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Rationality of Verbal Subsets in Solvable Groups

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Algebra and Logic Aims and scope

A verbal subset of a group G is a set w[G] of all values of a group word w in this group. We consider the question whether verbal subsets of solvable groups are rational in the sense of formal language theory. It is proved that every verbal subset w[N] of a finitely generated nilpotent group N with respect to a word w with positive exponent is rational. Also we point out examples of verbal subsets of finitely generated metabelian groups that are not rational.

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References

  1. A. Myasnikov and V. Roman’kov, “On rationality of verbal subsets in a group,” Theory Comput. Syst., 52, No. 4, 587-598 (2013).

    Article  MathSciNet  MATH  Google Scholar 

  2. R. H. Gilman, “Formal languages and infinite groups,” in DIMACS, Ser. Discr. Math. Theor. Comp. Sci., 25, Am. Math. Soc., Providence, RI (1996), pp. 27-51.

  3. V. A. Roman’kov, Rational Subsets in Groups [in Russian], Omsk Univ., Omsk (2014).

    Google Scholar 

  4. H. Neumann, Varieties of Groups, Springer, Berlin (1967).

    Book  MATH  Google Scholar 

  5. D. Segal, Words: Notes on Verbal Width in Groups, London Math. Soc. Lect. Notes Ser., 361, Cambridge Univ. Press, Cambridge (2009).

  6. V. A. Roman’kov, “Equations over groups,” Groups Complex. Cryptol., 4, No. 2, 191-239 (2012).

    MathSciNet  MATH  Google Scholar 

  7. A. W. Anissimov and F. D. Seifert, “Zur algebraischen Charakteristik der durch kontext-freie Sprachen definierten Gruppen,” Elektr. Inform.-verarb. Kybern., 11, 695-702 (1975).

    MathSciNet  MATH  Google Scholar 

  8. P. Hall, “Nilpotent groups. Notes of lectures given at the Canadian Mathematical Congress” (summer seminar, Univ. Alberta, Edmonton, 12-30 August, 1957), Queen Mary College Math. Notes, Queen Mary College (Univ. London), London (1969).

  9. M. I. Kargapolov and Yu. I. Merzlyakov, Fundamentals of Group Theory [in Russian], Nauka, Moscow (1972).

  10. S. Moran and J. Williams, “Completions of certain nilpotent groups,” J. Aust. Math. Soc., Ser. A, 28, 461-470 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  11. V. A. Roman’kov, “Width of verbal subgroups in solvable groups,” Algebra and Logic, 21, No. 1, 41-49 (1982).

    Article  MathSciNet  MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Dostoevskii Omsk State University, pr. Mira 55-A, Omsk, 644077, Russia

    V. A. Roman’kov

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  1. V. A. Roman’kov
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Correspondence to V. A. Roman’kov.

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Supported by Russian Science Foundation, project No. 16-11-10002.

Translated from Algebra i Logika, Vol. 57, No. 1, pp. 57-72, January-February, 2018.

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Roman’kov, V.A. Rationality of Verbal Subsets in Solvable Groups. Algebra Logic 57, 39–48 (2018). https://doi.org/10.1007/s10469-018-9477-6

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  • DOI: https://doi.org/10.1007/s10469-018-9477-6

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