Polynomial Time Categoricity and Linear Orderings

Abstract

Let ℒ = < {c_i}_i∈S, {R_i}_i∈T, {f_i}_i∈U> be a recursive language, i.e. assume that S, T and U are initial segments of the natural numbers N = (0, 1, 2, ,…), c_i is a constant symbol for each i ∈ S, and there are partial recursive functions t and u such that for all i ∈ T, R_i is a t(i)-ary relation symbol and for all j ∈ U, f is a u(i)-ary function symbol. A structure A = <A, {c_i ^A}_i∈S, {R_i ^A}_i∈T, {f_i ^A}_i∈U> is said to be a recursive structure if A, the universe of A, is a recursive subset of N, R_i ^A is a recursive relation for each i ∈ T, and f_i ^A is a partial recursive function from A^u(i) into A for each i ∈ U. Two recursive structures A and A′ over ℒ are recursively isomorphic, denoted by A ≈ _r A′, if there is a partial recursive function f which maps A onto A’ which is an ℒ-isomorphism from A onto A′. We say that a recursive structure A over ℒ is recursively categorical if any recursive structure A′ over ℒ which is isomorphic to A is recursively isomorphic to A. The notion of a recursively categorical structure was first defined by Mal’cev [M] and, in the Russian literature, such structures are called autostable. Recursively categorical structures have been widely studied in the literature of recursive algebra and recursive model theory. For example, general semantic conditions for when a decidable model is recursively categorical were given by Nurtazin [Nu] and similar results were found by Ash and Nerode [AN] for models in which one can effectively decide all Σ₁-formulas. Recursively categorical structures for various theories have been classified: Boolean algebras independently by Goncharov [Go] and La Roche [L], linear orderings independently by Dzgoev [GO] and Remmel [R], Abelian p-groups by Smith [S], and decidable dense two-dimensional partial orderings by Manaster and Remmel.