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Reverse Mathematics on Lattice Ordered Groups

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Reverse Mathematics on Lattice Ordered Groups 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 T hm to study. 2. Look at a textbook proof of T hm and find a set S of axioms for Z2 that are suitable for the proof of T hm. 3. See if it is possible to prove the axioms of S using T hm. (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 T hm, this tells us that S is the weakest axiom system capable of proving T hm. (If a strictly weaker system S0 proved T hm, S0 would then imply S, an impossibility!) In this case we have a proof of T hm from S and a reversal of T hm to S, and we say the two are equivalent over RCA0. If, on the other hand, attempts to prove S from T hm aren’t working out, we look for a new proof of T hm that uses weaker axioms Sˆ and repeat the process, trying 3 to prove Sˆ using T hm. 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. 1 In order of increasing strength, they are: RCA0, WKL0, ACA0, ATR0, and Π1-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 0 comprehension scheme is limited to ∆1 formulas. (We also include axioms allow- 0 ing Σ1 induction.) The next strongest, WKL0, consists of all the axioms of RCA0 plus the Weak K¨onig’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.
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