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EQUIVALENCE BETWEEN FRAÏSSÉ’S CONJECTURE AND JULLIEN’S THEOREM.
, 2004
"... We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a th ..."
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Cited by 7 (4 self)
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We say that a linear ordering L is extendible if every partial ordering that does not embed L can be extended to a linear ordering which does not embed L either. Jullien’s theorem is a complete classification of the countable extendible linear orderings. Fraïssé’s conjecture, which is actually a theorem, is the statement that says that the class of countable linear ordering, quasiordered by the relation of embeddablity, contains no infinite descending chain and no infinite antichain. In this paper we study the strength of these two theorems from the viewpoint of Reverse Mathematics and Effective Mathematics. As a result of our analysis we get that they are equivalent over the basic system of RCA0+Σ1 1IND. We also prove that Fraïssé’s conjecture is equivalent, over RCA0, to two other interesting statements. One that says that the class of well founded labeled trees, with labels from {+, −}, and with a very natural order relation, is well quasiordered. The other statement says that every linear ordering which does not contain a copy of the rationals is equimorphic to a finite sum of indecomposable linear orderings. While studying the proof theoretic strength of Jullien’s theorem, we prove the extendibility of many linear orderings, including ω2 and η, using just ATR0+Σ1 1IND. Moreover, for all these linear orderings, L, we prove that any partial ordering, P, which does not embed L has a linearization, hyperarithmetic (or equivalently ∆1 1) in P ⊕ L, which does not embed L.
UP TO EQUIMORPHISM, HYPERARITHMETIC IS RECURSIVE
"... Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. On the way to our main result we prove that a linear ordering has Hausdorff rank less than ω CK 1 if and only if it is equimorphic to a ..."
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Cited by 6 (3 self)
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Two linear orderings are equimorphic if each can be embedded into the other. We prove that every hyperarithmetic linear ordering is equimorphic to a recursive one. On the way to our main result we prove that a linear ordering has Hausdorff rank less than ω CK 1 if and only if it is equimorphic to a recursive one. As a corollary of our proof we prove that given a recursive ordinal α, the partial ordering of equimorphism types of linear orderings of Hausdorff rank at most α ordered by embeddablity, is recursively presentable.
COMPUTABLE LINEARIZATIONS OF WELLPARTIALORDERINGS
, 2007
"... We analyze results on wellpartialorderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every wellpartialorder has a linearization of maximal order type. We show that such a lineariz ..."
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We analyze results on wellpartialorderings from the viewpoint of computability theory, and we answer a question posed by Diana Schmidt. We obtain the following results. De Jongh and Parikh showed that every wellpartialorder has a linearization of maximal order type. We show that such a linearization can be found computably. We also show that the process of finding such a linearization is not computably uniform, not even hyperarithmetically.
OPEN QUESTIONS IN REVERSE MATHEMATICS
, 2010
"... The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discu ..."
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Cited by 4 (0 self)
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The objective of this paper is to provide a source of open questions in reverse mathematics and to point to areas where there could be interesting developments. The questions I discuss are mostly known and come from somewhere in the literature. My objective was to compile them in one place and discuss them in the context of related work. The list is definitely not comprehensive, and my
Slicing the Truth: On the Computability Theoretic and Reverse Mathematical Analysis of
 Institute for Mathematical Sciences, National University of Singapore, World Scientific
"... In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these ar ..."
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In this expository article, we discuss two closely related approaches to studying the relative strength of mathematical principles: computable mathematics and reverse mathematics. Drawing our examples from combinatorics and model theory, we explore a variety of phenomena and techniques in these areas. We begin with variations on König’s Lemma, and give an introduction to reverse mathematics and related parts of computability theory. We then focus on Ramsey’s Theorem as a case study in the computability theoretic and reverse mathematical analysis of combinatorial principles. We study Ramsey’s Theorem for Pairs (RT22) in detail, focusing on fundamental tools such as stability, cohesiveness, and Mathias forcing; and on combinatorial and model theoretic consequences of RT22. We also discuss the important theme of conservativity results. In the final section, we explore several topics that reveal various aspects of computable mathematics and reverse mathematics. An appendix contains a proof of Liu’s recent result that RT22 does not imply Weak König’s Lemma. There are exercises and open questions throughout the article. ∗Please send any corrections to
COUNTABLY COMPLEMENTABLE LINEAR ORDERINGS
, 2006
"... We say that a countable linear ordering L is countably complementable if there exists a linear ordering L, possibly uncountable, such that for any countable linear ordering B, L does not embed into B if and only if B embeds into L. We characterize the linear orderings which are countably complementa ..."
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We say that a countable linear ordering L is countably complementable if there exists a linear ordering L, possibly uncountable, such that for any countable linear ordering B, L does not embed into B if and only if B embeds into L. We characterize the linear orderings which are countably complementable. We also show that this property is equivalent to the countable version of the finitely faithful extension property introduced by Hagendorf. Using similar methods and introducing the notion of weakly countably complementable linear orderings, we answer a question posed by Rosenstein and prove the countable case of a conjecture of Hagendorf, namely, that every countable linear ordering satisfies the countable version of the totally faithful extension property.
ON THE EQUIMORPHISM TYPES OF LINEAR ORDERINGS.
, 2006
"... A linear ordering (also known as total ordering) embeds into another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to be equimorphic if they can be embedded in each other. This is an equivalence relation, and we call the equivalence classes equimorphism types. ..."
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A linear ordering (also known as total ordering) embeds into another linear ordering if it is isomorphic to a subset of it. Two linear orderings are said to be equimorphic if they can be embedded in each other. This is an equivalence relation, and we call the equivalence classes equimorphism types. We
INTERVAL ORDERS AND REVERSE MATHEMATICS
, 2006
"... Abstract. We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order i ..."
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Abstract. We study the reverse mathematics of interval orders. We establish the logical strength of the implications between various definitions of the notion of interval order. We also consider the strength of different versions of the characterization theorem for interval orders: a partial order is an interval order if and only if it does not contain 2⊕2. We also study proper interval orders and their characterization theorem: a partial order is a proper interval order if and only if it contains neither 2⊕2 nor 3⊕1. Interval orders are a particular kind of partial orders which occur quite naturally in many different areas and have been widely studied. A partial order P = (P, ≤P) is an interval order if the elements of P can be mapped to nonempty intervals of a linear order L so that p <P q holds iff every element of the interval associated to p precedes every element of the interval associated to q. The linear order L and the map from P to intervals are called an interval representation of P. The basic reference on interval orders is Fishburn’s monograph [9]. The name “interval order ” was introduced by Fishburn ([8]), although the notion
COMPUTABLY ENUMERABLE PARTIAL ORDERS
"... Abstract. We study the degree spectra and reversemathematical applications of computably enumerable and cocomputably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the latter is strictly s ..."
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Abstract. We study the degree spectra and reversemathematical applications of computably enumerable and cocomputably enumerable partial orders. We formulate versions of the chain/antichain principle and ascending/descending sequence principle for such orders, and show that the latter is strictly stronger than the latter. We then show that every ∅ ′computable structure (or even just of c.e. degree) has the same degree spectrum as some computably enumerable (coc.e.) partial order, and hence that there is a c.e. (coc.e.) partial order with spectrum equal to the set of nonzero degrees. 1.
LINEAR EXTENSIONS OF ORDERS INVARIANT UNDER ABELIAN GROUP ACTIONS
"... Abstract. Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when th ..."
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Abstract. Let G be an abelian group acting on a set X, and suppose that no element of G has any finite orbit of size greater than one. We show that every partial order on X invariant under G extends to a linear order on X also invariant under G. We then discuss extensions to linear preorders when the orbit condition is not met, and show that for any abelian group acting on a set X, there is a Ginvariant linear preorder ≤ on the powerset PX such that if A is a proper subset of B, then A < B (i.e., A ≤ B but not B ≤ A). 1. Linear orders Szpilrajn’s Theorem [9] (proved independently by a number of others) says that given the Axiom of Choice, any partial order can be extended to a linear order, where a relation ≤ ∗ extends a relation ≤ provided that x ≤ y implies x ≤ ∗ y. There has been much work on what properties of the partial order can be preserved in the linear order (e.g., [1, 5, 11]) but the preservation of symmetry under a group acting on partially ordered set appears to have been neglected. Suppose a group G acts on a partially ordered set (X, ≤) and the order is Ginvariant, where a relation ≼ on X is Ginvariant provided that for all g ∈ G and x, y ∈ X, we have x ≼ y if and only if gx ≼ gy. It is natural to ask about the conditions under which ≤ extends to a Ginvariant linear order. We shall answer this question in the case where G is abelian. Then we will discuss extensions where the condition is not met. In the latter case, there will still be an extension to a linear preorder (total, reflexive and transitive relation) that preserves strict comparisons. Finally, we will apply the results to show that for any abelian group G acting on a set X, there is a Ginvariant linear preorder on the powerset PX preserving strict set inclusion. Throughout the paper we will assume the Axiom of Choice and all our proofs will be elementary and selfcontained. An orbit of g ∈ G is any set of the form {gnx: n ∈ Z}. An obvious necessary condition for X to have a Ginvariant linear order is that no element of G have any finite orbit of size greater than 1. Surprisingly, this