Reverse Mathematics on Lattice Ordered
Groups
Alexander S. Rogalski, Ph.D.
University of Connecticut, 2007
Several theorems about lattice-ordered groups are analyzed. RCA0 is sufficient to
prove the induced order on a quotient of `-groups and the Riesz Decomposition
Theorem. WKL0 is equivalent to the statement “An abelian group G is torsion
free if and only if it is lattice-orderable.” ACA0 is equivalent to the existence of
various substructures: the join of two convex `-subgroups, the convex closure of
an `-subgroup, the polar subgroup X⊥ of an `-subgroup X, and a sequence of
values {V (g) : g 6= e}. The standard proof of Holland’s Embedding Theorem uses
ACA0. Holland’s Theorem is equivalent to the existence of a sequence of excluding
prime subgroups {P (g) : g 6= e}, and the existence of such a sequence is provable
in WKL0 when G is abelian.
Reverse Mathematics on Lattice Ordered
Groups
Alexander S. Rogalski
B.S., Marlboro College, Marlboro, VT, 2002
M.S., University of Connecticut, Storrs, CT, 2004
A Dissertation
Submitted in Partial Fullfilment of the
Requirements for the Degree of
Doctor of Philosophy
at the
University of Connecticut
2007
Copyright by
Alexander S. Rogalski
2007
APPROVAL PAGE
Doctor of Philosophy Dissertation
Reverse Mathematics on Lattice Ordered
Groups
Presented by
Alexander S. Rogalski, B.S., M.S.
Major Advisor
David Reed Solomon
Associate Advisor
Manuel Lerman
Associate Advisor
Joseph Miller
University of Connecticut
2007
ii
ACKNOWLEDGEMENTS
First, I wish to acknowledge Old Joe, my banjo. I brought Old Joe to the annual
Math Department picnic, where Reed excitedly introduced himself and suggested
that we play music sometime. We managed to accomplish this a few times, despite
the pressing obligations of academia and graduate studies. Reed truly deserves
my thanks for being a superb advisor. Tricia, you are wonderful, and I thank you
for believing in me when I feared it would take another year to finish. I also thank
our daughter, Georgia, for being born and giving me, among other numerous joys,
a considerable motivation to graduate. Last but not least, I wish to thank my
family for their considerable and constant support.
iii
TABLE OF CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Reverse Math . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Lattices, Lattice-ordered groups . . . . . . . . . . . . . . . . . . . . . 7
1.5 Fundamental Examples . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2. Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.1 Definition of `-group . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Definition of an `-group for Reverse Math . . . . . . . . . . . . . . . 19
2.3 Basic computation in `-groups . . . . . . . . . . . . . . . . . . . . . . 20
2.4 The Riesz Decomposition Theorem . . . . . . . . . . . . . . . . . . . 26
3. More Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.1 Identifying groups which are lattice-orderable . . . . . . . . . . . . . 30
3.2 Convex `-subgroups and Right Cosets . . . . . . . . . . . . . . . . . . 34
4. Convexity Results and Reversals for Substructure Existences . 38
4.1 Closure Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Existence of the subgroup generated by A, B. . . . . . . . . . . . . . 44
4.3 Existence of the convex closure of an `-subgroup H . . . . . . . . . . 45
iv
4.4 Existence of the convex `-subgroup generated by convex `-subgroups
A, B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4.2 Verification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.5 Existence of the Polar X⊥. . . . . . . . . . . . . . . . . . . . . . . . . 54
5. Prime Subgroups and Values . . . . . . . . . . . . . . . . . . . . . 62
5.1 Prime Subgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2 Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.3 Existence of a Sequence of Values . . . . . . . . . . . . . . . . . . . . 67
5.4 Existence of a Sequence of Excluding Primes . . . . . . . . . . . . . . 70
6. Holland’s Embedding Theorem . . . . . . . . . . . . . . . . . . . . 78
6.1 Summary of Original Proof of Holland’s Theorem . . . . . . . . . . . 79
6.2 Proof of Holland’s Theorem using excluding primes . . . . . . . . . . 81
6.3 A Reversal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
Bibliography 86
v
LIST OF FIGURES
4.1 The Root System Γ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.1 Initial Subtree of T ∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
vi
Chapter 1
Introduction
1.1 Background
It is a relatively common occurrence for a student of mathematics to read or write
a proof which, at a key step, uses Zorn’s Lemma or an equivalent principle to
prove the existence of a set with certain desired properties. Until one is used to
doing so, it can seem odd to call on a result from set theory in the middle of a
proof which otherwise only requires results in Algebra. One might wonder, “Does
one really need something as strong as Zorn’s Lemma to build this set, or could
it be done directly?”
Reverse Mathematics is a subfield of logic which tries to answer questions
like this by finding exactly which set-theoretic axioms are truly necessary to prove
a theorem. The usual axioms of set theory, ZFC or ZF, are quite strong. We
can make finer distinctions by restricting ourselves to “countable” mathematics
and axiom systems which, though weaker, are still able to prove many classical
theorems of mathematics. More formally, the setting for Reverse Math is the
1
2
language of second order arithmetic Z2. In this language, we have symbols +, ·,
<, 0, and 1, and the usual axioms defining them in the natural numbers N, set
membership ∈, and two types of variables: number variables which are intended
to range over N and set variables which are intended to range over subsets X ⊂ N.
A typical investigation in Reverse Math goes like this:
1. Pick a theorem Thm to study.
2. Look at a textbook proof of Thm and find a set S of axioms for Z2 that are
suitable for the proof of Thm.
3. See if it is possible to prove the axioms of S using Thm. (This is why it’s
called Reverse Math.)
We need some amount of set theory to do such a proof as in step 3, and
we typically work in a weak base system called RCA0, which strikes a balance
between being strong enough to allow basic proofs and weak enough to keep the
set-theoretic impact reasonably minimal. Generally, one of two things happens.
If we are able to prove S from Thm, this tells us that S is the weakest axiom
system capable of proving Thm. (If a strictly weaker system S ′ proved Thm, S ′
would then imply S, an impossibility!) In this case we have a proof of Thm from
S and a reversal of Thm to S, and we say the two are equivalent over RCA0. If,
on the other hand, attempts to prove S from Thm aren’t working out, we look
for a new proof of Thm that uses weaker axioms S and repeat the process, trying
3
to prove S using Thm.
The “full-blown” set existence axiom scheme for Z2 consists of axioms
∃X∀n(n ∈ X ↔ ϕ(n)), where ϕ is any formula in the language Z2 not men-
tioning X. This scheme basically says: if we have a formula in Z2, the set of
numbers which satisfy it exists. This full collection is too strong to be interesting
for Reverse Math, but it contains five particular subsystems (subcollections of
axioms) and most theorems successfully analyzed are equivalent to one of them.
In order of increasing strength, they are: RCA0, WKL0, ACA0, ATR0, and Π11-CA0.
In this dissertation, only the first three come into play.
RCA0 is the usual base system over which we prove equivalences, and its set
comprehension scheme is limited to ∆01 formulas. (We also include axioms allow-
ing Σ01 induction.) The next strongest, WKL0, consists of all the axioms of RCA0
plus the Weak Konig’s Lemma axiom saying “If T is an infinite binary tree, then
T has an infinite path”. WKL0 is sometimes sufficient where a standard proof
uses Zorn’s Lemma, though there are notable cases where it does not suffice. For
example, in the theory of commutative rings, the existence of a nontrivial prime
ideal is equivalent to WKL0. The existence of a prime ideal is usually proved as
a corollary to the existence of a maximal ideal. Since the existence of a maximal
ideal requires ACA0, this is a case where a set-theoretically “simpler” existence
proof for a prime ideal needed to be found. ACA0, or Arithmetical Comprehension,
is the strongest subsystem used in this study, obtained by allowing set comprehen-
4
sion using arithmetical formulas – those which may have any number of number
quantifiers in its definition but no set quantifiers.
The subsystem of axioms to which a given theorem is equivalent is a measure
of the set-theoretic complexity of the theorem. A theorem which requires only
weak set existence axioms is considered less complex than one which requires
a lot of complicated assumptions, i.e., a very strong axiom system. Knowing
which axioms are necessary is particularly of interest to those working on related
problems in Computable Algebra or Combinatorics.
For instance, an effective analogue of a theorem is likely true if the theorem
is provable in RCA0. As an example, RCA0 suffices to prove that the quotient
of a commutative ring by a maximal ideal is a field. Thus, given such a ring
and a maximal ideal in a computable presentation, one can effectively form the
quotient field. In contrast, maximal ideals generally require ACA0 and it is not
the case that, given a computable commutative ring, one can always effectively
find a maximal ideal. Knowing that maximal and prime ideals respectively require
ACA0 and WKL0 also enables one to make distinctions between the computational
“difficulty” of obtaining a maximal or prime ideal in a computable ring. That is,
it is both set-theoretically easier and computationally easier to obtain a prime
ideal than a maximal one.
This particular project in Reverse Math concerns theorems about lattice-
ordered groups. These groups are not necessarily exotic. For example: an abelian
5
group G is lattice-orderable if and only if it is torsion-free. In Chapter 3, we see
that this statement is equivalent to WKL0. Taking a step back, there are three
main types of ordered groups, so called because they have both an algebraic group
structure and a partial order structure which respects the group operations. The
most general kind is a partially-ordered group, or p.o.-group. If the partial order
is a lattice order, G is a lattice-ordered group, or `-group. If the order is a total
(linear) order, G is a totally ordered group, or o-group. We have the following
class containments:
o-groups ( `-groups ( p.o.-groups
In [6], Solomon investigated theorems about p.o.-groups and o-groups. With `-
groups in intermediate position between the other two, an immediate interest was
what connections there were between the theory of lattice-ordered groups and
the theories of the other two types, and whether or not there was a stronger
resemblance to either side. There seem to be relatively few places where the
theories of the three types of ordered groups can be compared directly, but these
few are enough to suggest that `-groups are no more like o-groups than p.o.-groups.
For example, Theorem 3.7 is an instance of `-groups being like o-groups and unlike
p.o.-groups, whereas in Corollary 4.10, the relationship is reversed.
The results presented follow an exploratory trend, beginning with an inves-
tigation of how best to formalize `-groups in Z2, then moving on to the existence
of the various types of subobjects which occur in the classical study of `-groups,
6
and culminating with an analysis of the more substantial Holland’s Embedding
Theorem.
Holland’s Embedding Theorem 1. Any `-group is `-isomorphic to an `-
subgroup of the `-group of order-preserving permutations of a totally ordered set.
The standard proof of Holland’s Theorem makes use of a sequence of values:
convex subgroups which are maximal with respect to excluding a specific element.
As shown in Chapter 5, the existence of such a sequence is equivalent to ACA0.
However, in Chapter 6, we show that Holland’s theorem is equivalent to an alter-
native assumption and, when G is abelian, is provable in the weaker subsystem
WKL0.
1.2 Notation
We will use the symbols ∧,∨ for lattice meet and join, and use the symbol “&”
and the word “or” in formulas for conjunction and disjunction, respectively.
We fix an enumeration Φ0, Φ1, . . . of all partial computable functions.
1.3 Reverse Math
With the Reverse Math scope limited to RCA0, WKL0, and ACA0, we routinely
use the following lemmas when proving equivalences over RCA0.
Lemma 1.1. The following are pairwise equivalent over RCA0 [5].
7
1. WKL0
2. (Σ01 separation) Let φi(n), i = 0, 1 be Σ0
1 formulas in which X does not occur
freely. If ¬∃n(φ0(n) & φ1(n)) then
∃X∀n((φ0(n) → n ∈ X) & (φ1(n) → n 6∈ X)).
3. If f, g : N → N are one-to-one with ∀m∀n(f(m) 6= g(n)) then
∃X∀m(f(m) ∈ X & g(m) 6∈ X).
Lemma 1.2. The following are pairwise equivalent over RCA0 [5].
1. ACA0
2. Σ01 comprehension, i.e., ∃X∀n(n ∈ X ↔ φ(n)) restricted to Σ0
1 formulas
φ(n) in which X does not occur freely.
3. For all one-to-one functions f : N → N there exists a set X ⊆ N such that
∀n(n ∈ X ↔ ∃m(f(m) = n)), i.e., X is the range of f .
1.4 Lattices, Lattice-ordered groups
Definition 1.3. A lattice is a partially ordered set such that every pair of ele-
ments has a least upper bound (join) and greatest lower bound (meet). The meet
and join of elements a, b are denoted a ∧ b and a ∨ b, respectively.
8
Definition 1.4. A partially-ordered group or p.o.-group is a group G that
is also a partial order and satisfies a condition that the group operation preserves
the order on G, that is
a ≤ b → ∀g(ag ≤ bg & ga ≤ gb).
If the partial order on G is a lattice order, then G is a lattice-ordered group or
`-group. If the order on G is a total (linear) order, then G is a totally-ordered
group or o-group.
1.5 Fundamental Examples
Example 1.5. Continuous functions on a topological space
From [1]: let X be a topological space and C(X) the additive group of real-
valued continuous functions. We make C(X) an `-group by providing it with its
usual pointwise order: f ≤ g if and only if f(x) ≤ g(x), for all x ∈ X.
Example 1.6. (Restricted) Hahn groups
From [1]: let Γ be a root system. That is, Γ is a partially ordered set
for which {α : α ≥ γ} is totally ordered, for any γ ∈ Γ. Let {Hγ : γ ∈ Γ}
be a collection of o-groups indexed by Γ. Consider functions v on Γ for which
v(γ) ∈ Hγ, for all γ ∈ Γ. Given such a function v, the subset of Γ where v is
9
not zero is called its support. Let Σ(Γ, Hγ) be the set of all such functions whose
support is finite. This is a group under addition. Furthermore, if we define an
element to be positive if it is positive at each maximal element of its support, then
Σ(Γ, Hγ) is an `-group, called a restricted Hahn group on Γ. (The general Hahn
group on a root system does not require finite support. Rather, the support of
each element must have the ascending chain condition.)
Example 1.7. The group of finite sequences of integers
Let G be the group⊕
ω Z. That is, G is the group of finite sequences of
integers, with the group operation of componentwise addition. (If two strings
differ in length, we append zeros to the shorter one until the lengths match.) We
say an element of G is positive if each component is greater than or equal to
zero. Thus, the meet and join of two elements f, g is computed by taking the
componentwise minimum (maximum, resp.) of the two. G may be viewed as
a restricted Hahn group on the root system Γ consisting solely of a countably
infinite antichain, with each Hγ = Z equipped with its usual order as an o-group.
Of course, G may also be viewed as the additive group of finite-support functions
from N to Z equipped with the pointwise order.
Example 1.8. Permutations of a Linear Order (and conventions)
From [4]: let L be a totally ordered set, and let Aut(L) be the set of order-
preserving bijections from L to L. Then Aut(L) is an `-group under composition,
10
and lattice operations ∨,∧ are defined by the rules:
[f ∨ g] (l) = max{f(l), g(l)}
[f ∧ g] (l) = min{f(l), g(l)}
In terms of the induced ≤ relation, we have f ≤ g ⇐⇒ f(l) ≤ g(l) for all l.
Example 1.9. Torsion-Free abelian group generated by {xi, yi}i∈N.
The formal presentation for this group is from Solomon as in [6], which is
summarized below for reference.
Let G be the free abelian group on the generators {xi, yj}i∈N. Formally,
elements of G are quadruples 〈I, q, J, p〉 where I and J are finite subsets of N and
p and q represent functions
q : I → Z\0 and p : J → Z\0
The element 〈I, q, J, p〉 is denoted∑i∈I
cixi +∑j∈J
djyj. The elements represented
by 〈I, q, J, p〉 and 〈I ′, q′, J ′, p′〉 are equal if and only if the four components are
equal. The sum (∑i∈I
qixi +∑j∈J
pjyj
)+
(∑k∈K
rkxk +∑l∈L
slyl
)
11
is∑m∈M
tmxm +∑n∈N
unyn where M = (I ∪K)\{z ∈ I ∩K | qz + rz = 0} and
tm =
qm if m ∈ I\K
rm if m ∈ K\I
qm + rm if m ∈ I ∩K.
N and un are defined similarly. The identity 1G is represented by 〈∅, ∅, ∅, ∅〉, and
if g is represented by 〈I, q, J, p〉 then g−1 is the sum∑i∈I
−qixi +∑j∈J
−pjyj.
Example 1.10. The group Fin∗Z
Fin∗Z is a formalization of Example 1.7 in RCA0. With FinZ as the for-
malization of the set of all finite integer sequences in RCA0, we let Fin∗Z = {α ∈
FinZ : α(|α| − 1) 6= 0}, i.e., the set of finite integer sequences which do not end
in 0. The group operation +Fin∗Zconsists of componentwise addition with the
removal of any trailing zeros afterwards. If the strings are of different lengths, we
pad the shorter one with zeros on the end until the lengths match, then add. The
identity is the empty string, and the inverse of α ∈ Fin∗Z, denoted α−1 or −α, is
obtained by changing the sign of every nonzero entry occurring in α.
Fin∗Z is an `-group under the pointwise order, and the meet and join can be
calculated directly by (after appending zeros on the right until the lengths match)
taking the minimum and maximum, respectively, in each component.
Chapter 2
Preliminary Results
An important step in working with a class of objects in Z2 is to choose a work-
able definition, especially when the object’s axioms involve quantifiers. We begin
with the definitions of partially and linearly ordered groups, which have fairly
straightforward axioms.
Definition 2.1. (RCA0) A group is a set G ⊆ N along with a constant, 1G (or
0G or e), and an operation, ·G, which obey the usual group axioms.
∀a, b, c ∈ G(a ·G (b ·G c) = (a ·G b) ·G c)
∀a ∈ G(1G ·G a = a ·G 1G = a)
∀a ∈ G∃a−1 ∈ G(a ·G a−1 = a−1 ·G a = 1G)
Definition 2.2. (RCA0) A partial order is a set X with a binary relation ≤X
12
13
satisfying the following axioms.
∀x ∈ X(x ≤X x)
∀x, y ∈ X(x ≤X y & y ≤X x → x = y)
∀x, y, z ∈ X(x ≤X y & y ≤X z → x ≤X z)
Definition 2.3. (RCA0) A partially ordered group is a pair (G,≤G) where
G is a group, ≤G is a partial order on G and, the group operation preserves the
partial order on G. That is, for any a, b, c ∈ G, if a ≤G b then
(a ·G c ≤G b ·G c) & (c ·G a ≤G c ·G b).
If the order on G is a linear order, then G is an ordered group or o-group.
2.1 Definition of `-group
Classically, a lattice-ordered group or `-group is defined as a partially ordered
group (G,≤) where each pair of elements has unique greatest lower bound (meet)
and least upper bound (join), i.e., the partial order is a lattice order.
Syntactically, the meet of a and b can be defined by a Π01 formula:
x = (a ∧ b) ↔ ∀z([z ≤ a & z ≤ b] → z ≤ x).
So can the join:
x = (a ∨ b) ↔ ∀z([z ≥ a & z ≥ b] → z ≥ x).
14
To say that the meet and join exist for every pair a, b is then a Π03 statement:
∀a, b∃x∀z([z ≤ a & z ≤ b] → z ≤ x)
∀a, b∃y∀z[z ≥ a & z ≥ b] → z ≥ y).
The obvious concern, then, when deciding on a definition for `-groups in
RCA0, is whether or not these usual axioms for meet and join will be useful in
actually calculating meet and join. Will knowing that every pair of elements has
a meet, say, enable us to calculate it by a ∆01 function? Or is it necessary to
explicitly require functions for meet and join as part of the definition in RCA0?
Considering (N, REC), the standard model of RCA0, the following theorem shows
that it is not possible in general to define meet and join by ∆01 functions.
Theorem 2.4. There is a computable p.o.-group (G,≤) which is a lattice-ordered
group, but for which the meet ∧ and join ∨ operations are not computable. Fur-
thermore, they are as complicated as the halting problem.
Proof. Imitating a construction by Downey and Kurtz in [2], we construct our
computable group G = ∪sGs in stages. Each Gs will be a finite set of integers
with an associated representation function σs. For all elements n ∈ Gs, σs(n) will
be an element of Fin∗Z, a finite string of integers with no trailing zeros. We will
define n ◦s m = k if σs(n) +FinZ σs(m) = σs(k). The operation ◦ will be ∪s∈ω◦s,
that is, once n ◦s m = k is defined, we will have n ◦t m = k for all t ≥ s. Similarly,
we define the partial order ≤s, whose union will be the partial order on G. The
15
partial order will satisfy the property n ≤G m ⇐⇒ σs(n) ≤s σs(m). That is,
once σs(n), σs(m) are defined, the partial order is defined and will be preserved
throughout the construction.
The construction of G will be done accomplishing the following:
• For some m(s) ∈ ω we will assign to all p ≤ m(s) a string σs(p).
• If t > s, then m(t) ≥ m(s).
• If p, q, r ≤ m(s) and σs(p) +FinZ σs(q) = σs(r), then for all t > s, we ensure
that σt(p) +FinZ σt(q) = σt(r).
• If p, q ≤ m(s) and σs(p) ≤ σs(q), then for all t > s, we ensure that σt(p) ≤
σt(q). We do the same for 6≤.
• For all p, lims∈ω σs(p) exists and is a finite string.
Additionally, G will have a computable sequence of pairs 〈ae, be〉, such that
ae ∧ be 6= 0G ⇐⇒ Φe(e) ↓. We will meet the following requirements:
• Group Closure
Ce: If e = 〈p, q〉, then p ◦ q is defined.
• Group Inverses
Ip: There exists a q such that p ◦ q = 0 (0 denotes the zero of ω and the
identity of G.)
16
• Coding Requirements
Ne: ae ∧ be can compute whether Φe(e) halts.
We order the requirements as follows: N0 ≥ C0 ≥ I0 ≥ N1 ≥ · · ·
Definition 2.5. We say
1. Ce requires attention at stage s + 1 if e = 〈p, q〉 for some p, q ≤ m(s) and
there is no r ≤ m(s) such that σs(p) +FinZ σs(q) = σs(r).
2. Ip requires attention at stage s + 1 if p ≤ m(s) and there is no q ≤ m(s)
such that σs(p) +FinZ σs(q) = 0.
3. Ne requires attention at stage s + 1 if Ne is unrealized and either
(a) for no t ≤ s are there a p, q such that σs(p) = 03e1, σs(q) = 03e01 (in
which case Ne is inactive); or
(b) there exists p, q as above and Φe,s(e) ↓. (we say Ne is active)
These p, q are the pair 〈ae, be〉.
The requirements Ne will achieve their goal as follows. When Ne is active
and unrealized, we have introduced elements ae, be whose meet corresponds to the
identity 0 ∈ Fin∗Z. Once Φe(e) halts, we change the mapping on ae and be in a
way that’s compatible with all the p.o.-group requirements so that ae∧be becomes
a new nonidentity element.
17
Construction.
Stage 0. Set p(0) = 0 and σ0(0) = 0 ∈ Fin∗Z. The zero of N is to be the
identity of 〈G, ◦〉.
Stage s + 1. Let R be the requirement of highest priority that requires
attention.
Case 1: R is Ce with e = 〈p, q〉. Define m(s+1) = m(s)+1. Define σs+1(r)
for r ≤ m(s + 1) via:
σs+1(r) =
σs(r) if r < m(s + 1),
σs(p) +FinZ σs(q) if r = m(s + 1).
Case 2: R is Ip. Let m(s + 1) = m(s) + 1. Define σs(r) for r ≤ m(s + 1)
via:
σs+1(r) =
σs(r) if r < m(s + 1),
−σs(p) if r = m(s + 1).
Case 3: R is Ne and Ne is inactive. Let m(s + 1) = m(s) + 2 and define
σs+1 for r ≤ m(s + 1) via:
σs+1(r) =
σs(r) if r < m(s),
03e1 if r = m(s + 1),
03e01 if r = m(s + 2).
At this point, Ne becomes active.
18
Case 4: R is Ne and Ne is active. Thus, there are p, q ≤ m(s) such that
σs(p) = 03e1, σs(q) = 03e01 and Φe,s(e) ↓. Let m(s + 1) = m(s) + 1, and define
σs+1 for r < m(s + 1) via:
σs+1(r)(x) =
σs(r)(x) if x 6= 3e + 2
σs(r)(3e) + σs(r)(3e + 1) if x = 3e + 2
and define σs+1 for r = m(s + 1) by
σs+1(r) = 03e001.
Ne is now realized.
This has the effect of re-mapping the generators 03e1 → 03e101 and 03e01 →
03e011, and defining an element mapped to the new meet, 03e001. This of course
means that the lengths of strings may change at some stage, but each string can
only get at most two digits longer.
For example, suppose we have σs(r) = 03e23, where Φe,s ↑. If there is a
t > s such that Φe,t ↓, then for some stage t′ ≥ t we will have σt′(r) = 03e235, as
a linear combination of the redefined generators.
This will allow us to compute 0′ from a meet function, since ae ∧ be 6=
0G ⇐⇒ Φe(e) ↓.
Verification.
The limit of σs exists, because the map will change at most dk3e times for a
string of length k. The partial order and group operations are preserved by the
19
redefinition in Case 4, because it is a linear mapping. The injectivity of lims σs
follows by construction and is preserved by the redefinition in Case 4.
Thus we have an injection of G into the finite sequences of integers as a p.o.
group. G is a computable p.o-group by the construction. Nonuniformly, we see
that G is isomorphic to the subgroup of Fin∗Z generated by the strings
{03e1, 03e01 : e 6∈ HALT} ∪ {03e1, 03e01, 03e001 : e ∈ HALT}.
This is also an `-subgroup of Fin∗Z, so the order ≤ on G ends up being a lattice
order in the classical sense. Because of the way that G is constructed, no steps
need to be taken to ensure that meets and joins exist, although requirements to
that effect could be easily added.
2.2 Definition of an `-group for Reverse Math
Because of the connection between RCA0 and computable mathematics, the above
result means that in order to work with meet and join in `-groups in RCA0, we
need to explicitly include them in the formal definition. Thus, we consider an
`-group to be given by a group G together with a lattice order ≤ and functions ∧
and ∨ giving the meet and join operations.
Definition 2.6. An `-group is a tuple (G, e, ·,∧,∨) such that under the partial
order defined by x ≤ y ⇔ x∨y = y, (G, e, ·,≤) satisfies the axioms of a p.o.-group
20
and the following identities:
L.1. x ∨ x = x x ∧ x = x
L.2. x ∨ y = y ∨ x x ∧ y = y ∧ x
L.3. (x ∨ y) ∨ z = x ∨ (y ∨ z) (x ∧ y) ∧ z = x ∧ (y ∧ z)
L.4. (x ∨ y) ∧ x = x (x ∧ y) ∨ x = x
We sometimes take the partial (lattice) order ≤ as included, though it is
computed directly from ∨. It is easy to give direct proofs that the lattice ax-
ioms imply the poset axioms of reflexivity, antisymmetry, and transitivity via the
equivalence x ≤L y ⇔ x ∨L y = y using the lattice axioms L.1, L.2, and L.3,
respectively. Verifying that the join is the least upper bound uses L.3. Verifying
the meet requires using L.4. to redefine the above equivalence using ∧, then uses
L.3 as in the join. Thus, it is possible to prove in RCA0 that if (G,∨,∧) satisfies
L.1 through L.4, then it is a lattice under the induced partial order.
2.3 Basic computation in `-groups
The following theorems have proofs that can be easily formalized in RCA0. (See
pages 15–18 of [4].)
Theorem 2.7. (RCA0) [Multiplication Across Meet and Join] Let (G, ·, e,∧,∨)
be such that (G, ·, e) is a group, (G,∧,∨) is a lattice, and in G the identities
u(x ∨ y)v = uxv ∨ uyv , u(x ∧ y)v = uxv ∧ uyv
21
hold. Then G is a lattice-ordered group under the partial order of the lattice
(G,∧,∨). If G is a lattice-ordered group, then the above identities hold.
Proof. See proof of Theorem 2 on pages 14–15 of [4].
Definition 2.8. An element g of an `-group is said to be positive if e ≤ g, or
equivalently, g ∧ e = e. Negative elements are defined similarly.
Theorem 2.9. (RCA0) In any `-group G the inequality a ∧ bc ≤ (a ∧ b)(a ∧ c) is
satisfied for any positive elements a, b, c. Furthermore, if a ∧ b = e then a ∧ bc =
a ∧ c.
Proof. From page 15 of [4]. If a, b, c ≥ e then a ≤ a2, a ≤ ac, a ≤ ba and
(a∧ b)(a∧ c) = a2 ∧ ba∧ ac∧ bc ≥ a∧ bc. The second statement then follows from
the first.
Corollary 2.10. (RCA0) For all m,n ∈ N, a ∧ b = e → an ∧ bm = e
Proof. We use the special cases where c = a or c = b to build up by induction.
Theorem 2.11. (RCA0) In any `-group, the following hold:
(x ∨ y)−1 = x−1 ∧ y−1 , (x ∧ y)−1 = x−1 ∨ y−1
x(x ∧ y)−1y = x ∨ y , x(x ∨ y)−1y = x ∧ y
Proof. See proof of Proposition 1 on page 16 of [4].
Theorem 2.12. (RCA0) In an `-group G, the following holds: ∀n ≥ 1(xn ≥ e →
x ≥ e).
22
To prove this universal statement in RCA0 requires several lemmas formaliz-
ing the standard proof. We define, by primitive recursion, a function f(x, n) = xn,
which we will use transparently by writing exponents, and two functions m, and
m+ with the following inductive definitions.
m(x, 1) = x ∧ e, m(x, n + 1) = xn+1 ∧m(x, n) for n ≥ 1
m+(x, 1) = x, m+(x, n + 1) = xn+1 ∧m+(x, n) for n ≥ 1
Intuitively, m(x, n) = xn ∧ xn−1 ∧ · · · ∧ x ∧ e, and m+(x, n) = xn ∧ xn−1 ∧ · · · ∧ x.
Lemma 2.13. (RCA0) ∀x∀n ≥ 1(m(x, n) = m+(x, n) ∧ e)
Proof. The proof is by induction on n. When n = 1, m(x, 1) = m+(x, 1) ∧ e by
definition. Suppose the lemma holds for n.
m(x, n + 1) = xn+1 ∧m(x, n)
= xn+1 ∧m+(x, n) ∧ e (by induction hypothesis)
= m+(x, n + 1) ∧ e (by definition of m+)
Lemma 2.14. (RCA0) ∀x∀n ≥ 1(x ·m(x, n) = m+(x, n + 1))
23
Proof. Again, proof by induction on n. When n = 1,
x ·m(x, 1) = x · (x ∧ e)
= x2 ∧ x
= x2 ∧m+(x, 1)
= m+(x, 2).
Supposing the lemma for n,
x ·m(x, n + 1) = x(xn+1 ∧m(x, n))
= xn+2 ∧ x ·m(x, n)
= xn+2 ∧m+(x, n + 1) (by induction hypothesis)
= m+(x, n + 2).
Lemma 2.15. (RCA0) ∀x∀n ≥ 1((x ∧ e)n = m(x, n))
Proof. Again, a proof by induction on n. The case n = 1 is trivial.
24
Suppose we have the lemma for n.
(x ∧ e)n+1 = (x ∧ e)(x ∧ e)n
= (x ∧ e) ·m(x, n) (by induction hypothesis)
= (x ·m(x, n)) ∧ (e ·m(x, n))
= (m+(x, n + 1) ∧m(x, n) (by Lemma 2.14)
= xn+1 ∧m+(x, n) ∧m(x, n)
= xn+1 ∧m+(x, n) ∧m+(x, n) ∧ e (by Lemma 2.13)
= xn+1 ∧m+(x, n) ∧ e
= xn+1 ∧m(x, n) (by Lemma 2.13)
= m(x, n + 1)
Now we return to proof of the theorem: ∀x∀n ≥ 1(xn ≥ e → x ≥ e)
Proof. The proof is now direct, and the case n = 1 is trivial. Otherwise, suppose
xn+1 ≥ e.
25
(x ∧ e)n+1 = m(x, n + 1) (by Lemma 2.15)
= xn+1 ∧m(x, n)
= xn+1 ∧m+(x, n) ∧ e
= (xn+1 ∧ e) ∧m+(x, n)
= m+(x, n) ∧ e (By hypothesis, xn+1 ≥ e, so xn+1 ∧ e = e.)
= m(x, n) = (x ∧ e)n
Thus, if we suppose (x ∧ e)n+1 ≥ e, then the above calculation shows that
(x ∧ e)n+1 = (x ∧ e)n. Multiplying both sides by (x ∧ e)−n, we obtain x ∧ e = e,
so x ≥ e.
Definition 2.16. For x ∈ G, we define x+ = (x ∨ e), x− = (x ∧ e), and |x| =
(x ∨ x−1).
Theorem 2.17. (RCA0) In an `-group G, x = x+x−, x+∧(x−)−1 = e, x+(x−)−1 =
(x−)−1x+.
Proof. See proof of Proposition 5 on page 17 of [4].
Theorem 2.18. (RCA0) In an `-group G, the following hold:
(xy)+ ≤ x+y+ , (xy)+ = x(x ∧ y−1)−1
∀n((xn)+ = (x+)n) , ∀n((xn)− = (x−)n)
26
Proof. See proof of Proposition 6 on page 18 of [4].
Theorem 2.19. (RCA0) In an `-group G, the following hold:
|x| = x+(x−)−1
∀n(|xn| = |x|n)
Proof. See proof of Proposition 7 on page 18 of [4].
Note that since x+ is positive and x− is negative, Theorem 2.19 implies that
absolute values are positive.
Theorem 2.20. (RCA0) In an `-group, |xy| ≤ |x||y||x|, and |x ∨ y| ≤ |x||y|.
Proof. From page 18 of [4]. Both are brief. First, |x|−1|y|−1|x|−1 ≤ |x|−1|y|−1 ≤
xy ≤ |x||y| ≤ |x||y||x|. This shows that xy and (xy)−1 are both bounded above
by |x||y||x|. Then |xy| = (xy) ∨ (xy)−1 ≤ |x||y||x|. Second, |x ∨ y| = (x ∨ y) ∨
(x ∨ y)−1 = (x ∨ y) ∨ (x−1 ∧ y−1) ≤ x ∨ y ∨ x−1 ∨ y−1 = |x| ∨ |y| ≤ |x||y|. (To see
this last inequality, just consider the fact that |x| ≤ |x||y| and |y| ≤ |x||y|.)
2.4 The Riesz Decomposition Theorem
Riesz Decomposition Theorem 1. Suppose h1, . . . , hn are positive elements of
the `-group G. If e ≤ g ≤ h1h2 · · ·hn, then g = g1g2 · · · gn, where e ≤ gi ≤ hi.
Proof. A proof of this Theorem (see page 3 of [1]) goes by induction on n. The
base case n = 1 is trivial. For the induction step, suppose e ≤ g ≤ h1h2 · · ·hn.
27
Let g1 = g ∧ h1 and note that e ≤ g1 ≤ h1. Then
k = g−11 g = (g−1 ∨ h−1
1 )g = e ∨ (h−11 g) ≤ h2 · · ·hn,
Now, by induction k = g2 · · · gn with e ≤ gi ≤ hi. But g = g1k, and e ≤ g1 ≤
h1.
Theorem 2.21. The Riesz Decomposition Theorem is provable in RCA0
Proof. We formalize the proof given above in RCA0. Suppose we are given an
`-group G. Let R = {〈n, g, x〉 : x is the code for an n-tuple 〈h1, . . . , hn〉 s.t. each
hi ≥ e and g ≤ h1 · h2 · · ·hn}. Let Q = {〈n, g, x, y〉 : 〈n, g, x〉 ∈ R & y codes an
n-tuple 〈g1, . . . , gn〉 s.t. e ≤ gi ≤ hi for each i and g = g1g2 · · · gn}.
By primitive recursion on n, we define a function f : R → N such that for
all 〈n, g, x〉 ∈ R, 〈n, g, x, f(n, g, x)〉 ∈ Q.
Case n = 1. If 〈1, g, x〉 ∈ R, then x = 〈g〉 and e ≤ g so 〈1, g, x, g〉 ∈ Q. So we
define f(〈1, g, x〉) = g.
Case n > 1. We define f(〈n, g, x〉) = 〈(g∧h1)〉?f(n−1, (g∧h1)−1g, 〈h2, . . . , hn〉),
where x = 〈h1, . . . , hn〉 and ? denotes concatenation and subsequent coding of
lists of elements.
By the induction hypothesis, f(n − 1, (g ∧ h1)−1g, 〈h2, . . . , hn〉) returns a
y such that (g ∧ h1)−1g = g2g3 · · · gn where y = 〈g2, . . . , gn〉. Then g = (g ∧
h1)g2 · · · gn〉, so the code 〈(g ∧ h1), g2, . . . , gn〉 returned by this process is correct.
Let Π(x) be the product of the elements coded by x, and let P (n) be the
28
formula ∀g∀x(〈n, g, x〉 ∈ R → g = Π(f(n, g, x))) By our definition of f , and by
calculations similar to those in the first proof, we have ∀n(P (n) → P (n + 1)), so
by Π01 induction:
∀n∀g∀x(〈n, g, x〉 ∈ R → g = Π(f(n, g, x)))
The following criterion for an `-subgroup will be convenient for later verifi-
cations.
Theorem 2.22. (RCA0) A subgroup H of the `-group G is an `-subgroup iff
∀a ∈ H(a ∨ e ∈ H).
Proof. From [4]: The first implication is obvious. Conversely, let H be a subgroup
of G and suppose ∀a ∈ H(a ∨ e ∈ H). Then x−1y ∨ e ∈ H for any x, y ∈ H and
x(x−1y∨ e) = y∨x ∈ H. Now that we have joins, x∧ y = (x−1 ∨ y−1)−1 ∈ H.
The following Lemma will be referred to in Chapter 5, and we put it here
with the other results about basic computations.
Lemma 2.23. (RCA0) |a| ∧ |b| ≤ |a ∧ b|
Proof. By writing out the absolute values and distributing, we obtain the equali-
29
ties
|a| ∧ |b| = (a ∧ b) ∨ (a ∧ b−1) ∨ (a−1 ∧ b) ∨ (a−1 ∧ b−1)
|a ∧ b| = (a ∧ b) ∨ (a−1 ∨ b−1).
Note that each is a join of (a ∧ b) with another element. Furthermore,
(a ∧ b−1) ≤ (a−1 ∨ b−1)
(a−1 ∧ b) ≤ (a−1 ∨ b−1)
(a−1 ∧ b−1) ≤ (a−1 ∨ b−1)
where the first is justified by noting that (a−1 ∨ b−1) ≥ b−1 ≥ (a ∧ b−1), and the
rest are similarly justified. The conclusion then follows by the principle x ≤ y →
a ∨ x ≤ a ∨ y.
Chapter 3
More Results
3.1 Identifying groups which are lattice-orderable
Since an `-group is first and foremost a group, it is worth asking the question
“Can I tell if a given group G can be made into an `-group?” This question is
partially answered by the following theorem about o-groups from [1].
Theorem 3.1. For an abelian group G, the following are equivalent:
1. G is totally orderable
2. G is lattice orderable
3. G is torsion-free
In the context of Reverse Math, the equivalences of the orderability state-
ments with the torsion-free statement are somewhat unbalanced in that one di-
rection can be done in RCA0 and the other cannot.
Lemma 3.2. (RCA0) If G is a (potentially non-abelian) group, then “lattice-
orderable → torsion free”
30
31
Proof. This follows from Theorem 2.12. Suppose we have some torsion element x
with xn = e Then, in particular xn ≥ e, so by Theorem 2.12, x ≥ e. On the other
hand, we must also have x−n = e, and a similar argument shows that x−1 ≥ e,
and thus x = e.
The equivalence of “t.f. abelian → totally orderable” and WKL0 was shown
in a paper by Hatzikiriakou and Simpson [3]. Since totally ordered groups are auto-
matically lattice ordered groups, we have WKL0 ` “t.f. abelian → lattice orderable.”
Since we already know that the implication for totally-orderable groups is equiv-
alent to WKL0, the remaining question is then: Does the weaker statement
“t.f. abelian → lattice orderable” imply WKL0? As it turns out, this is the case.
Theorem 3.3. (RCA0) The following are equivalent.
1. Every torsion-free abelian group is lattice-orderable.
2. WKL0
Proof. By the comments above, we have 2 → 1. Now, we prove 1 → 2.
Let u, v be one-to-one functions with disjoint ranges. We will form a sep-
arating set. Let G be the free abelian group on generators ai, bi modulo the
relations (2k + 3) · au(k) = bu(k) and (2k + 3) · bv(k) = av(k). (We use 2k + 3 so
that we do not have equality when k = 0.) Since torsion free and abelian implies
lattice-orderable, there is some lattice order with ∧,∨,≤ defined on G. We modify
the formal definition from Example 1.9 to get a formal representation of G. The
32
technique of introducing relations is adapted from [3]. Any g ∈ G has a unique
normal form
g =∑i∈I
qiai +∑j∈J
pjbj
where the coefficients are reduced according to the relations above. In terms of
the quadruple 〈I, q, J, p〉 which formally represents g, we require that
∀i ∈ I¬∃k ≤ (|qi| − 2)(i = u(k))
&
∀j ∈ J¬∃k ≤ (|pj| − 2)(j = v(k))
Since these are bounded-quantifier formulas, set of these normal forms exists
in RCA0. To see why these requirements give us a unique normal form, suppose
that u(k) = i. Then we can start generating normal forms like
ai, 2ai, . . . , kai, (k + 1)ai
−ai,−2ai, . . . ,−kai,−(k + 1)ai.
With positive coefficients, once we get to (k + 2)ai, we have the relation (2k +
3)ai = bi which gives us the equality (k + 2)ai = bi − (k + 1)ai. The bounds
on the coefficients appearing in the requirements above force us to use the latter
representation for the normal form.
33
We formally work with G′, defined as the set of all elements of G that are
in normal form. Then, for the G′ group sum on normal forms g′ +G′ h′, we take
the G-sum g′+G h′ and, if necessary, search for the normal form in G′ obtained by
reducing g′ +G h′ according to the relations. Since g′ and h′ were in normal form
before adding, this process may only change a given qi by 2k+3, where u(i) = k,
and change the corresponding pi by one, and similarly for the relations involving
the function v.
That is, after adding the normal-form elements g′ +G h′ to get∑
i∈I qiai +∑j∈J pjbj, the associated normal form has components 〈I ′, q′, J ′, p′〉, where each
component can be calculated as demonstrated below.
If, for example, ∃k ≤ (|qi| − 2)(i = u(k)), then we reduce to a normal form
by cases.
Case 1: qi > 0. Then we take qiai+
0G︷ ︸︸ ︷(bi − (2k + 3)ai) to get (qi−2k−3)ai+(pi+1)bi.
Since the two summands were in normal form, we have qi ≤ 2k +2. Thus k +3 ≤
qi ≤ 2k + 2, so −k ≤ qi − 2k − 3 ≤ −1, and the result is in normal form.
Coincidental cancellation may result in differences between I, J and I ′, J ′.
Case 2: qi < 0 is handled similarly.
Because this process uses only bounded quantification, including quantifying
over (codes for) finite sets, the function taking 〈I, q, J, p〉 to the normal form
〈I ′, q′, J ′, p′〉 may be defined by a ∆01 formula, so G′ exists as a group in RCA0.
Having done this, we refer to G′ as G.
34
Claim. G is torsion free.
The claim follows from the fact that u and v have disjoint ranges. Essen-
tially, a reduction of m · ai to bi may only happen if i ∈ ran(u) and a reduction
of m · bi to ai may only happen if i ∈ ran(v).
Now, we return to the proof of a reversal. The theorem asserts that there
is a lattice order on G, which implies there are meet and join functions ∨,∧.
This, in turn, lets us work with the absolute value |x| = x ∨ x−1. Then the set
{n : |an| ≤ |bn|} separates the ranges of u, v. Suppose u(k) = n. Then we have
a2k+3n = bn and |an|2k+3 = |bn|. Since |x| ≥ e for all x, it follows that |an| ≤ |bn|. It
is easy to show that for every n, |an| 6= |bn|, so the complement of the separating
set contains {n : |an| > |bn|}, which is the range of v. Note that |an| and |bn| are
only related by < or > if n is in the range of u or v. By Lemma 1.1, the existence
of a separating set implies WKL0.
3.2 Convex `-subgroups and Right Cosets
In the context of `-groups, a type of `-subgroup called a convex `-subgroup plays
an important role, analogous to that of normal subgroups in the context of general
group theory. Cosets of a normal subgroup inherit the group structure in a natural
way and, similarly, cosets of a convex `-subgroup inherit the lattice structure in
a natural way.
35
Definition 3.4. An `-subgroup C is convex if for every c1, c2 ∈ C and every x,
c1 ≤ x ≤ c2 → x ∈ C.
Theorem 3.5. (RCA0) The subgroup C is a convex `-subgroup iff ∀c ∈ C(|x| ≤
|c| → x ∈ C).
Proof. On page 23 of [4] there is a quick proof which can be done in RCA0.
Theorem 3.6. If C is a convex `-subgroup of the `-group G, then the right cosets
of C in G form a lattice under the partial order given by Cg ≤ Ch ↔ ∃c ∈ C(g ≤
hc).
This same condition gives us the induced order in the context of cosets of
a convex subgroup of a p.o.-group or an o-group. As shown in [6], the existence
of the induced order on G/C is equivalent to ACA0 when G is a p.o.-group, and
provable in RCA0 when G is an o-group. In this regard, it turns out that `-groups
are closer to o-groups than p.o.-groups.
Theorem 3.7. (RCA0) Let G be an `-group. Let H be a convex `-subgroup of G.
The induced order on the set of right cosets G/H exists.
Proof. Let a, b be elements of G. The induced order on G/H as originally given
is: Ha ≤ Hb ↔ (∃h ∈ H)a ≤ hb. Note that by choosing h = e, it is apparent that
a ≤ b → Ha ≤ Hb. Also note the following identity, which holds for all elements
x, y.
x = (e ∨ xy−1)(x ∧ y)
36
To verify this identity, we distribute. First,
(e ∨ xy−1)(x ∧ y) = [(e ∨ xy−1)x] ∧ [(e ∨ xy−1)y]
= (x ∨ xy−1x) ∧ (y ∨ x)
≥ x
Then, distributing the other way,
(e ∨ xy−1)(x ∧ y) = [e(x ∧ y)] ∨ [xy−1(x ∧ y)]
= (x ∧ y) ∨ (xy−1x ∧ x)
≤ x
Claim. (e ∨ ab−1) ∈ H → Ha ≤ Hb
Proof. Let (e ∨ ab−1) ∈ H. Then a = (e ∨ ab−1)(a ∧ b), by the identity above.
Therefore, Ha = H(a ∧ b) ≤ Hb.
Claim. Ha ≤ Hb → (e ∨ ab−1) ∈ H
Proof. Fix h ∈ H s.t. a ≤ hb. Then we have
(a ∧ b) ≤ a ≤ hb
(a ∧ b)b−1 ≤ ab−1 ≤ h
(ab−1 ∧ e) ≤ ab−1 ≤ h
e ∨ (ab−1 ∧ e) ≤ (e ∨ ab−1) ≤ e ∨ h
e ≤ (ab−1 ∨ e) ≤ e ∨ h
37
Since e, h ∨ e belong to H and H is convex, we have ab−1 ∨ e ∈ H.
By these two claims, we can decide the induced order on cosets by the
quantifier free condition aH ≤ bH ↔ (e ∨ ab−1) ∈ H.
If C is normal then G/C has the familiar quotient group structure described
by Ca ·Cb = C(ab). Even when C is not normal, by convexity alone, we have the
induced order on the right cosets, as defined above, and this gives us the properties
Ca∨Cb = C(a∨ b) and Ca∧Cb = C(a∧ b) as discussed in the following theorem
from [4].
Theorem 3.8. (RCA0) Let C be a convex subgroup of G. Then C(x∧y) = Cx∧Cy
and C(x ∨ y) = Cx ∨ Cy under the induced order.
Proof. We prove the case for joins, and the case for meets is similar.
By definition of the induced order, it follows that Cx ≤ C(x∨y) and Cy ≤ C(x∨y).
Let Cz ≥ Cx, Cz ≥ Cy. for some z ∈ G. Then there exist c1, c2 ∈ C such that
c1z ≥ x, c2z ≥ y. Let c = c1 ∨ c2 ∈ C. Then cz ≥ c1z ≥ x and cz ≥ c2z ≥ y, so
cz ≥ x∨ y. Thus Cz ≥ C(x∨ y) and C(x∨ y) is the least upper bound of Cx,Cy
in G/C.
Chapter 4
Convexity Results and Reversals for Substructure
Existences
4.1 Closure Operations
To each kind of subgroup we associate a closure operation.
Definition 4.1. Let A be a subset of the `-group G. Let
• 〈A〉 denote the subgroup of G generated by A,
• 〈A〉` denote the `-subgroup of G generated by A,
• and CL(A) denote the convex `-subgroup of G generated by A.
To form 〈A〉, one must close under inverses and group composition. To form
〈A〉`, it is sufficient to further close under join, since x∧ y = (x−1 ∨ y−1)−1. From
this, one may form CL(A) by adding all elements bounded above and below by
elements of 〈A〉`.
To work with these closures in RCA0, we define a mechanism of generation
by primitive recursion. First we define a Σ01(A) mechanism for generating 〈A〉. Let
38
39
A be a subset of the `-group G, and suppose G is enumerated as {g0 = e, g1, . . .}.
Let
• 〈A〉0 = {0}
• 〈A〉t+1 = {gk ∈ G : gk ≤N t & ∃gi, gj ∈ 〈A〉t s.t.
( gk ∈ 〈A〉t
or gk ∈ A
or g−1i = gk
or gk = gigj)}
Then we define 〈A〉 as the set {g ∈ G : ∃n(g ∈ 〈A〉n)}, which is Σ01(A). Thus,
given an `-group G and a subset A ⊂ G, ACA0 is sufficient to prove the existence
of 〈A〉.
By adding the disjunction “or gk = gi ∨ gj”, one defines 〈A〉`. By further
adding the disjunction “or gi ≤ gk ≤ gj”, one defines CL(A).
Note that the definition of the closure is Σ01 in all three cases. While ACA0
is equivalent to the existences of 〈A〉, 〈A〉`, and CL(A), given A, many of the
interesting properties of these sets can be verified in RCA0 if they exist. For the
next lemma, we will use the following notation which generalizes to the other
forms of closure.
• g ∈ 〈A〉 means ∃n(g ∈ 〈A〉n)
40
• 〈A〉 ⊆ 〈B〉 means ∀g(∃n(g ∈ 〈A〉n) → ∃n(g ∈ 〈B〉n)).
• 〈A〉 is a subgroup of G means ∀g, h(g, h ∈ 〈A〉 → (g−1 ∈ 〈A〉 & g ·h ∈ 〈A〉)).
Lemma 4.2. (RCA0) Using the abbreviations above, for all A, B ⊆ G,
1. 〈A〉 is a subgroup of G.
2. If A is a subgroup of G then 〈A〉 = A.
3. If A ⊆ B then 〈A〉 ⊆ 〈B〉.
4. If A ⊆ B and B is a subgroup of G then 〈A〉 ⊆ B.
Moreover, the Lemma is also true if one replaces all instances of the word “sub-
group ” with “`-subgroup ” or “convex `-subgroup” and uses the appropriate closure
operation.
Proof. Statement (1) is clear by the construction of 〈A〉. For (2), the containment
A ⊆ 〈A〉 follows from the construction. The reverse containment is shown by the
statement ∀n(〈A〉n ⊆ A) which is proved by Σ00 induction. There are three cases
to consider for an element added at the (n + 1)th step. If gk is added because it
belongs to A, then it is obviously in A already. If gk is added because it is the
inverse of an element in 〈A〉n, then that element, by the induction hypothesis,
belonged to A, and thus so did gk since A is a subgroup. The third case, that gk
is a product of elements in 〈A〉n, is similar.
41
Similarly, to prove (3) one uses the statement “A ⊆ B → ∀n(〈A〉n ⊆ 〈B〉n)”
which is proved by Σ00 induction. Finally, (4) follows from (2) and (3).
The generalizations to `-subgroup closure and convex closure are proved in
the same way.
We also state the following lemma using the abbreviations described above.
Lemma 4.3. (RCA0) Let S be a subset of the `-group G. Then we have the
following:
• y ∈ S → 〈S〉 = 〈S ∪ {y−1}〉
• x, y ∈ S → 〈S〉 = 〈S ∪ {xy}〉
• x, y ∈ S → 〈S〉` = 〈S ∪ {x ∨ y}〉`
• x, y ∈ S & x ≤G z ≤G y → CL(S) = CL(S ∪ {z})
Furthermore, the first three implications are also valid for the more specific closure
operations.
Proof. We prove only the first statement. The rest are similar. First, suppose
y ∈ S. Then we have the containments S ⊆ S ∪ y−1 ⊆ 〈S〉. To be explicit about
the second containment, suppose y−1 is not already in S. If y = ga and y−1 = gb,
with a < b, then the latter will be a member of 〈S〉b by construction. If the
inequality is reversed, then the latter will be a member of 〈S〉a+1. By arguments
like those in Lemma 4.2, we want to show that 〈S〉 ⊆ 〈S ∪ y−1〉 ⊆ 〈〈S〉〉 = 〈S〉.
42
Formally, the containments can be stated as ∃n(x ∈ 〈S〉) → ∃n(x ∈ 〈S ∪ y−1〉n)
and ∃n(x ∈ 〈S ∪ y−1〉n) → ∃m∃n(x ∈ 〈〈S〉n〉m), while the equality can be stated
∃m∃n(x ∈ 〈〈S〉n〉m) ↔ ∃n(x ∈ 〈S〉n).
Lemma 4.4. (RCA0) Let g be an element of the `-group G. Then CL({g}) =
{h ∈ G : ∃n > 0(|h| ≤ |g|n)}, i.e. ∀h[∃n(h ∈ CL({g})n ↔ ∃n > 0(|h| ≤ |g|n))].
Proof. Fix h ∈ G. The statement ∃n > 0(|h| ≤ |g|n) → ∃n(h ∈ CL({g})n) follows
from the definition of CL({g}).
For the other direction, instead of proving the implication ∃n > 0(|h| ≤
|g|n) → ∃n(h ∈ CL({g})n), which is neither Σ01 nor Π0
1, we use a bounded quan-
tifier in the second part. Let f(n) = 3n, so that f(n + 1) = 3 · f(n). As in a
classic “ ε3” argument, this choice will make sense in the future. We show by Π0
1
induction on n that ∃n(h ∈ CL({g})n) → |h| ≤ |g|f(n)). For n = 0, we have
CL({g})0 = {0}, |0| = 0 ≤ |g|1, and 0 < 1 ≤ f(0) = 1. For the induction step,
suppose h ∈ CL({g}n+1. Then we have several cases, corresponding to the ways
an element can enter the convex closure.
• If h = g then |h| ≤ |g|1 and 1 ≤ f(n + 1).
• If h ∈ CL({g})n, then |h| ≤ |g|k for some 0 < k ≤ f(n) < f(n + 1).
• If ∃gi ∈ CL({g})n with h = g−1i , then |h| = |g−1
i | = |gi| ≤ |g|k for some
0 < k ≤ f(n) < f(n + 1).
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• If ∃gi, gj ∈ CL({g})n such that either h = gigj, h = gi ∨ gj, or gi < h < gj,
then we have 3 subcases.
– If h = gigj then, by Theorem 2.20, |h| ≤ |gi| · |gj| · |gi| ≤ |g|f(n) · |g|f(n) ·
|g|f(n) = |g|3f(n) = |g|f(n+1).
– If h = gi ∨ gj then |h| ≤ |gi| · |gj| ≤ |g|2f(n) < |g|f(n+1).
– Suppose gi < h < gj. Then |h| = (h ∨ h−1) ≤ (gj ∨ h−1) ≤ (gj ∨
g−1i ) ≤ (|gi| ∨ |gj|). Since gi, gj ∈ CL({g})n, |gi| ∨ |gj| ≤ |g|f(n). Thus,
|h| ≤ |g|f(n) < |g|f(n+1).
Lemma 4.5. (RCA0). Let g1, . . . gn be positive elements of the `-group G. Then
CL({g1, . . . , gn}) = CL({∏
1≤j≤n
gj}).
Proof. Both containments are proved using an n-fold application of Lemma 4.3.
First, since each gj is positive, e ≤ gi ≤∏
j gj. This proves the “⊆” containment.
For the other direction, observe that the product∏
1≤j≤n
gj must be contained in
CL({g1, . . . , gn}), since the latter is a subgroup containing each factor.
Theorem 4.6. (RCA0) If A, B are convex `-subgroups and 〈A, B〉 exists, then
CL(A ∪B) = 〈A, B〉.
Proof. The method of proof on page 8 of [1] uses the Riesz Decomposition The-
orem and Theorem 2.20, and can be adapted to work in RCA0. We sketch the
44
proof. Let C = 〈A, B〉 := 〈A ∪ B〉. By Lemma 4.2, C is a subgroup. We use the
criterion in Theorem 3.5 to show C is a convex `-subgroup. Let c ∈ C, g ∈ G,
and |g| ≤ |c|. Then there is a finite collection of elements ci, i ≤ n which belong
to either A or B such that
c = c0c1 · · · cn.
Then g+ ∨ (g−)−1 = |g| ≤ |c0c1 · · · cn|, and by repeated use of Theorem 2.20 one
obtains the inequality
|c0c1 · · · cn| ≤ |c0||c1| · · · |cn||cn−1| · · · |c1| · · · |c0|.
Then, by the Riesz Decomposition Theorem, g+ and (g−)−1 (and hence g−) can
be expressed as products of elements which belong to A or B by convexity. Since
g = g+g−, g ∈ C.
4.2 Existence of the subgroup generated by A, B.
Theorem 4.7. (RCA0) The following are equivalent.
1. For any group G and subgroups A, B ⊆ G, the subgroup 〈A, B〉 exists.
2. ACA0
Proof. 〈A, B〉 can be formed in ACA0 since it has a Σ01 definition. To show
the reversal, let f be a one-to-one function. Let G be the torsion free abelian
group generated by {xi, yi}i∈N. The formal presentation for this group as sums
45
∑i∈I qiai +
∑j∈J pjbj was defined in Example 1.9 and we subsequently refer to
that notation below.
Let A be the set of elements of the form∑
i∈I qixi + 0, i.e., those for which
J = p = ∅. Clearly, A is a subgroup of G. Then let B be the subgroup of G
generated by all elements of the form xi + yf(i). The exact condition for a formal
sum to belong in B can be stated as the following conjunction:
∀i ∈ I∃j ∈ J(j = f(i) & pi = qj)
&
∀j ∈ J∃i ∈ I(j = f(i) & pi = qj).
Since I and J are finite, the quantifiers are bounded, so both A and B may be
formed in RCA0.
Clearly, if ∃i(f(i) = n) then the elements xi and xi + yn belong to A and
B, respectively. In particular, the group sum −1xi + (xi + yn) = yn belongs to
〈A, B〉. On the other hand, assume that yn = a + b for some a ∈ A, b ∈ B. Then
yn =∑
i′∈I′ pi′xi′ + (∑
i∈I pixi +∑
i∈I piyf(i)). The parts involving x terms must
cancel, leaving I ′ = I. Since only yn appears in the left hand side, we must have
I = {m} where f(m) = n and pn = 1.
Then yn ∈ 〈A, B〉 ↔ ∃i(f(i) = n), and ran(f) = {n : yn ∈ 〈A, B〉}.
4.3 Existence of the convex closure of an `-subgroup H
Theorem 4.8. (RCA0)
46
1. For any `-group G and `-subgroup H ⊆ G, CL(H) exists.
2. ACA0.
Proof. Since CL(H) has Σ01 definition with H as a parameter, it is clear that
2 → 1. For the other direction, let G be Fin∗Z, the `-group of finite-support
functions from N to Z in the pointwise order. Let f be a one-to-one function.
The set A of all strings σ such that all three of the following hold
1. ∀x < |σ|(σ(x) = 0 or 1 or 2)
2. ∀x < |σ|(σ(x) = 1 → ∃i ≤ x[x = 2i & 2f(i) + 1 < |σ| & σ(2f(i) + 1) = 2])
3. ∀x < |σ|(σ(x) = 2 → ∃i[2i < |σ| & x = 2f(i) + 1 & σ(2i) = 1])
is definable by a bounded-quantifier formula, so exists in RCA0.
In other words, if ai = 02i−11 and bn = 02n1, then the generators in A are
all strings of the form ai + 2bf(i). The subgroup H = 〈A〉 may also be described
as all elements of the form∑i∈I
ci · ai +∑i∈I
2ci · bf(i) for some finite set I, and the
set A represents elements for which ci = 1. Thus H exists in RCA0. Clearly, if
f(i) = n then 0 < bn < 2bn < ai + 2bf(i), so bn must belong to CL(H). On
the other hand, suppose bn ∈ CL(H). Then there is some finite set I such that
0 ≤ bn ≤∑i∈I
ci · ai +∑i∈I
2ci · bf(i). Since Fin∗Z has the pointwise order and
bn has the form 02n1, the sum bounding bn must have a positive number in the
(2n + 1)th place. This can’t come from the∑
ci · ai part, so must come from the
47
the∑
2ci · bf(i) part. This means ∃i(f(i) = n). Then bn ∈ CL(H) if and only if
n ∈ ran(f).
4.4 Existence of the convex `-subgroup generated by convex
`-subgroups A, B.
In the setting of totally ordered groups, the question is trivial, since CL(A, B) is
simply the larger of the two or, equivalently, their union.
The first version of this proof uses computable algebra, and is followed by a
more direct proof typical of Reverse Math. In all but the easiest reversals, it seems
easier to work in the computable setting, and build an computable encoding object
by a step-by-step construction, and to then use the process of the construction
to help define the object in more general terms using ∆01 formulas. Theorem 4.6
allows us to simplify both versions of the proof by looking at 〈A, B〉 instead of
CL(A, B).
Theorem 4.9. There is a computable abelian `-group G with computable convex
`-subgroups A, B such that A + B ≥1 0′.
Corollary 4.10. (RCA0) The following are equivalent.
1. For any convex `-subgroups A, B of the `-group G, the subgroup 〈A, B〉 exists.
2. ACA0.
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Proof. As in the proof of Theorem 2.4, we construct an injection from a com-
putable `-group G into the `-group D =⊕
ω Z, equipped with the pointwise
order. Elements of D are finite sequences of integers with the last entry nonzero.
The image of G in D will be a (nonconvex) `-subgroup of D, yet classically iso-
morphic to the full group. G will contain computable convex `-subgroups A, B
such that A + B=〈A, B〉 is ≥1 0′.
We construct our computable group G = ∪sGs in stages. Each Gs will be
a finite set of integers, and representation function σs : Gs → D. For all elements
n ∈ Gs, σs(n) will be a finite string of integers with no trailing zeros. We will
define n ◦s m = k if σs(n) + σs(m) = σs(k). The operation ◦ will be ∪s∈ω◦s.
Similarly, we define ∧s,∨s whose union will be the usual meet and join on G. The
partial order on G can be recovered from ∨ and ∧ since a ∧ b = b ↔ b ≤ a.
The construction of G will be done accomplishing the following:
• For some m(s) ∈ ω we will assign to all p ≤ m(s) a string σs(p).
• If t > s, then m(t) ≥ m(s).
• If p ≤ m(s), then σs(p) = σt(p) for all t ≥ s.
Additionally, G will have computable convex `-subgroups A, B. Once σs(p) is
defined, the membership of p in A or B will be computable from σs(p). We
will have p ∈ A ⇐⇒ σs(p)(n) = 0 for each odd n up to |σs(p)|, and p ∈
49
B ⇐⇒ σs(p)(n) = 0 for each even n up to |σs(p)|. We will meet the following
requirements:
• Group Closure
Ce: If e = 〈p, q〉, then p ◦ q is defined.
• Group Inverses
Ip: There exists a q such that p ◦ q = 0 (0 denotes the zero of ω and the
identity of G.)
• Meet and Join
Me, Je: If e = 〈p, q〉, then p ∧ q, p ∨ q are defined.
• A∨B non-computability requirements
Ne: 〈A, B〉 can compute whether φe(e) halts.
We order the requirements as follows: N0 ≥ C0 ≥ M0 ≥ J0 ≥ I0 ≥ N1 ≥ · · ·
Definition 4.11. We say
1. Ce requires attention at stage s + 1 if e = 〈p, q〉 for some p, q ≤ m(s) and
there is no r ≤ m(s) such that σs(p) + σs(q) = σs(r).
2. Me requires attention at stage s + 1 if e = 〈p, q〉 for some p, q ≤ m(s) and
there is no r ≤ m(s) such that σs(p) ∧ σs(q) = σs(r), and similarly for Je.
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3. Ip requires attention at stage s + 1 if p ≤ m(s) and there is no q ≤ m(s)
such that σs(p) + σs(q) = 0
4. Ne requires attention at stage s + 1 if Ne is unrealized and either
(a) for no t ≤ s is there a p such that σs(p) = 02e11 (in which case we say
Ne is inactive); or
(b) there exists p as above and φe,s(e) ↓ (in which case we say Ne is active).
4.4.1 Construction
Stage 0. Set p(0) = 0 and σ0(0) = 0. The zero of ω is to be the identity of 〈G, ◦〉.
Stage s + 1. Let R be the requirement of highest priority that requires
attention.
Case 1: R is Ce with e = 〈p, q〉. Define m(s+1) = m(s)+1. Define σs+1(r)
for r ≤ m(s + 1) via:
σs+1(r) =
σs(r) if r < m(s + 1),
σs(p) + σs(q) if r = m(s + 1).
Case 2: R is Ip. Let m(s + 1) = m(s) + 1. Define σs(r) for r ≤ m(s + 1)
via:
σs+1(r) =
σs(r) if r < m(s + 1),
−σs(p) if r = m(s + 1).
51
Case 3: R is Me with e = 〈p, q〉. Define m(s+1) = m(s)+1. Define σs+1(r)
for r ≤ m(s + 1) via:
σs+1(r) =
σs(r) if r < m(s + 1),
σs(p) ∧ σs(q) if r = m(s + 1).
Case 4: R is Je. Similar to case 3.
Case 5: R is Ne and Ne is inactive. Let m(s + 1) = m(s) + 1 and define
σs+1 for r ≤ m(s + 1) via:
σs+1(r) =
σs(r) if r < m(s + 1),
02e11 if r = m(s + 1).
At this point, Ne becomes active.
Case 6: R is Ne and Ne is active. Thus, there is a p ≤ m(s) such that
σs(p) = 02e11 and φe,s(e) ↓. Let m(s + 1) = m(s) + 2, and define σs+1 for
r ≤ m(s + 1) via:
σs+1(r) =
σs(r) if r ≤ m(s)
02e1 if r = m(s) + 1
02e+11 if r = m(s) + 2
This creates elements of G in A and B, respectively, whose group sum is p, thereby
putting p ∈ 〈A, B〉. Ne is now realized. Notice that this will ensure 〈A, B〉 ≥1 0′:
To determine if e ∈ 0′, find p such that σ(p) = 02e11. Then e ∈ 0′ ⇐⇒ p ∈
〈A, B〉.
52
4.4.2 Verification
G is clearly a computable `-group by the construction. A, B, and 〈A, B〉 are easily
verified as convex `-subgroups. For example, A contains the elements p such that
σ(p) is zero in all odd-numbered components. The operations of group sum, meet
and join on two elements of A cannot introduce a nonzero entry into an odd-
numbered component with respect to the map σ, so A is an `-subgroup. As for
the convexity of A: suppose a ∈ A and 0 ≤ |g| ≤ |a|. It follows that σ(g) is zero
in all odd-numbered components, and so g ∈ A.
The verification for B, 〈A, B〉 is very similar, except that in the case of
〈A, B〉, we say p ∈ 〈A, B〉 ⇐⇒ σ(p) has nonzero entries only in components
numbered 2e or 2e + 1 where φe(e) ↓.
Now, we give a formal proof of the reverse math corollary.
Theorem 4.12. (RCA0) The following are equivalent.
1. ACA0
2. If A, B are convex `-subgroups of G, then
∃X∀t(t ∈ X ↔ ∃n(t ∈ 〈A ∪B〉n)).
Proof. Since X has a Σ01 definition, 1 → 2. Now we show 2 → 1. Fix a one-to-one
function f . We define an injection σ : N → Fin∗Z by primitive recursion. The
53
group G will be N with +G given by g +G h = k ↔ σ(g) +Fin∗Zσ(h) = σ(k), and
we define αe = 02e11, βe = 02e1, andγe = 02e+11.
• Let σ(0) = 0 (as a string)
• For r = 5m+1, set σ(r) = αe where e is least such that ∀k < r(σ(r) 6= αe).
(This will be the default action, if the test condition fails in other stages.)
• For r = 5m + 2, let 〈i, j〉 be N-least such that i, j < r and ∀k < r(σ(i) +
σ(j) 6= σ(k)). If such a pair exists, define σ(r) = σ(i) +Fin∗Zσ(j), otherwise
perform the default action.
• For r = 5m + 3, let 〈i, j〉 be N-least such that i, j < r and ∀k < r(σ(i) ∧
σ(j) 6= σ(k)). If such a pair exists, define σ(r) = σ(i) ∧ σ(j), otherwise perform
the default action.
• For r = 5m+4, let i be N-least such that i < r and ∀k < r(σ(i)+Fin∗Zσ(k) 6=
0. If such i exists, define σ(r) = σ(i)−1, otherwise perform the default action.
• For r = 5m + 5 look for N-least e < r s.t. ∃k < r(σ(k) = αe) & ¬∃k <
r(σ(k) = βe or σ(k) = γe) & ∃k < r(f(k) = e). Then we define σ(r) = βe.
Then we define the sets
A = {n : ∀s(2s + 1 ≤ |σ(n)| → σ(n)[2s + 1] = 0)}
B = {n : ∀s(2s ≤ |σ(n)| → σ(n)[2s] = 0)}
It is easy to verify that A, B are both convex `-subgroups. Let e ∈ N. By definition
of σ, there is some k s.t. σ(k) = αe. If e ∈ ran(f), then by definition of σ, there
are m, n s.t. σ(n) = βe, σ(m) = γe. Then m ∈ A and n ∈ B. Then σ(n)+σ(m) =
54
αe = σ(k), so k ∈ 〈A, B〉. On the other hand, suppose k ∈ 〈A, B〉 = A+B. Then
there exist m ∈ A, n ∈ B s.t. m + n = k i.e., σ(m) + σ(n) = αe. But, by the
definitions of αe and A, B, this means that all entries of σ(m), σ(n) must be zero
except at 2e + 2 and 2e + 1, respectively. This implies that σ(n) = βe, which can
only happen if ∃j < r(f(j) = e), by definition of σ. Thus we define a function
φ(e) by φ(e) = k ↔ k ≤ 5e + 1 & σ(k) = αe. By the construction, φ is total and
definable in RCA0. By the verification, ran(f) = {e : φ(e) ∈ 〈A ∪B〉}.
4.5 Existence of the Polar X⊥.
Definition 4.13. Two elements a, b of an `-group are said to be orthogonal if
a ∧ b = e.
Definition 4.14. Let M be a subset of the `-group G. The polar of M , denoted
M⊥, is defined as the set {g ∈ G : ∀x ∈ M(|x| ∧ |g| = e)}.
Theorem 4.15. (RCA0) If it exists, M⊥ is a convex subgroup of G.
Proof. Adapted from [1]. We verify that M⊥ is a subgroup and is convex. Suppose
g, h are positive elements of M⊥. Let m ∈ M . Then
e = |m| ∧ g = |m| ∧ (|m| ∧ h)g = |m| ∧ (|m|g) ∧ (hg) = |m| ∧ hg.
Since the definition of M⊥ uses absolute value, inverses are automatically included,
and the identity satisfies the definition. Thus, checking composition is enough to
55
show that M is a subgroup. For convexity, suppose b ∈ M⊥ and |a| ≤ |b|. Then
|a| ≤ |b| → e ≤ |m| ∧ |a| ≤ |m| ∧ |b|.
Since b ∈ M⊥, |m| ∧ |b| = e, and therefore |m| ∧ |a| = e. So a ∈ M⊥.
Note that if M is finite, M⊥ has a Π00 definition, so exists in RCA0. In
particular, for each g ∈ G, g⊥ = 〈g〉⊥ exists in RCA0. Such polar subgroups are
called principal polars. In general, though, M⊥ is Π01(M) and can be complicated.
When seeking a reversal for the existence of M⊥, it is worth considering
how much structure M should have – when M is finite, the case is simple. When
M is infinite, is proving the existence of M⊥ any simpler if M is already, say, a
subgroup, `-subgroup, or even convex `-subgroup? The following shows that, in
general, it is not.
Theorem 4.16. (RCA0) Let G be an `-group. The following are equivalent.
1. For every M ⊆ G, M⊥ exists.
2. For every convex `-subgroup M ⊆ G, M⊥ exists.
3. ACA0
Proof. 1 ⇒ 2 is trivial, and 3 ⇒ 1 by the arithmetical definition of M⊥. It
remains to show 2 ⇒ 3. We will construct an abelian `-group G with a convex
`-subgroup M such that M⊥ computes the range of a given one-to-one function.
Group theoretically, G is a restricted Hahn group on the root system Γ which may
56
be described as ω incomparable copies of an ω∗ ordering.
...
!
Fig. 4.1: The Root System Γ
We formalize this group in RCA0: let G be the free abelian group on gen-
erators aij. That is, an element of G is a reduced finite sum∑
1≤i≤m,1≤j≤n
cijaij,
where cij ∈ Z and ∃i∃j(cmj 6= 0 & cin 6= 0.) That is, the normal form for an
element has coefficients indexed by an m × n array, where m, n are as small as
possible in the sense that the last column and last row must each contain some
nonzero coefficient. We may think of each aij as representing the jth element of
the ith column of Γ. The underlying formal representation uses triples 〈m, n, c〉
where c is a function from {〈i, j〉 : 1 ≤ i ≤ m, 1 ≤ j ≤ n} to Z. As with Example
1.9, the group operation is componentwise addition, with cancellation. (If adding
elements of different sizes, one may perform the operation with oversized arrays
(using extra 0s) and then put the result in normal form by eliminating unneces-
sary rows and columns.) Thus, the inverse of the element∑
i≤m,j≤n
cijaij is given by∑i≤m,j≤n
−cijaij, and the group identity is represented by 〈∅, ∅, ∅〉.
Let g = 〈m, n, c〉 be an element of G and let Sg = {i ≤ m : ∃j ≤ n(cij 6= 0)},
57
that is, Sg is the set of indices i such that g has a nonzero coefficient in the ith
column. An element in a Hahn group is positive if it is so at each maximal element
of its support. Thus g = 〈m, n, c〉 is positive iff the set
Smaxg = {cij : i ∈ Sg, j ≤ n & cij 6= 0 & ∀k < j(cik = 0)}
contains only positive numbers. This allows us determine the partial order ≤G.
For example, a ≤ b ↔ e ≤ ba−1, i.e., Smaxba−1 only contains positive numbers.
Equivalently, suppose we have two elements c, d with coefficients cij, dij defined
on an x× y array. Then
c ≤ d ↔ ∀i ≤ x[∃j ≤ y(cij 6= dij) → [dik > cik where k = µs(cis 6= dis)]].
This reasoning behind this is that when attempting to compare the two elements
c, d by testing to see if cd−1 is positive, the maximal elements of the support of
cd−1 correspond to the least-numbered row in each column such that differing
entries can be found.
So far, G is a p.o.-group in RCA0. Classically, we know it to be an `-group.
To verify that it is an `-group in RCA0, we need to be able to calculate the join of
elements a = 〈m, n, c〉 and b = 〈p, q, d〉. Putting some formality aside, we describe
an algorithm which can be used to calculate a ∨ b and claim that a function that
uses said algorithm to calculate joins is Σ00 definable.
First, letting x = max(m, p), y = max(n, q) we work with the “improper”
58
representations a∗ = 〈x, y, c∗〉 and b∗ = 〈x, y, d∗〉 where
c∗ij :=
cij if 1 ≤ i ≤ m & 1 ≤ j ≤ n
0 if m < i ≤ x or n < j ≤ y
and d∗ is defined similarly.
We build the join a∨ b by using c∗ and d∗ to construct a coefficient function
h for an x× y array, and then, if necessary, we reduce the array size to eliminate
unnecessary rows or columns of zeros. For each i ≤ x, one compares the ith
columns of c∗ and d∗, and defines the ith coefficient column of h according to three
cases.
1. If ∀j ≤ y(c∗ij = d∗ij), then ∀j ≤ y we define h∗ij := c∗ij.
2. If, at the least j such that c∗ij 6= d∗ij, c∗ij > d∗ij, then ∀j ≤ y we define h∗ij := c∗ij.
3. If, at the least j such that c∗ij 6= d∗ij, c∗ij < d∗ij, then ∀j ≤ y we define h∗ij := d∗ij.
Essentially, this procedure compares the ith columns as if they were elements
of the lexicographically ordered direct sum⊕
ω Z, and takes the larger of the two
for the ith column of the join. This follows from the Hahn order on the root
system Γ, which in this case ends up being a “pointwise” order on the columns of
Γ, where each column is given a lexicographic ordering.
To see that the resulting h gives us a join for 〈x, y, c∗〉 and 〈x, y, d∗〉, we
consider each column. Fix some i, and consider the ith column. If c∗ = d∗ in
59
the ith column, then any “larger” column would be unsuitable for a least upper
bound, and any “smaller” column wouldn’t work for an upper bound.
If c∗ > d∗ at the first disagreement in the ith column, then the ith column of
c∗ − d∗ will be positive at the maximal element of its support. Since we use c∗ to
define h in this case, we have h = c∗ and h > d∗ in the ith column. Any “larger”
column would not contribute to a least upper bound, and any “smaller” column
would result in an element that was no longer an upper bound of 〈x, y, c∗〉.
The remaining case is similar. Thus, after reducing, one gets the join of
a∨ b in normal form with no extraneous rows or columns of zeros. This algorithm
gives rise to a Σ00 definition for the join function, so RCA0 proves that G exists
and is an `-group.
Now, on to the reversal. Let f be a one-to-one function. Let M = {〈m,n, c〉 ∈
G : ∀i ≤ m∀j ≤ n(cij 6= 0 → ∃k ≤ j(f(k) = i))}.
Claim. M is a convex `-subgroup of G.
Proof. The question of inverses is easy, since the support of the coefficient function
is identical for an element and its inverse. The support of the coefficient function
for a group sum is contained in the union of the supports of its summands’ co-
efficient functions, all of which must satisfy the condition in the definition of M .
Thus, we have that M is at least a subgroup. Suppose that x, y ∈ M . Since the
columns of x∨ y are columns from either x or y, all of which satisfy the condition
of M , the coefficients of x ∨ y must satisfy the conditions of M . So M is closed
60
under meet (and join), and is an `-subgroup.
To verify convexity, suppose that |x| ≤ |m| and m ∈ M . Suppose, for a
contradiction, that x 6∈ M . Then ∃i, j(|x|ij 6= 0 & ∀k ≤ j(f(k) 6= i)). Without
loss of generality, take j to be the least so that this is true for the given i, that is,
∀k < j(|x|ik = 0). Since |x| is positive, |x|ij, as the maximal element of the support
of the ith column, must be positive. Then, since |m| ≥ |x|, either ∀s(|m|is = |x|is),
(which is impossible, since it implies that m 6∈ M) or, at the first place they differ
in the ith column, |m|is > |x|is. This would imply that either |m|ij > |x|ij > 0, or
for some k < j(|m|ik > 0). However, ∀k ≤ j(f(k) 6= i) → ∀k ≤ j(|m|ik = 0), so
neither case is possible, and we have our contradiction.
Then the convex `-subgroup M is definable by a bounded-quantifier formula,
so exists in RCA0.
Let αuv denote the element 〈u, v, c〉, where
cij =
1 if i = u & j = v
0 otherwise.
Suppose f(j) = i. Then αij ∈ M , so |αi0|∧|αij| = |αij| 6= e and αi0 6∈ M⊥. On the
other hand, suppose αi0 ∈ M⊥. Then, for all m ∈ M, |αi0|∧|m| = e. In particular,
no αij is in M , since |αi0| ∧ |αij| = |αij|. But [(∀j)αij 6∈ M ] → i 6∈ ran(f). Thus
we have i ∈ ran(f) ⇐⇒ αi0 6∈ M⊥.
The reversals of this chapter can be summarized as follows:
61
Theorem 4.17. (RCA0) Let G be an `-group. The following are equivalent:
1. ACA0
2. The existence of the subgroup generated by two subgroups A, B.
3. The existence of the convex `-subgroup generated by two convex `-subgroups
A, B.
4. The existence of the convex `-subgroup generated by a (nonconvex) `-subgroup
A.
5. The existence of the polar subgroup A⊥ = {g ∈ G : (∀a ∈ A)|g| ∧ |a| = 1G}
Chapter 5
Prime Subgroups and Values
5.1 Prime Subgroups
Definition 5.1. The convex `-subgroup P is prime if
∀x∀y[x ∧ y ∈ P → x ∈ P or y ∈ P ].
The following lemma from [4] will be useful.
Lemma 5.2. (RCA0) Let g be an element of the `-group G and P be a convex
`-subgroup of G. Then the set C = {x ∈ G : |g| ∧ |x| ∈ P} is a convex `-subgroup
of G.
62
63
Proof. Let x, y ∈ C. Then e ≤ |x| ∧ |g| = h1 ∈ P and e ≤ |y| ∧ |g| = h2 ∈ P .
|xy−1| ∧ |g| ≤ |x||y||x| ∧ |g| (by Thm. 2.20)
= |x||y||x| ∧ |x||y||g| ∧ |g|( since |x||y||g| ≥ |g|)
= |x||y|(|x| ∧ |g|) ∧ |g|
= |x||y|h1 ∧ |g|
≤ |x||y|h1 ∧ |g|h1
= (|x||y| ∧ |g|)h1
= (|x||y| ∧ |x||g| ∧ |g|)h1( since |x||g| ≥ |g|})
= (|x|(|y| ∧ |g|) ∧ |g|)h1
= (|x|h2 ∧ |g|)h1
≤ (|x|h2 ∧ |g|h2)h1
= (|x| ∧ |g|)h2h1 = h1h2h1 ∈ P.
Thus, we have e ≤ |xy−1| ∧ |g| ≤ h1h2h1, so |xy−1| ∧ |g| ∈ P and xy−1 ∈ C.
C is therefore a subgroup of G. According to Theorem 2.22, the subgroup C is
an `-subgroup iff ∀x ∈ C(x ∨ e ∈ C). By Theorem 2.20 |x ∨ e| ≤ |x|. Thus
e ≤ |x∨e|∧ |g| ≤ |x|∧ |g| ∈ P , so x∨e ∈ C and C is an `-subgroup. To show C is
convex, let e ≤ x ≤ h where h ∈ C. Thus |x| ≤ |h| so e ≤ |x| ∧ |g| ≤ |h| ∧ |g| ∈ P ,
so |x| ∧ |g| ∈ P and x ∈ C.
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Theorem 5.3. (RCA0) For a convex subgroup P of the `-group G, the following
are equivalent:
1. P is prime.
2. x ∧ y = 0 → x ∈ P or y ∈ P .
3. G/P is totally ordered under the induced order.
4. If A, B are convex `-subgroups containing P then A ⊆ B or B ⊆ A.
Note that RCA0 is sufficient to form the quotient group from (3). The proofs
of 1 → 2, 2 → 3, 3 → 4 are in [1], whereas 4 → 1 is from [4].
Proof.
1 → 2: Trivial.
2 → 3: Given g, h ∈ G, we have
((g ∨ h)g−1) ∧ ((g ∨ h)h−1) = (g ∨ h)(g−1 ∧ h−1) = (g ∨ h)(g ∨ h)−1 = e.
WLOG, by 2, (g ∨ h)h−1 ∈ P . Then Ph = P (g ∨ h) ≥ Pg.
3 → 4: Suppose A, B are convex `-subgroups containing P s.t. A 6⊂ B and B 6⊂ A.
By taking absolute values if necessary, choose positive a ∈ A\B and positive
65
b ∈ B\A. Then, WLOG, Pa ≥ Pb, so there exists p ∈ P s.t. pa ≥ b. Since
P ⊂ A, pa ∈ A and by convexity we have b ∈ A, a contradiction.
4 → 1: We prove ¬1 → ¬4. Suppose P is not prime, so there exists a, b not in
P with a ∧ b ∈ P . Form the Σ0 sets A = {x ∈ G : |x| ∧ |b| ∈ P}, B = {x ∈ G :
|x| ∧ |a| ∈ P}. Let x = a. By Lemma 2.23, |a| ∧ |b| ≤ |a ∧ b|. Since a ∧ b ∈ P ,
|a ∧ b| ∈ P , and |a| ∧ |b| ∈ P by convexity, and we have x ∈ A. On the other
hand, if x = a then x 6∈ B. Thus we have A 6⊆ B and the proof of B 6⊆ A is
similar. By the definitions of A and B and the convexity of P , it follows that
P ⊆ A and P ⊆ B. By Lemma 5.2, A and B are convex `-subgroups. Since
A and B contain P and are incomparable under inclusion, we have shown the
contrapositive, ¬1 → ¬4.
5.2 Values
Definition 5.4. Let g be a nonidentity element of an `-group G. A value of g,
denoted V (g), is a convex `-subgroup maximal w.r.t not containing g.
In general, values are not unique. The notation V (g) serves mainly to denote
that V is maximal with respect to excluding g.
The most natural way to describe the maximality of a value would be to
say V is a value of g if it is a convex subgroup not containing g such that for
every convex `-subgroup W ) V, g ∈ W . Unfortunately, this statement is Π11(V ).
66
We can describe the maximality, however, in terms of convex closure, which is
arithmetical.
Definition 5.5. Let g be a nonidentity element of the `-group G. A value of g is
a set V such that
• V is a convex `-subgroup (which is an arithmetical condition).
• g 6∈ V .
• ∀x 6∈ V ∃n(g ∈ CL(V ∪ {x})n), that is, the convex `-subgroup generated by
V ∪ {x} contains g.
Thus, whether or not a set V is a value of g is decidable in ACA0.
Theorem 5.6. (RCA0) Values are prime.
Proof. The proof is adapted from [4] : Let V be a value of g. Suppose there are
elements a, b > e such that a ∧ b = e and a 6∈ V . Form the set C = {x ∈ G :
|x| ∧ |g| ∈ V }. By Lemma 5.2, C is a convex `-subgroup. Since |x| ∧ |g| ≤ |x| and
V is convex, every element of V belongs to C.
Suppose, for a contradiction, that there exists an element y ∈ C\V . By the
definition of V as a value, g ∈ CL(V ∪ {y}). By Lemma 4.2, CL(V ∪ {y}) ⊂
CL(C) = C. Then g ∈ C so, by definition of C, |g|∧|g| = |g| ∈ V , a contradiction.
So, V = C.
Now form the set A = {x ∈ G : |x| ∧ b ∈ V }. By Lemma 5.2, A is a convex
`-subgroup. Since |x| ∧ b ≤ |x| and V is convex, every element of V belongs to
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A. Furthermore, we have assumed a 6∈ V , but a ∈ A. So A strictly contains
V , and since V is a value, there is an n s.t. g ∈ CL(V ∪ {a})n, the nth-stage
approximation to CL(V ∪ {a}). As above, by properties of the convex closure, A
must contain g, so |g| ∧ b ∈ V . This implies that b ∈ C, and since C = V , we
have b ∈ V . To recap, we have shown that (a ∨ b = e & a 6∈ V ) → b ∈ V , which
implies V is prime.
5.3 Existence of a Sequence of Values
As mentioned above, values are not generally unique. However, in an o-group,
the convex subgroups form a chain, so there is only one value for each element.
Furthermore, in an o-group, any two elements are comparable.
Definition 5.7. We say x � y if ∀n(|x|n < |y|), and say x ≈ y ↔ ¬(x �
y or y � x). Equivalently, x ≈ y ↔ ∃n([|x|n ≥ |y|&|y|n ≥ |x|]).
Lemma 5.8. (RCA0) If a, b are elements of an o-group, then exactly one of the
three relations a � b, a � b, a ≈ b must hold.
We will need the following:
Lemma 5.9. (RCA0) Let A be a subset of an o-group G such that ∀x ∈ A(x � y).
Then ∀x ∈ G∀n ∈ N(x ∈ CL(A)n → x � y).
Proof. In the context of o-groups, join is trivial to compute since a∨b = max≤G(a, b).
The result follows by an induction on the definition of CL(A). Since join is trivial,
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the only interesting case is the group operation, which is also resolved easily.
Lemma 5.10. (RCA0) Let G be an o-group. Suppose V is a value of g. Then
x ∈ V iff x � g.
Proof. Let x ∈ V . Suppose ¬(x � g). Then x ≈ g or x � g. Then, since V is
a convex `-subgroup, x generates g ∈ V , a contradiction. For the other direction,
suppose x � g and x 6∈ V . By Lemma 5.9, ∀h∀n(h ∈ CL(V ∪{x})n → h � g). In
particular, ∀n(g 6∈ CL(V ∪{x})). Along with the fact that x 6∈ V , this violates the
third condition of Definition 5.5, so V is not a value and we have our contradiction.
Corollary 5.11. (RCA0) Let G be an o-group. Let x, y be distinct elements of
G, and V (x), V (y) be values of x, y, respectively. Then x ≈ y ↔ x 6∈ V (y) & y 6∈
V (x).
Now, suppose one had a set K coding a sequence of values Vi for each
nonidentity gi ∈ G. By the corollary above, K can compute the set of pairs R =
{〈i, j〉 : gi ≈ gj} and thus also compute a set of archimedean class representatives
C = {i : (∀s <N i)〈i, s〉 6∈ R}. In [7], Solomon & Downey proved that the existence
of a set of archimedean class representatives for an abelian o-group is equivalent
to ACA0. Thus, the existence of a sequence of values implies ACA0.
Theorem 5.12. (RCA0) The following are equivalent.
1. For every o-group G the sequence of values Vi = V (gi) exists.
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2. ACA0
Corollary 5.13. (RCA0) The existence of a sequence of values for an `-group
implies ACA0.
Together with this corollary, the following theorem constitutes an equiva-
lence of the existence of a sequence of values to ACA0.
Theorem 5.14. (RCA0) ACA0 implies the existence of a sequence of values for
an `-group.
Proof. Let G be enumerated as g0, g1 . . .. Let A be (the code for) a finite subset of
G. Using the mechanism of convex closure defined earlier, we define a bounded-
quantifier relation Ψ(A, i, s) such that gi ∈ CL(A) ↔ ∃sΨ(A, i, s). Informally, s
represents a stage of the construction of CL(A) such that gi ∈ CL(A)s. In ACA0,
we may form the set S of pairs 〈A, i〉 such that A is a (code for a) finite set, and
∀s¬Ψ(A, i, s). We then define a function f(j, n) : N2 → {0, 1}. We define by
primitive recursion on n:
Case n=0: f(j, 0) =
1 if 〈{g0}, j〉 ∈ S
0 if 〈{g0}, j〉 6∈ S.
Case n+1: f(j, n + 1) =
1 if 〈({gk : k ≤ n & f(j, k) = 1} ∪ {gn+1}, j〉 ∈ S
0 if 〈({gk : k ≤ n & f(j, k) = 1} ∪ {gn+1}, j〉 6∈ S.
Claim. For each j, the set W = {gk : f(j, k) = 1} is a value of gj.
70
Proof. Each successive element gk is included or excluded by f based solely on
whether its inclusion would eventually cause gj to be generated. By Lemma 4.3,
the convex closure of a set of elements is unaffected by adding group inverses,
compositions, joins, and positive elements bounded above by a member of that
set. For example, if f(j, 5) = 1, and g6 = g−15 , then f(j, 6) = 1 because including
inverses will not cause any new elements (particularly gj) to enter the convex
closure. The other criteria for a convex `-subgroup are satisfied similarly. Thus
W is a convex `-subgroup. By the definition of f , gj 6∈ W . Also by the definition
of f , any element not in W is excluded specifically because including it would
generate gj.
5.4 Existence of a Sequence of Excluding Primes
Definition 5.15. An excluding prime for g, denoted P (g), is a prime convex
`-subgroup not containing g.
Since values are prime, a value V (g) is a maximal excluding prime for g.
However, it is easy to show that there are, in general, non-maximal excluding
primes. Take the o-group obtained by lexicographically ordering Z⊕ Z⊕ Z. The
element (1, 0, 0) is excluded by the convex `-subgroups generated by (0, 1, 0) and
(0, 0, 1), both of which are prime, as are all convex subgroups of an o-group.
However, the latter is a proper subgroup of the former.
71
Lemma 5.16. (RCA0). Let G be an abelian `-group, with G = {g0 = e, g1, . . .}.
Then there is a uniform sequence of infinite trees 〈T1, T2, . . .〉 such that ∀m ≥
1, f ∈ [Tm] ↔ f is the characteristic function of a prime subgroup not containing
gm.
Proof. For each σ ∈ 2<ω we write gi ∈ σ ↔ σ(i) = 1 and abuse this notation by
writing things like |gi| ∈ σ to mean σ(k) = 1, where gk = |gi|. For each σ ∈ 2<ω,
define
Sσ = {k < |σ| : gk ∈ σ or
∃i < |σ|(gi ∈ σ & (gi · gk = e or |gk| ≤ |gi|)) or
∃i, j < |σ|(gi, gj ∈ σ & (gk = gi · gj or gk = gi ∧ gj))}.
Note that τ ⊆ σ → Sτ ⊆ Sσ. We define the relations R,Q :
R(σ) := ∀i, j[(i < j < |σ| & gi ∧ gj = e) → (gi ∈ σ or gj ∈ σ)].
Q(σ) := ∀k < |σ|(gk ∈ Sσ → k ∈ σ).
Note that τ ⊆ σ → (R(σ) → R(τ) & Q(σ) → Q(τ)).We say σ is m-acceptable
if and only if R(σ) & Q(σ) & |gm| 6∈ σ. Then τ ⊆ σ & σ is m-acceptable →
τ is m-acceptable. For m ≥ 1, we define Tm = {σ ∈ 2<ω : σ is m-acceptable}.
Claim. If f ∈ [Tm] then f is the characteristic function of a prime subgroup not
containing gm.
72
Let f ∈ [Tm]. Then f(|gm|) = 0, since f � n is m-acceptable for all n. Since
Q(f � n) for each n, f codes a convex `-subgroup and since R(f � n) for each n,
this convex `-subgroup is prime.
If I can prove in RCA0 that Tm is infinite, WKL0 will prove that there is a
path, and hence a prime subgroup not containing gm.
Lemma 5.17. (RCA0) Every tree Tx, x ≥ 1, is infinite. Formally, ∀n(Tx has a
node of length n).
Proof. Suppose not. Fix x ≥ 1 and let n be N-least such that Tx has no node of
length n. If n is less than the index of |gx|, then we are done, since the string
1n satisfies Q, R and is x-acceptable, hence belongs to Tx. So assume n ≥ the
index of |gx|. Let Y = {〈i, j〉 : i < j < n & gi ∧ gj = e}. We will construct a
node σ of length n that is x-acceptable, so belongs to Tx. To satisfy R, σ must
contain at least one element from every pair in Y . Intuitively, it is easy to find a
convex `-subgroup excluding a nonidentity element – one may choose the identity
subgroup or consider a principal polar. The difficulty arises when we try to also
require it to be prime. Thus, our first priority is to choose one element from each
pair in Y . The next priority is to make sure that Q is satisfied.
Claim. (RCA0) If Y contains m pairs, there is a way to choose one element from
each pair so that our choices do not generate |gx|, in the sense of satisfying Q.
Suppose the m pairs of Y are written as 〈ai,0, ai,1〉, with 0 ≤ i ≤ m− 1. A
73
binary string of length m can then represent a choice of one element from each
pair. Given τ ∈ 2m, we form στ with |στ | = n by bounded Σ01 comprehension:
define
Sστ = {k : k < n & gk ∈ CL({ai,τ(i) : i < m)}}
and then let στ (k) = 1 ⇐⇒ k ∈ Sστ .
To prove the claim, we need to show that there is a τ ∈ 2m such that στ ∈ 2n
is x-acceptable. For τ with |τ | ≤ m, let gτ =∏|τ |−1
i=0 ai,τ(i). Then the meet of all
possible products with one factor from the first k pairs in Y is∧
τ∈2k
gτ . First we
prove by induction that, for k ≤ m,∧
τ∈2k
gτ = e. Case k = 1. We have one pair
a0,0, a0,1 such that a0,0 ∧ a0,1 = e. In this case, the only possibilies for gτ are a0,0
and a0,1, which clearly meet to the identity.
Suppose we have proved the result for k < m, and consider the case k + 1.
By the distributive law,
[(∧
τ∈2k
gτ ) · ak,0] =∧
τ∈2k
(gτ · ak,0) = (∧
τ∈2k+1:τ(k)=0
gτ ).
Similarly,
[(∧
τ∈2k
gτ ) · ak,1] = (∧
τ∈2k+1:τ(k)=1
gτ ).
Thus, ∧τ∈2k+1
gτ = [(∧
τ∈2k
gτ ) · ak,0] ∧ [(∧
τ∈2k
gτ ) · ak,1].
By distribution again, this equals (∧
τ∈2k
gτ ) · (ak,0 ∧ ak,1). The first factor is equal
to e by induction, and the second is equal to e by definition of Y . This proves
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that∧
τ∈2k+1
gτ = e. By induction,∧
τ∈2m
gτ = e.
Since G is abelian, it follows that (gτ )p = Πm−1
i=0 (ai,τ(i))p. By repeated appli-
cation of Corollary 2.10, we have∧
τ∈2m
(gτ )p = e.
Suppose, for a contradiction, that every way of choosing one element from
each pair in Y forced the generation of |gx|, that is, for all τ ∈ 2m, |gx| ∈ Sστ .
By Lemmas 4.4 and 4.5, there is a p ≥ 0 such that ∀τ ∈ 2m(|gx| ≤ (gτ )p). Thus
|gx| ≤∧
τ∈2m(gτ )p = e, a contradiction since gx 6= e. Therefore, there is at least
one τ which does not generate |gx|. This completes the proof of the claim. To
finish the proof of the lemma, fix any τ ∈ 2m such that |gx| 6∈ Sστ . Clearly στ is
x-acceptable, and |στ | = n, a contradiction. Therefore, each tree is infinite.
So, we have a uniform sequence of infinite trees such that a path through
Tm codes a prime convex `-subgroup not containing gm. What we really want,
however, is a set uniformly coding a sequence of excluding prime subgroups. We
accomplish this by coding all the trees Tm into a single tree T ∗ such that a path
through T ∗ uniformly computes a path through each Tm.
Theorem 5.18. (WKL0) Let e = g0, g1, . . . be an enumeration of the `-group
G. Then there is a set K such that ∀x ≥ 1, Kx is a prime subgroup of G not
containing gx.
Lemma 5.19. (RCA0) If 〈T1, . . .〉 is a sequence of infinite trees, then there is an
infinite tree T ∗ and a Σ00 function φ(f, i) s.t.∀f ∈ [T ∗](φ(f, i) is a path in Ti).
75
We define, by induction, the tree T ∗ and a labeling function l : T ∗ → N×2<ω.
The root node λ of T ∗ is labeled l(λ) = 〈1, λ〉. Induction step: If σ ∈ T ∗ and
l(σ) = 〈i, τ〉, then σ ∗ j ∈ T ∗ ↔ τ ∗ j ∈ Ti. Furthermore, we define
l(σ ∗ j) =
〈n + 1, λ〉 if i = 1
〈i− 1, τ ∗ k〉 if i > 1
where n = max{j : (∃σ ( σ)π1(l(σ)) = j} and τ , k have the property that if σ is
the longest proper substring of σ such that π1(l(σ)) = i−1, then l(σ) = 〈i−1, τ〉,
and σ ∗ k ⊆ σ. Here, π1 denotes the projection function on the first component.
(See Figure 5.1.) To prove that T ∗ is infinite, we describe a method of producing
nodes longer than a specified height. Let m > 1 be fixed. Since each Ti is infinite,
there exist σi ∈ Ti for i = 1, . . . ,m such that |σi| = m. If we run through the
construction substituting σi for Ti, we will be able to produce a node in T ∗ of
length greater than m.
Given {σi : 1 ≤ i ≤ m}, let f be the function which on input 〈σ, l(σ)〉
returns the pair 〈σ ∗ j, l(σ ∗ j)〉 as defined in the construction above, replacing Ti
with σi, or more precisely, the tree consisting of all strings contained in σi.
Suppose fk(〈1, λ〉) = 〈µ, 〈i, τ〉〉. Then f(〈µ, 〈i, τ〉〉) has two components.
The first component is found by looking in σi for the string τ . We extend µ
by the next bit of σi after τ , resulting in µ ∗ σi(|τ | + 1) The second component
76
has two cases. If i = 1, then the second component is 〈n + 1, λ〉, where n =
max{j : (∃σ ( µ)π1(l(σ)) = j}. If i > 1, then we look for the longest proper
substring σ ( µ whose label has first component i − 1. Because we are working
with the non-branching tree generated from σi instead of the whole of Ti, we know
that l(σ) = 〈i − 1, σi−1 � j〉 for some j. In this case, the second component is
〈i− 1, σi−1 � j + 1〉.
This construction “visits” T1, then T2, T1, then T3, T2, T1, etc. So, just given
σi ∈ Ti of length m we will be able to define at least (m−1)(m)2
iterates of f . Since
each iteration produces a longer node of T ∗, this is enough to produce a node
longer than m bits.
Claim. If f is a path in T ∗, then f computes a path in Ti for each i.
Proof. By primitive recursion, we can define a function g(i, j) such that g(i, j) is
the level in T ∗ coding the jth level of Ti. For any f ∈ [T ∗], l(f � g(i, j)) has the
form 〈i, τ〉, where |τ | = j. Thus, f(g(i, j)) is the jth bit of the path in Ti coded
by f , and the function hi(j) := f(g(i, j)) gives a path through Ti.
Now, RCA0 proves that T ∗ exists and that paths through Ti correspond to
prime subgroups of G not containing gi. WKL0 proves that T ∗ has a path, and we
have seen that such a path can uniformly compute paths through each Ti. Thus,
we have proved Theorem 5.18.
77
Initial Subtree of T !
!, "1, !#
"2, !#
0
"1, "0##
01
"3, !#
010
"2, "1##
0101
"1, "00##
01011
"4, !#
010110
"3, !#
011
"2, "1##
0111
"1, "01##
01111
"2, !#
1
"1, "1##
11
T2
T1
T1
Information Used
T1 T2 T3 T4
Fig. 5.1: Initial Subtree of T ∗
Chapter 6
Holland’s Embedding Theorem
The central object of study in this chapter is the following theorem:
Holland’s Embedding Theorem 2. Let G be a lattice ordered group. Then
there is an embedding of G into the group of order-preserving permutations of
some linear order.
At face value, this theorem purports both the existence of a linear order
and the group of order-preserving permutations of that linear order. For this to
be non-trivial, the linear order must of course be infinite. Moreover, the per-
mutation group is then a third-order object which we cannot define directly in
Z2. We carefully sidestep this obstacle by restating the theorem. Specifically,
using the purported linear order L, we show that there is a uniform way of
obtaining, for each g ∈ G, a specific function fg : L → L which is in fact
an order-preserving permutation, and verify all the desired properties of the
embedding without explicitly forming the range of the embedding as a set.
78
79
Holland’s Embedding Theorem 3. If G is an `-group, then there exists a
linear order L and a function f(g, l) : G × L → L such that for each g ∈ G, the
function f(g, l), abbreviated fg, is an order preserving bijection on L, and satisfies
the properties
• ∀l ∈ L(fe(l) = l).
• [∀l ∈ L(fg(l) = fh(l))] ↔ g = h.
• ∀g, h ∈ G, l ∈ L(fg∧h(l) = min{fg(l), fh(l)}).
• ∀g, h ∈ G, l ∈ L(fg∨h(l) = max{fg(l), fh(l)}).
• ∀g, h ∈ G, l ∈ L(fgh(l) = fh(fg(l))}.
The first two properties above indicate, respectively, that the group identity
induces the identity permutation, and that each element induces a distinct permu-
tation. The last three ensure that the embedding respects the `-group structure
of G – that is, that if one forms the meet, join, or group product of two elements,
then one gets the same permutation as though one had taken the meet, join, or
composition of the permutations induced by those elements individually.
6.1 Summary of Original Proof of Holland’s Theorem
1. Fix a sequence of values {Vg}, one for each g 6= e.
80
2. Since values are prime, the induced order on each collection of right cosets
G/Vg is linear.
3. The collections of cosets G/Vg may be ordered lexicographically to form a
linear order L.
4. Each element of G naturally induces an order-preserving permutation of L
by right multiplication.
5. The `-group structure of G naturally embeds into the `-group structure of
order-preserving permutations of L.
6. Proof is relatively constructive and can be done in RCA0 except for obtain-
ing a sequence of values Vg which, as we saw in Chapter 5, is equivalent to
ACA0.
There are two qualities of values that are used in the proof, namely, that
they are prime and that each element is excluded from some value – that is, there
is no use of values’ maximality except for proving that values are prime. This
observation motivated the definition of excluding primes in Chapter 5 and led to
a couple of questions, namely: “Is the existence a sequence of excluding primes
equivalent to something weaker than ACA0?” and “Can Holland’s Theorem be
proved using excluding primes instead of values?”
The answer to the first question is “Yes, but so far only for abelian `-groups”,
as was shown in Chapter 5. We now address the second question.
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6.2 Proof of Holland’s Theorem using excluding primes
Let G be an `-group. For g 6= e, let P (g) denote a prime subgroup not containing
g. Suppose we are given a set K = {〈x, y〉 : y ∈ P (x)} coding a sequence of
excluding primes. (If G is abelian, then by Theorem 5.18, K can be formed in
WKL0.) By Theorem 3.7, RCA0 is sufficient to establish the induced orders on the
sets of right cosets G/P (g) for each nonidentity g. Thus, given K we can form
K = {〈x, y〉 : y is the N-least element of a right coset in G/P (x)}.
Since P (g) is prime for each g, each column of K (representing cosets
G/P (g)) is linearly ordered under the induced order <G/P (g), by Theorem 5.3.
Thus, we may totally order K: 〈a, b〉 < 〈x, y〉 iff a <N x or a = x and b <G/P (x) y
in RCA0. The set K plays the role of the linear order L in the statement of
Holland’s Theorem.
Lemma 6.1. (RCA0) If P is a prime subgroup of an `-group G, then an element
g ∈ G induces an order-preserving permutation fg of G/P , defined by fg(Px) =
Pxg.
Proof. Since P is prime, G/P is totally ordered under the induced order. Let the
induced order be represented by �. Since fg(Pyg−1) = Py, it is clear that the
map is surjective. Suppose fg(Px) = fg(Py). Then Pxg = Pyg. It follows that
∃p ∈ P (x = py), so Px = Py, and we have shown fg is injective. Now, we show
it is order-preserving. Suppose Px � Py. By the definition of the induced order,
82
this means ∃p ∈ P (x ≤ py). Then xg ≤ pyg, so Pxg � Pyg.
Lemma 6.2. (RCA0) Given an `-group G and sequence of excluding primes K,
there exists a function f(g, 〈x, y〉) : G × K → K so that for all g ∈ G, f(g, k) is
an order-preserving permutation of K.
Proof. We define f(g, 〈x, y〉) using right multiplication: f(g, 〈x, y〉) = 〈x, z〉,
where z is the least coset representative of yg in Kx∼= G/P (x). By Lemma
6.1, this mapping is a bijection of each column Kx which preserves each induced
order <G/P (x). Thus fg preserves the order on K.
Lemma 6.3. (RCA0) The function f(g, k) respects the `-group structure of G,
and the only element which induces the identity permutation of K is the identity
of G.
Proof. We need to establish the following:
1. ∀g, h ∈ G(fg ◦ fh = fhg).
2. ∀g ∈ G(fg = idK ↔ g = e).
3. ∀g, h ∈ G(fg∧h = fg ∧ fh = min{fg, fh}).
4. ∀g, h ∈ G(fg∨h = fg ∨ fh = max{fg, fh}).
(1) Follows directly from the definition of f(g, k) by right multiplication.
(2) It is clear that fe is the identity on K. On the other hand, suppose g 6= e.
83
Then we have a column in K corresponding to G/P (g). Since g 6∈ P (g), fg per-
mutes Kg nontrivially, so cannot be the identity on K.
(3) By Theorem 3.8, for any convex subgroup C, C(g ∧ h) = Cg ∧ Ch. In
the case that C is prime, the induced order is linear so the meet is the mini-
mum. Let k = 〈x, y〉 ∈ K. fg∧h(〈x, y〉) = 〈x, y(g ∧ h)∗〉 = 〈x, (yg ∧ yh)∗〉 =
〈x, minG/P (x){yg, yh}∗〉 = minK{fg(k), fh(k)}. The asterisk ∗ reminds us that
technically these equalities use the least coset representative, e.g., yg∗ is the N-
least coset representative of P (x)yg.
(4) Similar to (3).
Note: the last two criteria feature a consequence of the standard lattice
order on the group of automorphisms of a linear order as mentioned in Example
1.8.
While we do not explicitly form the set of automorphisms of the linear
order K, we produce a function f(g, k) which, practically speaking, provides an
`-embedding of G into Aut(K). We have:
Theorem 6.4. (RCA0 + Existence of a sequence of excluding primes) ` Holland’s
Theorem.
By Theorem 5.18, we then have:
Corollary 6.5. WKL0 ` Holland’s Embedding Theorem for abelian `-groups.
This Corollary is certainly a successful result – the standard assumption of
84
a sequence of values requires ACA0 even for abelian `-groups and, in that con-
text, the strictly weaker assumption of a sequence of excluding primes suffices for
the proof of Holland’s Theorem. Without having a reversal, however, this raises
another question: “Might Holland’s Theorem be proven by still weaker assump-
tions?”
6.3 A Reversal
Theorem 6.6. (RCA0) Let G be an `-group. The following are equivalent:
1. Holland’s Theorem
2. The existence of a sequence of prime subgroups P (g) s.t. ∀g 6= e(g 6∈ P (g)).
Proof. We have just seen that 2 → 1. We now show that 1 → 2. If G is an `-group
then by Holland’s Theorem there is a linear order L and a function f(g, l) : L →
L satisfying all the properties mentioned in the revised statement of Holland’s
Embedding Theorem for Z2. Suppose we are given such G, L, f . Define Fl = {g ∈
G : f(g, l) = l}, the set of g which “fix” l. It is easy to check that for each l, Fl is
a prime subgroup of G.
• e ∈ Fl because fe = idL.
• x ∈ Fl ↔ x−1 ∈ Fl.
• x, y ∈ Fl → xy ∈ Fl, since fxy(l) = fy(fx(l)) = fy(l) = l.
85
• x ∈ Fl → x∨e ∈ Fl, since fx∨e(l) = max{fx(l), fe(l)} = l (here we are using
criterion 2.22).
• Suppose e ≤ x ≤ g and g ∈ Fl. Then fe(l) ≤ fx(l) ≤ fg(l), so l ≤ fx(l) ≤ l
and x ∈ Fl. Hence Fl is a convex `-subgroup.
• Suppose a ∧ b = e. Then fa∧b(l) = fe(l) = l. But we also have fa∧b(l) =
min{fa(l), fb(l)}, so either a ∈ Fl or b ∈ Fl, and therefore Fl is a prime
subgroup.
Furthermore, for each g 6= e, fg is nontrivial, so there is an l so that g 6∈ Fl.
Let l be the N-least such that f(g, l) 6= l. Then g 6∈ Fl.
This is an effective way of finding l s.t. g 6∈ Fl. Since Fl has a quantifier-free
definition, given L we can form in RCA0 a set K = {〈x, y〉} such that Kx codes a
prime subgroup not containing gx for x ≥ 1.
Bibliography
[1] Marlow Anderson and Todd Feil. Lattice-Ordered Groups: an introduction. D.Reidel Publishing Company, Dordrecht, Holland, 1988.
[2] R.G. Downey and Stuart A. Kurtz. “Recursion Theory and Ordered Groups”.Annals of Pure and Applied Logic, 32:137–151, 1986.
[3] K. Hatzikiriakou and S.G. Simpson. “WKL0 and Orderings of CountableAbelian Groups”. Contemporary Mathematics, 106:177–180.
[4] V.M. Kopytov and N.Ya. Medvedev. The Theory of Lattice-Ordered Groups,pages 1–55. Kluwer, Dordrecht, The Netherlands, 1994.
[5] Stephen G. Simpson. Subsystems of Second Order Arithmetic. Springer-Verlag,Berlin, Germany, 1999.
[6] D.R. Solomon. “Reverse Mathematics and Fully Ordered Groups”. NotreDame Journal of Formal Logic, 39(2):157–189, 1998.
[7] D.R. Solomon and Rod Downey. “Reverse Mathematics, Archimedean Classes,and Hahn’s Theorem”. In Stephen Simpson, editor, Reverse Mathematics2001, pages 147,163. AK Peters, 2005.
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