Primitive connected theories

Palyutin, E. A.

doi:10.1007/bf02681663

Cited by 10 publications

(10 citation statements)

References 6 publications

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“…A formula Φ(x,ȳ,z,c) defines an Abelian group structure (an AG-structure) on X if this formula endows X with an operation with respect to which X is an Abelian group. The next statement is implicitly contained in [2]. Claim 1 [2].…”

Section: Preliminary Informationmentioning

confidence: 96%

“…The next statement is implicitly contained in [2]. Claim 1 [2]. Let T be a primitive normal theory for a structure A and let some primitive formula define an AG-structure on A.…”

Section: Preliminary Informationmentioning

confidence: 97%

“…Information given in this section can be found in [1,2]. We write C to denote a saturated model of a complete theory T of a language L which elementarily contains all models of T under consideration.…”

Section: Preliminary Informationmentioning

confidence: 99%

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Polygons with primitive normal and additive theories

Stepanova¹

2008

Algebra Logic

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Section: Preliminary Informationmentioning

confidence: 96%

“…The next statement is implicitly contained in [2]. Claim 1 [2]. Let T be a primitive normal theory for a structure A and let some primitive formula define an AG-structure on A.…”

Section: Preliminary Informationmentioning

confidence: 97%

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Polygons with primitive normal and additive theories

Stepanova¹

2008

Algebra Logic

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“…We know from [5,7] that if a primitive normal structure A has additively connected primitive sets then a nontrivial group is primitive interpreted in the structure A. THEOREM 6. Let K be a categorical Horn class in which no nontrivial group is primitive interpreted.…”

Section: Definitionmentioning

confidence: 97%

“…An analog of the above definition in [7] dealt with so-called Δ-primitive sets. In categorical theories, there are no infinite descending chains of primitive sets, and so Δ-primitive sets will be primitive in the case under consideration.…”

Section: Definitionmentioning

confidence: 98%

Categorical horn classes. 2

Palyutin

2011

Algebra Logic

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Primitive connected and additive theories of polygons

Stepanova¹

2006

Algebr Logic

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We study into monoids S the class of all S-polygons over which is primitive normal, primitive connected, or additive, that is, the monoids S the theory of any S-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid, and that it is primitive connected iff S is a group. It is pointed out that there exists no monoid S with an additive class of all S-polygons.Additive and primitive connected theories were studied in [1,2]. These theories are a generalization of module theory. Like the latter, the theories in question admit quantifier elimination down to primitive formulas (see [1,2]). The class of additive theories is contained in the class of primitive connected ones. As distinct from the primitive connected theory, the additive theory has the property that on factors of any primitive copies w.r.t. some primitive equivalence, isomorphic Abelian groups can be defined using a primitive formula. This property of additive theories generalizes a known property of modules stating the following: in any module, primitive copies are conjugacy classes of some Abelian group. The additive and primitive connected theories are, by definition, primitive normal theories.In this paper, we look at monoids S the class of all S-polygons over which is primitive normal, primitive connected, or additive, that is, those monoids the theory of any S-polygon over which is primitive normal, primitive connected, or additive. It is proved that the class of all S-polygons is primitive normal iff S is a linearly ordered monoid (Thm. 1); there exists no monoid S with an additive class of all S-polygons (Thm. 2); the class of all S-polygons is primitive connected iff S is a group. PRELIMINARIESModel-theoretic information given under this section is borrowed from [1,2]. Let T be a complete theory of a language L. Fix some sufficiently large and sufficiently saturated model C of T , which we call a monster model, since it is assumed that all the models of T under consideration are elementary submodels of C. All elements, tuples of elements, and sets are taken from the monster model C. A tuple a 1 , . . . , a n of elements and a tuple x 1 , . . . , x n of variables are denoted byā andx, respectively. Instead ofā ∈ C n , we writē a ∈ C. Lets,t be tuples of elements or variables. We introduce the following notation: l(s) is the length of s; s is the set consisting of elements ofs;sˆt is a tuple obtained by ascribing the tuplet to the right of s. We write s ∈s in place of s ∈ s, and writes ∪t in place of s ∪ t . If Φ(x,ȳ) is a formula in L, A is a model of T ,ā is a tuple of elements of A, and l(ā) = l(ȳ), then Φ(A,ā) denotes the set {b | A |= Φ(b,ā)}.

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Primitive connected theories

Cited by 10 publications

References 6 publications

Polygons with primitive normal and additive theories

Polygons with primitive normal and additive theories

Categorical horn classes. 2

Primitive connected and additive theories of polygons

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