Keywords: admissible set, structure, semilattice of degrees of Σ-definability.Presentations of structures in admissible sets, as well as different relations of effective reducibility between the structures, are treated. Semilattices of degrees of Σ-definability are the main object of investigation. It is shown that the semilattice of degrees of Σ-definability of countable structures agrees well with semilattices of T -and e-degrees of subsets of natural numbers. Also an attempt is made to study properties of the structures that are inherited under various effective reducibilities and explore how degrees of presentability depend on choices of different admissible sets as domains for presentations.In the paper we deal with presentations of structures in admissible sets, and also with different effective reducibility relations between the structures. Semilattices of degrees of Σ-definability (Ershov semilattices) are the main object of our investigation. This concept can be viewed, on the one hand, as a natural generalization of oracle definability, that is, when a complex abstract object -a structure -plays the role of an oracle and of a result of computations. (The given approach can be conceived of as a theoretical model of object-oriented programming.) On the other hand, the concept of Σ-definability of a structure in an admissible set is an effectivization of one of the main notions in model theory, that of interpretability of one structure in another, and moreover, it generalizes the concept of constructivizability of structures on natural numbers.It will be shown that the semilattice of degrees of Σ-definability of countable structures agrees well with semilattices of T -and e-degrees of subsets of natural numbers. The concept of a structure having a degree, which is known in constructive model theory, is just a partial characteristic of complexity, since by no means all structures can have degrees. As distinct from this, degrees of Σ-definability, as well as the degrees of presentability of relatively distinct uniform and non-uniform effective reducibilities treated in the paper, are natural characteristics of complexity defined for any structure. We also make an attempt to study properties of the structures that are inherited under various effective reducibilities and look at how degrees of presentability depend on choices of different admissible sets as domains for presentations.
Keywords: admissible set, semilattice of degrees of Σ-definability.We show that the property of being locally constructivizable is inherited under Muchnik reducibility, which is weakest among the effective reducibilities considered over countable structures. It is stated that local constructivizability of level higher than 1 is inherited under Σ-reducibility but is not inherited under Medvedev reducibility. An example of a structure M and a relation P ⊆ M is constructed for which (M, P ) ≡ M but (M, P ) ≡ Σ M. Also, we point out a class of structures which are effectively defined by a family of their local theories.This paper is a continuation of [1, 2] and uses the same notation. PROPERTIES OF STRUCTURES INHERITED UNDER EFFECTIVE REDUCIBILITIESBelow we show that the condition of a structure M having a degree specified in [1, Thm. 7] is essential. To do this, we state necessary conditions for effective reducibilities between structures, in particular, conditions that are necessary for being Σ-definable.Structure M is said to be locally constructivizable [3] if Th ∃ (M,m) is computably enumerable (c.e.) for anym ∈ M <ω . As noted in [3], the local constructivizability of M is equivalent to the fact that for every tuplem ∈ M <ω , there exist a constructivizable structure N and a tuplen ∈ N <ω such that Th ∃ (M,m) = Th ∃ (N,n). For structures M and N, by writing M ∃ N we mean that for every tuplē m ∈ M <ω , there is a tuplen ∈ N <ω for which Th ∃ (M,m) e Th ∃ (N,n). In particular, if M is locally constructivizable then M ∃ N for any structure N.That a structure M is locally constructivizable if so is N with M Σ N was first mentioned in [3]. A direct generalization of this fact is the following: if M Σ N then M ∃ N. In order to state other necessary conditions of Σ-definability (which are used in proving negative results on Σ ), we consider some relevant notions. Definition 1. Structure M is locally constructivizable of level n (1 < n ω) if for every tuplem ∈ M <ω there exist a constructivizable structure N and a tuplen ∈ N <ω such that (M,m) ≡ HF n (N,n). A countable *
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