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  • 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 +, ·,

  • 3

    to prove Ŝ 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 Π 1 1-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 König’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

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