Recursively categorical linear orderings

Abstract

A recursive structure C is recursively categorical if every recursive structure i' isomorphic to C is recursively isomorphic to (i. We classify the recursively categorical linear orderings as precisely those recursive linear orderings L which have only finitely many elements with an immediate successor. A structure (E over a recursive language e is said to be recursive if A, the domain of i, is recursive and there is a uniform effective procedure for deciding whether 6& satisfies any atomic formula T(ao, . . , an) with ao,..., a, C A. Two recursive structures 6i and 6W' are recursively isomorphic, denoted 6i d r 6W, if there is a partial recursive isomorphismf from 6i onto 6'. We say a recursive structure 2i is recursively categorical if any recursive structure 6' isomorphic to 6i is recursively isomorphic to 6i. The notion of recursive categoricity was first defined for groups by Mal'cev [3] and is referred to in the Soviet literature as autostability. In recent years, the notion of recursive categoricity has been studied widely in the literature. General semantic conditions for when decidable models are recursively categorical was given by Nurtazin [4] and similar results were found by Ash and Nerode for models in which one can effectively decide all El-formulas. The recursively categorical structures for various theories have been classified, including: Boolean algebras, independently by Goncharov [2] and La Roche [5] (the first full proof in the literature is found in Remmel [7]); Boolean algebras with additional predicates for atoms or atoms and atomless elements by Remmel [8], [9]; Abelian p-groups by Smith [10]; and decidable dense two-dimensional partial orderings by Remmel and Manaster [4]. The main result of this paper is the classification of the recursively categorical recursive linear orderings as precisely those recursive linear orderings L which have only finitely many elements which have immediate successors. We end the paper with a brief discussion of the relationship between our classification and the known classification of recursively categorical Boolean algebras. Finally we should note that Theorem 1 to follow does not follow from the general work of Nurtazin or Ash and Nerode since Goncharov [2] has constructed recursive linear orderings K such that one cannot effectively decide the El-formulas in any recursive linear ordering isomorphic to K. Given a linear ordering L = , we say the pair (x, y) is a successivity of L, written x -*L Y, if x <L y and there is no z E A such that x <L Z <L Y. We let Received by the editors September 29, 1980. 1980 Mathematics Subject Classification. Primary 02F27; Secondary 06A05. Partially supported by NSF Grant MCS79-03406. ( 1981 American Mathematical Society 0002-9939/81 /0000-0485/$02.2 5