On Mass Problems of Presentability

Stukachev, A. I.

doi:10.1007/11750321_74

Cited by 3 publications

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“…Proof. Item (1) being obvious, we only consider (2). In fact, m, n ∈ χ(E) iff there exist pairwise distinct elements a 1 1 , .…”

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confidence: 99%

Degrees of presentability of structures. II

Stukachev

2008

Algebra Logic

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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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“…Proof. Item (1) being obvious, we only consider (2). In fact, m, n ∈ χ(E) iff there exist pairwise distinct elements a 1 1 , .…”

mentioning

confidence: 99%