Abstract
We deal with a class of rational subsets of a group, that is, the least class of its subsets which contains all finite subsets and is closed under taking union. a product of two sets, and under generating of a submonoid by a set. It is proved that the class of rational subsets of a finitely generated nilpotent group G is a Boolean algebra iff G is Abelian-by-finite. We also study the question asking under which conditions the set of solutions for equations in groups will be rational. It is shown that the set of solutions for an arbitrary equation in one variable in a finitely generated nilpotent group of class 2 is rational. And we give an example of an equation in one variable in a free nilpotent group of nilpotency class 3 and rank 2 whose set of solutions is not rational.
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References
R. H. Gilman, inFormal Languages and Infinite Groups, DIMACS Ser. Discr. Math. Theor. Comp. Sc., Vol. 25, Am. Math. Soc., Providence, RI (1996), pp. 27–51.
S. M. Gersten and H. Short, “Rational subgroups of biautomatic groups,”Ann. Math., II. Ser.,134, No. 1, 125–158 (1991).
R. C. Lyndon and P. E. Schupp,Combinatorial Group Theory, Springer, Berlin (1977).
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Supported by RFFR grant No. 98-01-00932.
Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 379–394, July–August, 2000.
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Bazhenova, G.A. Rational sets in finitely generated nilpotent groups. Algebr Logic 39, 215–223 (2000). https://doi.org/10.1007/BF02681647
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DOI: https://doi.org/10.1007/BF02681647