## Rational sets in finitely generated nilpotent groups.(Russian, English)Zbl 1054.20015

Algebra Logika 39, No. 4, 379-394 (2000); translation in Algebra Logic 39, No. 4, 215-223 (2000).
Summary: We deal with the 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 unions, products of two sets, and under generation 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 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.

### MSC:

 20F18 Nilpotent groups 20F05 Generators, relations, and presentations of groups 20F65 Geometric group theory 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20M35 Semigroups in automata theory, linguistics, etc. 20M05 Free semigroups, generators and relations, word problems
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