\(\exists\)-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.

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 |

##### Keywords:

existentially free groups; \(\exists\)-sentences; free groups of finite rank; \(\exists\)-theory; finitely generated groups; \(\exists\)-free groups; \(\omega\)-residually free groups; \(\omega\)-free commutative rings
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\textit{V. N. Remeslennikov}, Sib. Math. J. 30, No. 6, 998--1001 (1989; Zbl 0724.20025); translation from Sib. Mat. Zh. 30, No. 6(178), 193--197 (1989)

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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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