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A. Tsurkov

Publications and source records attributed to A. Tsurkov.

15 recordsLinked to original sources
arXiv2026

Categories and functors of universal algebraic geometry. Automorphic equivalence of algebras

Universal algebraic geometry allows considering of geometric properties of every universal algebra. When two algebras have same algebraic geometry? We must consider the categories of algebraic closed sets of these algebras to answer this question. The complete coincidence of these categories gives us a concept of the geometric equivalence of algebras. Some type of isomorphisms of these categories gives us a concept of the automorphic equivalence of algebras. This concept has been considered since article B. Plotkin, Algebras with the same (algebraic) geometry. Proceedings of the Steklov Institute of Mathematics. 242 (2003), 17--207. DOI: 10.1134/S0081543812070048. We will give by language of category theory one more elegant definition of this concept and recall some theorems related to this concept.

math.CT↗
arXiv2020

IBN-varieties of algebras

The concept of variety with IBN (invariant basic number) propriety first appeared in ring theory. But we can define this concept for arbitrary variety of universal algebras with arbitrary signature; see Definition 1.4. The proving of the IBN propriety of some variety is very important in universal algebraic geometry. This is a milestone in the study of the relation between geometric and automorphic equivalences of algebras of this variety. In this paper we prove very simple but very useful for studying of IBN proprieties of different varieties Theorems 2.1 and 3.2. We will consider applications of this theorem. We will consider many-sorted universal algebras as well as one-sorted. So all concepts and all results will by generalized for the many-sorted case.

math.RA↗
arXiv2019

Automorphisms of the category of finitely generated free groups of the some subvariety of the variety of all groups

In universal algebraic geometry the category of the finite generated free algebras of some fixed variety of algebras and the quotient group A/Y are very important. Here A is a group of all automorphisms of this category and Y is a group of all inner automorphisms of this category. In the varieties of all the groups, all the abelian groups (see B. Plotkin and G. Zhitomirski, 2006), all the nilpotent groups of the class no more then n (see A. Tsurkov, 2007) the group A/Y is trivial. B. Plotkin posed a question: "Is there a subvariety of the variety of all the groups, such that the group A/Y in this subvariety is not trivial?" A. Tsurkov hypothesized that exist some varieties of periodic groups, such that the groups A/Y in these varieties is not trivial. In this paper we give an example of one subvariety of this kind.

math.GR↗
arXiv2015

Automorphic Equivalence in the Varieties of Representations of Lie algebras

In this paper we consider the very wide class of varieties of representations of Lie algebras over the field k, which has characteristic 0. We study the relation between the geometric equivalence and automorphic equivalence of the representations of these varieties. We calculate the group, which measures the difference between the geometric equivalence and automorphic equivalence of representations of theses varieties. In Section 5, we present one example of the subvariety of the variety of all the representations of the Lie algebras over the field k, and two representations from these variety which are automorphically equivalent but not geometrically equivalent.

math.RA↗
arXiv2013

Automorphic Equivalence in the Classical Varieties of Linear Algebras

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this category we calculate the quotient group of the group of the all automorphisms of this category over the subgroup of the all inner automorphisms. This quotient group measures difference between the geometric equivalence and automorphic equivalence of algebras from this variety. In the all considered varieties of the linear algebras this group is generated by cosets which are presented by no more than two kinds of the strongly stable automorphisms of our category. One kind of automorphisms is connected to the changing of the multiplication by scalar and second one is connected to the changing of the multiplication of the elements of the algebras. We present some examples of the pairs of linear algebras such that the considered automorphism provides the automorphic equivalence of these algebras but these algebras are not geometrically equivalent. These examples are presented for the all considered above varieties of algebras and for both these kinds of the strongly stable automorphisms, when they exist in our group.

math.RA↗
arXiv2013

Automorphic Equivalence of Many-Sorted Algebras

In the first part of our paper (Sections 1, 2 and 3) we reprove results of B. Plotkin, G. Zhitomirski. On automorphisms of categories of free algebras of some varieties, Journal of Algebra, 306:2, (2006), 344 -- 367 for the case of many-sorted algebras. In the second part of our paper (Section 4) we apply the results of the first part to the universal algebraic geometry of many-sorted algebras and refine and reprove results of B. Plotkin, Algebras with the same (algebraic) geometry, Proceedings of the International Conference on Mathematical Logic, Algebra and Set Theory, dedicated to 100 anniversary of P.S. Novikov, Proceedings of the Steklov Institute of Mathematics, MIAN, 242 (2003), 127 -- 207 and A. Tsurkov, Automorphic equivalence of algebras, International Journal of Algebra and Computation. 17:5/6, (2007), 1263 -- 1271 for these algebras. In the third part of this paper (Section 5) we consider some varieties of many-sorted algebras. We prove that automorphic equivalence coincide with geometric equivalence in the variety of the all actions of semigroups over sets and in the variety of the all automatons. We also consider the variety of the all representations of groups and the all representations of Lie algebras. For both these varieties we give an examples of the representations which are automorphically equivalent but not geometrically equivalent.

math.CT↗
arXiv2012

Strongly Stable Automorphisms of the Categories of the Finitely Generated Free Algebras of the some Varieties of Linear Algebras

In this paper we consider some classical varieties of linear algebras over the field which has characteristic 0. For every considered variety we take a category of the finite generated free algebras of this variety. And for every this category we calculate the quotient group of the group of the all automorphisms of this category over the subgroup of the all inner automorphisms. This quotient group measures difference between the geometric equivalence and automorphic equivalence of algebras from this variety.

math.RA↗
arXiv2012

Automorphic equivalence of the representations of Lie algebras

We prove that if a field k is infinite, char(k)=0 and k has not nontrivial automorphisms then automorphic equivalence of representations of Lie algebras coincide with geometric equivalence. We achieve our result by consideration of 1-sorted objects. We suppose that our method can be perspective in the further researches.

math.RA↗
arXiv2006

The problem of the classification of the nilpotent class 2 torsion free groups up to the geometrically equivalence

In this paper we consider the problem of classification of the nilpotent class 2 finitely generated torsion free groups up to the geometric equivalence. By a very easy technique it is proved that this problem is equivalent to the problem of classification of the complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism. This result, allows us to once more comprehend the complication of the problem of the classification of the quasi-varieties of nilpotent class 2 groups. It is well known that the variety of a nilpotent class s (for every s) groups is Noetherian. So the problem of the classification of the quasi-varieties generated even by a single nilpotent class 2 finitely generated torsion free group, is equivalent to the problem of classification of complete (in the Maltsev sense) nilpotent torsion free finite rank groups up to the isomorphism.

math.GR↗
arXiv2006

Aupomorphisms of the category of the free nilpotent groups of the fixed class of nilpotency

This research was motivated by universal algebraic geometry. One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? For answer of this question (see [Pl],[Ts]) we must consider the variety, to which our algebras belongs, the category K of all finitely generated free algebras of our variety and research how the group of all the automorphisms of this category AutK are different from the group of the all inner automorphisms of this category InnK. An automorphism of the category is called inner, if it is isomorphic as functor to the identity automorphism. In the case when we consider a variety of all groups we have the classical results which let us resolve this problem by an indirect way. In [DF] proved that for every free group F the group Aut(AutF) coincides with the group Inn(AutF), from this result in [Fo] was concluded that Aut(EndF)=Inn(EndF) and from this fact by theorem of reduction [BPP] it can be concluded that AutK=InnK. In the case of nilpotent group by [Ks] we have, that if the number of generators of the free nilpotent class d group NF are bigger enough than d, then Aut(AutNF)=Inn(AutNF). But we have no description of Aut(EndNF) and so can not use the theorem of reduction. In this paper the method of verbal operations is used. This method was established in [PZ]. In [PZ] by this method was very easily proved that AutK=InnK in the case of free groups and in the case of free abelian groups. In this paper we will prove, by this method, that AutK=InnK free nilpotent groups of arbitrary fixed class of nilpotency.

math.GR↗
arXiv2006

Automorphic equivalence of one-sorted algebras

One of the central questions of universal algebraic geometry is: when two algebras have the same algebraic geometry? There are various interpretations of the sentence "Two algebras have the same algebraic geometry". One of these is automorphic equivalence of algebras, which is discussed in this paper, and the other interpretation is geometric equivalence of algebras. In this paper we consider very wide and natural class of algebras: one sorted algebras from IBN variety. The variety is called an IBM variety if two free algebras W(X), W(Y) of this variety are isomorphic if and only if the powers of sets X and Y coincide. In the researching of the automorphic equivalence of algebras we must study the group of automorphisms of the category of the all finitely generated free algebras of our variety and the group of its automorphisms. By [PZ, Theorem 2], if we deal with an IBN variety of one-sorted algebras, then every strongly stable (see Definition 3.1) automorphism of our category defines the other algebraic structure on every algebra H of our variety, such that this algebra with the new algebraic structure automorphically equivalent to the algebra H (Theorem 4.1), i.e., has the same algebraic geometry. From this we conclude the necessary and sufficient conditions for two algebras to be automorphically equivalent. We formulate these conditions by using the notion of geometric equivalence of algebras. It means that we reduce automorphic equivalence of algebras to the simpler notion of geometric equivalence. This paper is a continuation of the research which was started in [PZ].

math.GM↗
arXiv2005

Action Type Geometrical Equivalence of Representations of Groups

For every variety of algebras and every algebras in these variety we can consider an algebraic geometry. Algebras may be many sorted (not necessarily one sorted) algebras. A set of sorts is fixed for each variety. This theory can be applied to the variety of representations of groups over fixed commutative ring with unit. We consider a representation as two sorted algebra. We concentrate on the case of the action type algebraic geometry of representations of groups. In this case algebraic sets are defined by systems of action type equations and equations in the acting group are not considered. This is the special case, which cannot be deduced from the general theory. In this paper the following basic notions are introduced: action type geometrical equivalence of two representations, action type quasi-identity in representations, action type quasi-variety of representations, action type Noetherian variety of representations, action type geometrically Noetherian representation, action type logically Noetherian representation.

math.RT↗
arXiv2004

Geometrical equivalence of nilpotent torsion free groups

The variety of nilpotent groups is Noetherian. That is why two nilpotent class s groups are geometrically equivalent if and only if they have same quasi-identities ([Pl3]). Therefore, we can describe classes of geometrical equivalence of nilpotent groups instead of describing quasivarieties generated by single nilpotent group. The problem of geometrical equivalence can be researched by the technique of approximation of groups ([Pl2]). In this paper only nilpotent torsion free groups are considered. In the first part two sufficient conditions are presented (Theorem 1 and Theorem 2) when the nilpotent torsion free group is geometrically equivalent to its Mal'tsev completion and so it belongs to the same quasi-variety. In the second part of this paper some results are achieved by the easy methods of approximation of groups in the describing of classes of geometrical equivalence of nilpotent class 2 torsion free groups with the small rank of center.

math.GR↗