Abstract Cohesive powersof computable structures are effective analogs of ultrapowers, where cohesive sets play the role of ultrafilters. Let $\omega $ , $\zeta $ , and $\eta $ denote the respective order-types of the natural numbers, the integers, and the rationals when thought of as linear orders. We investigate the cohesive powers of computable linear orders, with special emphasis on computable copies of $\omega $ . If $\mathcal {L}$ is a computable copy of $\omega $ that is computably isomorphic to the usual presentation of $\omega $ , then every cohesive power of $\mathcal {L}$ has order-type $\omega + \zeta \eta $ . However, there are computable copies of $\omega $ , necessarily not computably isomorphic to the usual presentation, having cohesive powers not elementarily equivalent to $\omega + \zeta \eta $ . For example, we show that there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \eta $ . Our most general result is that if $X \subseteq \mathbb {N} \setminus \{0\}$ is a Boolean combination of $\Sigma _2$ sets, thought of as a set of finite order-types, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ , where $\boldsymbol {\sigma }(X \cup \{\omega + \zeta \eta + \omega ^*\})$ denotes the shuffle of the order-types inXand the order-type $\omega + \zeta \eta + \omega ^*$ . Furthermore, ifXis finite and non-empty, then there is a computable copy of $\omega $ with a cohesive power of order-type $\omega + \boldsymbol {\sigma }(X)$ .

Abstract We improve on and generalize a 1960 result of Maltsev. For a field F, we denote by $H(F)$ the Heisenberg group with entries in F. Maltsev showed that there is a copy of F defined in $H(F)$ , using existential formulas with an arbitrary non-commuting pair of elements as parameters. We show that F is interpreted in $H(F)$ using computable $\Sigma _1$ formulas with no parameters. We give two proofs. The first is an existence proof, relying on a result of Harrison-Trainor, Melnikov, R. Miller, and Montalbán. This proof allows the possibility that the elements of F are represented by tuples in $H(F)$ of no fixed arity. The second proof is direct, giving explicit finitary existential formulas that define the interpretation, with elements of F represented by triples in $H(F)$ . Looking at what was used to arrive at this parameter-free interpretation of F in $H(F)$ , we give general conditions sufficient to eliminate parameters from interpretations.

Abstract This paper calculates, in a precise way. the complexity of the index sets for three classes of computable structures: the class of structures of Scott rank , the class , of structures of Scott rank , and the class K of all structures of non-computable Scott rank. We show that I(K) is m-complete is m-complete relative to Kleene's and is m-complete relative to .

We consider only computable languages, and countable structures, with universe a subset of ω, which we think of as a set of constants. We identify sentences with their Gödel numbers. Thus, for a structure , the complete (elementary) diagram, Dc( ), and the atomic diagram, D( ), are subsets of ω. We classify formulas as usual. A formula is both Σ0 and Π0 if it is open. For n > 0, a formula, in prenex normal form, is Σn, or Πn, if it has n blocks of like quantifiers, beginning with ∃, or ∀. For a formula θ, in prenex normal form, we let neg(θ) denote the dual formula that is logically equivalent to ¬θ—if θ is Σn, then neg(θ) is Πn, and vice versa.

Abstract The relationship between ideals I of Turing degrees and groups of I-recursive automorphisms of the ordering on rationals is studied. We discuss the differences between such groups and the group of all automorphisms, prove that the isomorphism type of such a group completely defines the ideal I, and outline a general correspondence between principal ideals of Turing degrees and the first-order properties of such groups.