Recursive Linear Orders with Incomplete Successivities

Abstract

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.

Every Low Boolean Algebra is Isomorphic to a Recursive One

“…Nevertheless, Jockusch and Soare [11] used a kind of diagonal argument (rather than a coding argument) to show that there are low linear orderings not isomorphic to recursive ones. Jockusch and Soare's new technique has found several applications (see [6]). …”

Degree spectra of the successor relation of computable linear orderings

“…In fact, in this case L is computably categorical ( [5], [11]), that is, for every computable copy M of L, there is a computable isomorphism from L to M . Downey and Moses [4] constructed the other known singleton example: a linear ordering L having a successor relation with degree spectrum…”

“…This example is an immediate consequence of the following theorem. [4]). For any non-computable c.e.…”

On the Complexity of the Successivity Relation in Computable Linear Orderings

“…In this paper, we solve a long-standing open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees.…”