Elementary Theories

Ershov, Yu. L.; Lavrov, I. A.; Taimanov, A. D.; Taitslin, Michael A.

doi:10.1070/rm1965v020n04abeh001188

Cited by 109 publications

(51 citation statements)

References 13 publications

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“…Also, other properties that can be expressed in terms of the language of elementary group theory, such as the existence of an element not conjugate to its inverse or the representability of any element as a commutator have been studied (see, for example, [1,2,4]). The algorithmic undecidability of elementary theories was proved for many groups, and some examples are given in [5]. The conjecture that the elementary theory of the groups considered in the present paper is undecidable was put forward by Ryzhikov.…”

Section: Introductionmentioning

confidence: 94%

Undecidability of the elementary theory of groups of measure-preserving transformations

Mitin

1998

Math Notes

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Section: Introductionmentioning

confidence: 94%

Undecidability of the elementary theory of groups of measure-preserving transformations

Mitin

1998

Math Notes

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“…73. tThe systems ofaxioms of theories Qc and ClJ( are recursively denumerable. Therefore, these theories are recursively axiomatizable (see [2], Lemma 1.2.1. ).…”

mentioning

confidence: 98%

Deductive Validity and Reduction Classes

Lifshits¹

1969

Studies in Constructive Mathematics and Mathematical Logic

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In the present note we define the relation "the predicate formula _ is deductively valid in the theory _" which is analogous to the relation "the predicate formula _ is valid in the nonempty subject domain _." A method is proposed to prove statements of the type "a set _ ofpredicate formulas is areduction class" by using the property of the mentioned relation. The majority of the analysis is made in parallel for classical and constructive predicate calculus.1. Henceforth, the following concepts and notations are used. We denote by C and K respectively the classical and constructive predicate calculus without functional symbols. We consider ~,. 'lO.~ , • ' • • to be the subj ect variables of these calculuses. We will understand the cl ass i c a I (c 0 ns t r u c ti v e) th e 0 r y to be an arbitrary recursively axiomatized theory based on the classical (or constructive) predicate calculus (possibly with equality and functional symbols). We will consider ;:(.1' ~lI.' '" to be the subject variables of all such theories also. We will understand the formula (or term) to be a formula (or term) of any theory of the mentioned kind.

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“…For example, Tarski [21] proved that the theory of Boolean algebras is decidable and in a series of papers [20,7,25] it was shown that a variety of groups is decidable if and only if it contains no non-Abelian group.…”

Section: Introductionmentioning

confidence: 99%

A characterization of decidable locally finite varieties

McKenzie¹,

Valeriote²

1992

Proceedings of the International Conference on Algebra Dedicated to the Memory of A. I. Mal’cev

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We describe the structure of those locally finite varieties whose first order theory is decidable. A variety is a class of universal algebras defined by a set of equations. Such a class is said to be locally finite if every finitely generated member of the class is finite. It turns out that in order for such a variety to have a decidable theory it must decompose into the varietal product of three special kinds of varieties; a strongly Abelian variety; an affine variety; and a discriminator variety.

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Elementary Theories

Cited by 109 publications

References 13 publications

Undecidability of the elementary theory of groups of measure-preserving transformations

Undecidability of the elementary theory of groups of measure-preserving transformations

Deductive Validity and Reduction Classes

A characterization of decidable locally finite varieties

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