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$$\exists$$-free groups. (English. Russian original) Zbl 0724.20025
Sib. Math. J. 30, No. 6, 998-1001 (1989); translation from Sib. Mat. Zh. 30, No. 6(178), 193-197 (1989).
For a group $$G$$ let $$Th_{\exists}G$$ denote the set of all $$\exists$$-sentences true in $$G$$. If $$F_ 1$$ and $$F_ 2$$ are free groups of finite rank $$\geq 2$$, then $$Th_{\exists}F_ 1=Th_{\exists}F_ 2$$, because each of them contains the other. So one can speak about the $$\exists$$-theory $$Th_{\exists}F$$ of free non-abelian groups. Call a group $$G$$ $$\exists$$-free iff $$Th_{\exists}F=Th_{\exists}G$$. Call a group $$G$$ $$\omega$$-residually free if for every finite subset of non-unit elements of $$G$$ there exists a normal subgroup $$N$$ of $$G$$ such that $$G/N$$ is free and $$N$$ has empty intersection with the given finite subset.
In the paper under review it is proved that for a finitely generated group $$G$$ the following conditions are equivalent: 1. $$G$$ is $$\exists$$-free, 2. $$G$$ is $$\omega$$-residually free, 3. $$G$$ is residually free and does not contain $$F_ 2\times Z$$, 4. The relation of commutativity in $$G$$ is an equivalence. The analogs for $$\omega$$-free commutative rings are also proved.
Reviewer: G.A.Noskov (Omsk)

##### MSC:
 20F10 Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) 20E05 Free nonabelian groups 03C60 Model-theoretic algebra 20E26 Residual properties and generalizations; residually finite groups
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##### References:
 [1] G. S. Makanin, ?Solvability of the universal and positive theories of free groups,? Izv. Akad. Nauk SSSR,48, No. 4, 735-749 (1984). [2] V. N. Remeslennikov, Finitely Generated Groups Which are ?-Equivalent to a Free Group [in Russian], 11th All-Union Symposium on Group Theory, Sverdlovsk (1989). [3] B. Baumslag, ?Residually free groups,? Proc. London Math. Soc.,17, No. 3, 402-418 (1967). · Zbl 0166.01502
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