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Superpositions of free Fox derivations | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2022. № 56. DOI: 10.17223/20710410/56/3

Fox derivations are an effective tool for studying free groups and their group rings. Let Fr be a free group of finite rank r with basis {f1,..., fr}. For every i, the partial Fox derivations ∂/∂fi и ∂/∂fi-1 are defined on the group ring ℤ[Fr]. For k / 2, their superpositions Dfϵi = = ∂/∂fϵki о ... о ∂/∂fϵk1, ϵ = (ϵ1,..., ϵk) Є{±1}k, are not Fox derivations. In this paper, we study the properties of superpositions Dfϵi. It is shown that the restrictions of such superpositions to the commutant F′r are Fox derivations. As an application of the obtained results, it is established that for any rational subset R of F′r and any i there are parameters k and ϵ such that R is annihilated by Dfϵi.
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  • Title Superpositions of free Fox derivations
  • Headline Superpositions of free Fox derivations
  • Publesher Tomask State UniversityTomsk State University
  • Issue Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics 56
  • Date:
  • DOI 10.17223/20710410/56/3
Keywords
free group, group ring, Fox derivations, annihilators, rational subsets
Authors
References
Fox R.H. Free differential calculus I - Derivation in the free group ring. Ann. Math., 1953, vol. 57, pp. 547-560.
Crowell R. H. and Fox R. H.Introduction to Knot Theory. N.Y., Springer Verlag, 1963, X + 182 p.
Timoshenko E. I. Endomorfizmy i universal’nye teorii razreshimykh grupp [Endomorphisms and Universal Theories of Solvable Groups]. Novosibirsk, Novosibirsk State Technical University, 2011. 327p. (in Russian)
Roman’kov V. A. Essays in Algebra and Cryptology. Solvable Groups. Omsk, Dostoevsky Omsk State University, 2017. 267p.
Myasnikov A., Roman’kov V., Ushakov A., and Vershik A. The word and geodesic problems for free solvable groups. Trans. Amer. Math. Soc., 2010, vol. 362, no. 9, pp. 4655-4682.
Anashin V. and Khrennikov A. Applied Algebraic Dynamics (de Gruyter Expositions in Math., vol. 49). Berlin, N.Y., Walter de Gruyter GmbH & Co., 2009. 557p.
Anashin V. Noncommutative algebraic dynamics: Ergodic theory for profinite groups. Proc. Steklov Institute of Mathematics, 2009, vol. 265, pp. 30-58.
Anashin V. S. Uniformly distributed sequences in computer algebra, or how to construct program generators of random numbers. J. Math. Sci., 1998, vol. 89, no. 4, pp. 1355-1390.
Gilman R. H. Formal languages and infinite groups. Geometric and Computational Perspectives of Infinite Groups, Minneapolis, MN, and New Brunswick, NJ, DIMACS Ser. Discrete Math. Theoret.Comput. Sci., 1994, vol. 25, pp. 27-51.
Roman’kov V. A. Ratsional’nye podmnozhestva v gruppakh [Rational subsets in groups]. Omsk, Dostoevsky Omsk State University, 2014. 176 p. (in Russian)
Roman’kov V. A. Polycyclic, metabelian or soluble of type (FPgroups with Boolean algebra of rational sets and biautomatic soluble groups are virtually abelian. Glasgow Math. J., 2018, vol. 60, no. 1, pp. 209-218.
Roman’kov V. A. Rationality of verbal subsets in solvable groups. Algebra and Logic, 2018. vol. 57, no. 1, pp. 39-48.
Roman'kov V. and Myasnikov A. On rationality of verbal subsets in a group. Theory of Computing Systems, 2013, vol. 52, no. 4, pp. 587-598.
Superpositions of free Fox derivations | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2022. № 56. DOI: 10.17223/20710410/56/3
Superpositions of free Fox derivations | Prikladnaya Diskretnaya Matematika - Applied Discrete Mathematics. 2022. № 56. DOI: 10.17223/20710410/56/3
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