On Choice Sets and Strongly Non‐Trivial Self‐Embeddings of Recursive Linear Orders

Downey, Rodney G.; Moses, Michael

doi:10.1002/malq.19890350307

Cited by 21 publications

(13 citation statements)

References 3 publications

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“…We note that this result, which was obtained by the author via the construction of a counter example (linear ordering) in [Har14], solves a question mentioned by several authors including Fellner [Fel76], Lerman and Rosenstein [LR82] and Downey and Moses [DM89]. We also show that if computable A is (1) strongly η-like or is (2) η-like but contains no strongly η-like interval, then A has order type τ determined by a 0 -limitwise monotonic function F .…”

Section: Introductionsupporting

confidence: 67%

On limitwise monotonicity and maximal block functions

Harris

2015

COM

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General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Abstract. We prove the existence of a limitwise monotonic function g : N → N \ {0} such that, for any Π 0 1 function f : N → N \ {0}, Ran f = Ran g. Relativising this result we deduce the existence of an η-like computable linear ordering A such that, for any Π 0 2 function F : Q → N \ {0}, and η-like B of order type { F (q) | q ∈ Q }, B A . We prove directly that, for any computable A which is either (i) strongly η-like or (ii) η-like with no strongly η-like interval, there exists 0 -limitwise monotonic G : Q → N \ {0} such that A has order type { G(q) | q ∈ Q }. In so doing we provide an alternative proof to the fact that, for every η-like computable linear ordering A with no strongly η-like interval, there exists computable B ∼ = A with Π 0 1 block relation. We also use our results to prove the existence of an η-like computable linear ordering which is ∆ 0 3 categorical but not ∆ 0 2 categorical.

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Section: Introductionsupporting

confidence: 67%

On limitwise monotonicity and maximal block functions

Harris

2015

COM

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“…Remove all labels > k . 2. If m < k and there is no successivity labelled m , introduce one, placing it immediately to the right of Ls_x.…”

Section: Intrinsically Complete Successivitiesmentioning

confidence: 99%

“…This basic construction appears also in proofs of the main results in Watnick [15] and Downey and Moses [2] where it is modified to prove other results. Consequently we only sketch this basic construction, devoting our proof to a description of the modification necessary to arrange that the recursive linear order constructed has incomplete successivities.…”

Section: Discrete Linear Ordersmentioning

confidence: 99%

“…Consequently we only sketch this basic construction, devoting our proof to a description of the modification necessary to arrange that the recursive linear order constructed has incomplete successivities. We do present all the formal definitions necessary for the basic construction; Downey and Moses [2] offers a rigorous verification of its details. Theorem 2.…”

Section: Discrete Linear Ordersmentioning

confidence: 99%

“…We define intervals 1(a) for each node o and write I (a, s) for the version defined at stage s. If i e B and a is the unique node of length i that corresponds to a correct guess, then 7(rr) will be an interval of type co* + co. Every other 1(a) will be finite (or empty) and will become part of some 1(a) of type co*+co. For a more leisurely account of this basic construction see Theorem 1 of Downey and Moses [2].…”

Section: Discrete Linear Ordersmentioning

confidence: 99%

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Recursive Linear Orders with Incomplete Successivities

Downey¹,

Moses²

1991

Transactions of the American Mathematical Society

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Abstract. A recursive linear order is said to have intrinsically complete successivities if, in every recursive copy, the successivities form a complete set. We show (Theorem 1) that there is a recursive linear order with intrinsically complete successivities but (Theorem 2) that this cannot be a discrete linear oder. We investigate the related issues of intrinsically non-low and non-semilow successivities in discrete linear orders. We show also (Theorem 3) that no recursive linear order has intrinsically w «-complete successivities. IntroductionIn an addendum to [10] Remmel suggests that every recursive Boolean algebra with infinitely many atoms has a recursive copy whose set of atoms is incomplete. The result remains a conjecture. The corresponding result for linear orders is that every recursive linear oder has a recursive copy whose set of successivities is incomplete. (A successivity is a pair of adjacent elements.) We show in Theorem 1 that this is not true. Our proof uses a construction involving the novel idea of "separators" from Jockusch and Soare [8]. From initial wayward attempts to prove the converse to Theorem 1, we were able to salvage Theorem 2: every discrete linear oder has a recursive copy whose set of successivities is incomplete, and Theorem 3: every recursive linear order has a recursive copy whose set of successivities is wrf-incomplete. Theorems 1, 2, and 3 are presented in § §1, 2, and 3, respectively. In §2 we also present three results noting some of the peculiarities of discrete linear orders: there is a discrete recursive linear order none of whose recursive copies has low successivities; every discrete recursive linear order has a recursive copy with semi-low successivities; and every semi-low n, discrete linear oder has a recursive copy.Our terminology and notation are as presented in Soare [14] for general recursion theory and Rosenstein [ 12] for recursive linear orders. A linear order is discrete if every element has an immediate predecessor and an immediate successor. It is recursive if its universe is co (equivalently an r.e. subset of co) and it has a recursive order relation. We use 21, £ and in Theorem 1, s/ , 3? to 1980 Mathematics Subject Classification (1985. Primary 06A05, 03D45.The authors wish to express their thanks to the referee for many helpful suggestions.

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Automorphisms ofη-like computable linear orderings and Kierstead's conjecture

2016

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Abstract. We develop an approach to the longstanding conjecture of H.A. Kierstead concerning the character of strongly nontrivial automorphisms of computable linear orderings. Our main result is that for any η-like computable linear ordering B, such that B has no interval of order type η, and such that the order type of B is determined by a 0 ′ -limitwise monotonic maximal block function, there exists computable L ∼ = B such that L has no nontrivial Π 0 1 automorphism.

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On Choice Sets and Strongly Non‐Trivial Self‐Embeddings of Recursive Linear Orders

Cited by 21 publications

References 3 publications

On limitwise monotonicity and maximal block functions

On limitwise monotonicity and maximal block functions

Recursive Linear Orders with Incomplete Successivities

Automorphisms ofη-like computable linear orderings and Kierstead's conjecture

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