Valentina Harizanov

Let be an infinite computable structure, and let R be an additional computable relation on its domain A. The syntactic notion of formal hypersimplicity of R on , first introduced and studied by Hird, is analogous to the computability-theoretic notion of hypersimplicity of R on A, given the definability of certain effective sequences of relations on A. Assuming that R is formally hypersimple on , we give general sufficient conditions for the existence of a computable isomorphic copy of on whose domain the image of R is hypersimple and of arbitrary nonzero computably enumerable Turing degree

With every new recursive relation R on a recursive model , we consider the images of R under all isomorphisms from to other recursive models. We call the set of Turing degrees of these images the degree spectrum of R on , and say that R is intrinsically r.e. if all the images are r.e. C. Ash and A. Nerode introduce an extra decidability condition on , expressed in terms of R. Assuming this decidability condition, they prove that R is intrinsically r.e. if and only if a natural recursive-syntactic condition is satisfied. We show that, while a recursive non-intrinsically r.e. relation may have a two element degree spectrum, a non-intrinsically r.e. relation which satisfies the Ash–Nerode decidability condition has an infinite degree spectrum. We also study several related decidability conditions and their effects on the degree spectra, including some conditions which are sufficient to obtain every r.e. degree in a spectrum

A cohesive power of a structure is an effective analog of the classical ultrapower of a structure. We start with a computable structure, and consider its effective power over a cohesive set of natural numbers. A cohesive set is an infinite set of natural numbers that is indecomposable with respect to computably enumerable sets. It plays the role of an ultrafilter, and the elements of a cohesive power are the equivalence classes of certain partial computable functions determined by the cohesive set. Thus, unlike many classical ultrapowers, a cohesive power is a countable structure. In this paper we focus on cohesive powers of graphs, equivalence structures, and computable structures with a single unary function satisfying various properties, which can also be viewed as directed graphs. For these computable structures, we investigate the isomorphism types of their cohesive powers, as well as the properties of cohesive powers when they are not isomorphic to the original structure.

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.

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)$.