Quasivarieties of algebraic systems

Budkin, A. I.; Gorbunov, V. A.

doi:10.1007/bf01668420

Cited by 73 publications

(51 citation statements)

References 9 publications

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“…, g j l(j) ) = 1 holds in H, for every j ∈ J. Then the mapping x i → g i , i ∈ I, extends to a homomorphism of G into H. We will need a criterion determining whether or not a finitely presented group G belongs to a quasivariety qR, which is a partial case of the result in [10,Thm. 3] (see also [9]) saying the following: a finitely presented group G belongs to a quasivariety qR iff G is embeddable in a Cartesian product of the groups in qR.…”

Section: Preliminariesmentioning

confidence: 98%

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The quasivariety generated by a torsion-free Abelian-by-finite group

Budkin¹

2007

Algebr Logic

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Section: Preliminariesmentioning

confidence: 98%

“…For any group G in M, by N M (G) we denote the class of all those groups in M in which is the group G not embeddable. We also use the following well-known fact (see [9][10][11]):…”

Section: Preliminariesmentioning

confidence: 99%

The quasivariety generated by a torsion-free Abelian-by-finite group

Budkin¹

2007

Algebr Logic

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“…3 from [6], of belonging to a quasivariety qK~ for a finitely defined group: a finitely defined group G fails to belong to a quasivariety generated by the class/C if and only if there exists an element 9 E G, 9 ~ 1, such that, for any homomorphism ~o from G to any group in/C, we have ~o(9) = 1. Remark.…”

Section: (Vz)(vy)(vz)(vw)([[z Y] [Z W]] = 1) (Vz)(vy)([[z Y] Z mentioning

confidence: 98%

The lattice of quasivarieties of metabelian groups

Lenyuk¹

1996

Algebr Logic

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“…The conjunction of a finite number of quasi-identities is equivalent to some quasi-identity in the class of groups, We will require the following criterion for a finitely determined group ~ to belong to the quasi-variety ~ , which is a special case of Theorem 3 of [6]:…”

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Independent axiomatizability of quasi-varieties of soluble groups

Budkin¹

1991

Algebra and Logic

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The question of independent axiomatizability has been studied widely in various classes of algebraic systems; in the present paper it will be studied in the context of quasi-varieties of groups.Let ~ be the class of all Abelian groups, ~ the lattice of the quasi-varieties in which all groups are soluble. It is a natural to ask for the existence of independent bases of quasi-identities of the quasi-varieties which are covers of the quasi-variety ~ in the lattice ~ . The present paper is devoted to the solution of this question.In the paper we shall find all covers of the quasi-variety ~ in the lattice ~9 " For each of these covers we solve the question of existence of an independent basis of quasiidentities.We note that for the cover ~V~p ~ Z ( p a prime number) of the quasi-variety the question of independent axiomatizability is solved only modulo Artin's conjecture about the infinity of the set of prime numbers modulo which p is a primitive root.We mention the following result of the paper.Let H be a finitely generated non-Abelian ~-group which does not contain non-Abelian nilpotent subgroups and such that the last nonidentity term in the derived series is torsionfree, then the quasi-variety ~V ~H possesses an independent basis of quasi-identities.At present, "few" quasi-varieties are known which are generated by a finitely generated group and do not have an independent basis of quasi-identities. The first example of this kind was given in [I, cf. also 2, 3, 4]. In connection with this the following result is of interest:THEOREM. The set of quasi-varieties each of which is generated by a 2-generator group and does not possess an independent basis of quasi-identities has the power of the continuum.The present paper may be considered as a continuation of [5].

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Quasivarieties of algebraic systems

Cited by 73 publications

References 9 publications

The quasivariety generated by a torsion-free Abelian-by-finite group

The quasivariety generated by a torsion-free Abelian-by-finite group

The lattice of quasivarieties of metabelian groups

Independent axiomatizability of quasi-varieties of soluble groups

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