ZPrtsehr. f. rutitt~. Logik u i d Cruiidlagen d . bfath. Ba. 32, s. 461-412 ( 1~8 6 ) RELATIOXS ISTRIKSICALLY RECURSIVE IN LINEAR ORDERS by MICHAEL MOSES in Macomb, Illinois (U.S.A.)')A recursively enumerable (r.e.) relation R on a recursive structure 8 is said to be iirtrinsically r.e. on 8 if every isomorphism of 3 with a recursive structure carries R t o an r.e. relation. This concept was introduced in [l] by C . J. ASH and A. NERODE.It was shown in that paper that R is intrinsically r.e. if and only if it is formally r.e., where a relation is defined to be formally r.e. on 8 if it is equivalent in 8 to an r.e. disjunction V pn(ii, 3 ) of existential formulae with finitely many parameters. The rehult is proved under an "extra decidability assumption" that there is a recursive procedure for determining, given a n existential formula q,,(g, z) and elements a of 8, whether the implication pi(& 3) -+ R(3) is true in 8. It has been shown (GONCHAROV [ 2 ] , MANASSE [4]) that without this assumption formally r.e. is strictly stronger than intrinsically r.e. n < oA relation is intrinsically recursive if both it and its complement are intrinsically r.e.Relations that are intrinsically recursive have been considered in particular cases. I n [l], ASH and NERODE showed that the successivity relation S ( x , y ) and the block * ) These results \\-ere presented at the Conference on Iiogic imd Computation in 1984, held at :30* JIonash University, llelbo~trnc, Australia.

in Wellington (New Zealand)') I n [ l j DUSHNIK and MILLER prove that every countably infinite linear oder has a non-trivial self-embedding. The first part of their argument considers linear orders with an interval of type w or GO*. Mapping x to its immediate successor (resp. predecessor) if x is in this interval and the interval is of type w (resp. o*), and to itself otherwise, produces the required non-trivial self-embedding. The existence of an interval of type o* + w would similarly guarantee a non-trivial automorphism.I n [a] KIERSTEAD defines the maps described above to be fairly trivial in the sense that they are non-trivial but map every element x to an element y with [x, y ] finite. He defines a self-embedding to be strongly %on-trivial if it is neither trivial nor fairly trivial.Tbe above-mentioned argument from DUSHNIK and MJLLER [l] shows that every recursive linear order (i.e. with universe N and < a recursive relation) with a recursive subset consisting only of intervals of type o, or only of intervals of type w*, has a fairly trivial 17, self-embedding. We use the fact that the successor relation S(a, 6 ) defined byis 17, in every recursive linear order. As before, if the recursive subset consisted only of intervals of type w* + w, we are guaranteed a fairly trivial IT, automorphism.It follows that every recursive discrete linear order has a fairly trivialn, automorphism. (A linear order is discrete if every element has both an immediate predecessor and an immediate successor, or equivalently, if the linear order is of type (o* + w ) * z for some order type t.)This contrasts with the main result of this paper:Every reoursive discrete linear order has a recursive copy with no strongly non-trivial 17, self-embedding.This result proves the conjecture, stated in KIERSTEAD [4], that there is a recursive linear order of type (a* + w) * 9 with no strongly non-trivial17, automorphism.Notice that if f is it non-trivial self-embedding of a linear order, then there is an element 2 with x =# f ( x ) and with {x, f ( x ) , f 2 ( x ) , f3(x), . . .} defining a suborder of type o or w*. If f is strongly non-trivial, then there is such a suborder with the elements all in separate blocks. (A block is an equivalence class cF(a) = {b : [a, b] is finite} ; see ROSENSTEIN [6].) A choice set for a linear order is a subset consisting of precisely I ) The second author was supported by a VUW Post-Doctoral Fellowship in the Department of Mathematics.

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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