Arkady Tsurkov - Academia.edu (original) (raw)

Papers by Arkady Tsurkov

In the first part of our paper (Sections 1, 2 and 3) we reprove results of B. Plotkin, G. Zhitomi... more 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.

Linear Algebra and its Applications, 2005

We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to c... more We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.

International Journal of Algebra and Computation, 2007

This paper is motivated by the following question arising in universal algebraic geometry: when d... more This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The key problem is to study how far the group Aut Θ0 of all automorphisms of the category Θ0 is from the group Inn Θ0 of inner automorphisms of Θ0 (see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎. Let Θ = 𝔑d be the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category [Formula: see text], d ≥ 2 is inner.

In the first part of our paper (Sections 1, 2 and 3) we reprove results of B. Plotkin, G. Zhitomi... more 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.

Linear Algebra and its Applications, 2005

We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to c... more We prove that the problems of classifying triples of symmetric or skew-symmetric matrices up to congruence, local commutative associative algebras with zero cube radical and square radical of dimension 3, and Lie algebras with central commutator subalgebra of dimension 3 are hopeless since each of them reduces to the problem of classifying pairs of n-by-n matrices up to simultaneous similarity.

International Journal of Algebra and Computation, 2007

This paper is motivated by the following question arising in universal algebraic geometry: when d... more This paper is motivated by the following question arising in universal algebraic geometry: when do two algebras have the same geometry? This question requires considering algebras in a variety Θ and the category Θ0 of all finitely generated free algebras in Θ. The key problem is to study how far the group Aut Θ0 of all automorphisms of the category Θ0 is from the group Inn Θ0 of inner automorphisms of Θ0 (see [7, 10] for details). Recall that an automorphism ϒ of a category 𝔎 is inner, if it is isomorphic as a functor to the identity automorphism of the category 𝔎. Let Θ = 𝔑d be the variety of all nilpotent groups whose nilpotency class is ≤ d. Using the method of verbal operations developed in [8, 9], we prove that every automorphism of the category [Formula: see text], d ≥ 2 is inner.