of (Σ0). As (ΠG0) (cid:53)3 (Σ0), it Gwou4ld
G 3 G 3 G 3
follow that (Σ0) (Π0). Hence asking if there is a large subgroup would not
G 4 ≤ G 4
distinguish between the Σ0 and the Π0 outcomes.
β β
10 ANDERSEN,KACH,MELNIKOV,ANDSOLOMON
For even successor ordinals β 4, we will need additional auxiliary groups
≥
(Σ0) and (Π0).
H β H β
Definition 3.3. Foreachevencomputableordinalβ =δ+2(cid:96)+2 4,definerooted
≥
torsion-free abelian groups (Σ0) and (Π0) as follows.
H β H β
Define (Σ0) to be the group
H β
(cid:42) (cid:43)
(cid:77) (cid:77)
[Z]pβ ⊕ G(Σ0β−1)⊕ G(Π0β−1); qβ−t(r+rk),qβ−t(r+rk(cid:48)):k,k(cid:48),t∈ω ,
k∈ω k(cid:48)∈ω
with root r =1 in [Z] , where r is the root of the kth copy of (Σ0 ) and r is
the root of k th copypoβf (Π0 )k. G β−1 k(cid:48)
Define (Π(cid:48) 0) to be thGe grβo−u1p
H β
(cid:42) (cid:43)
(cid:77)
[Z]pβ ⊕ G(Σ0β−1); qβ−t(r+rk):k,t∈ω
k ω
∈
with root r =1 in [Z] , where r is the root of the kth copy of (Σ0 ).
These are illustratepdβin Figurek3. G β−1
It is now possible to define for S ω.
S
G ⊆
Definition 3.4. For each successor ordinal α 3 and set S ω, define a torsion-
≥ ⊆
free abelian group = α as follows.
GS GS
If α=δ+2(cid:96)+1 3, define to be the group
S
• ≥ G
:= (cid:77)(cid:2) (Σ0)(cid:3) (cid:77)(cid:2) (Π0)(cid:3) .
GS G α dn ⊕ G α dn
n S n S
∈ (cid:54)∈
If α=δ+2(cid:96)+2, define to be the group
S
• G
:= (cid:77)(cid:2) (Σ0)(cid:3) (cid:77)(cid:2) (Π0)(cid:3) .
GS H α dn ⊕ H α dn
n S n S
∈ (cid:54)∈
Thefollowingcrucialobservationwillbeexploitedinlatersectionswhenwewish
to express elements as sums of roots of subcomponents.
Remark 3.5. If is any group within this section, or any direct product of prime
G
closures of such groups, then we let R denote the set of (distinct) roots of the
G
recursively nested components of . (Some elements may serve as the root of more
G
than one component at different ordinal levels. For example, if β is odd, then the
root of a (Σ0 (m)) component of (Π0) is also the root of a [ (Σ0 )]
componentG. Hβo−w1ever, this root appearGs onβly once in R .) The impGortaβn−t2pouiβn−t1i,0s
G
that the set R is a basis for both and D( ).
G G G
4. Proof of Theorem 1.5
Havingdefined foreachsuccessorordinalα 3,itofcourseremainstoverify
S
G ≥
the desired properties. We state these explicitly.
Lemma 4.1. For each successor ordinal α 3, there is an effective enumeration
≥
Υ of computable Σc sentences such that =Υ if and only if n S.
{ n}n∈ω α GS | n ∈
Lemma 4.2. For each successor ordinal α 3, if S Σ0(X), then has an
≥ ∈ α GS
X-computable copy.