\(\exists\)-free groups and groups with length function.

*(English)*Zbl 0856.20002
Bokut’, L. A. (ed.) et al., Second international conference on algebra dedicated to the memory of A. I. Shirshov. Proceedings of the conference on algebra, August 20-25, 1991, Barnaul, Russia. Providence, RI: American Mathematical Society. Contemp. Math. 184, 369-376 (1995).

Let \(F_n\) be a free non-abelian group of rank \(n\). Since \(\text{Th}_\exists F_{n_1}=\text{Th}_\exists F_{n_2}\) for any pair \((n_1,n_2)\) one can speak of the \(\exists\)-theory of a non-abelian free group \(\text{Th}_\exists F\). A group \(G\) is said to be \(\exists\)-free if \(\text{Th}_\exists G=\text{Th}_\exists F\).

Let \((n,k)\) be a pair of natural numbers such that \(n\geq k\). By \(\Lambda_{nk}\) is denoted an ordered abelian group with the following properties: the group \(\Lambda_{nk}\) is the direct sum of \(n\) copies of \(Z\), and the total series of convex subgroups of \(\Lambda_{nk}\) contains \(k\) non-trivial terms. It is known [V. N. Remeslennikov, Ukr. Mat. Zh. 44, No. 6, 813-818 (1992; Zbl 0784.20015), and independently A. M. Gaglione and D. Spellman, Contemp. Math. 169, 277-281 (1994; see the following review Zbl 0856.20003)] that for every finitely generated \(\exists\)-free group \(G\) there exists a pair \((n,k)\) of natural numbers, and an abelian ordered group \(\Lambda_{nk}\) such that there exists a \(\Lambda_{nk}\)-valued Lyndon length function making \(G\) into a \(\Lambda_{nk}\)-free group in the sense of H. Bass.

A notion of \(\exists\)-free constructible group is introduced, and an example of an \(\exists\)-free constructible non-\(\exists\)-free group is presented. This group is \(\Lambda_{22}\)-free and hence is a counterexample to the question posed by A. M. Gaglione and D. Spellman in the paper mentioned above on the equivalence of the notions of \(\exists\)-free groups and \(\Lambda_{nk}\)-free groups for the class of all finitely generated groups.

For the entire collection see [Zbl 0824.00029].

Let \((n,k)\) be a pair of natural numbers such that \(n\geq k\). By \(\Lambda_{nk}\) is denoted an ordered abelian group with the following properties: the group \(\Lambda_{nk}\) is the direct sum of \(n\) copies of \(Z\), and the total series of convex subgroups of \(\Lambda_{nk}\) contains \(k\) non-trivial terms. It is known [V. N. Remeslennikov, Ukr. Mat. Zh. 44, No. 6, 813-818 (1992; Zbl 0784.20015), and independently A. M. Gaglione and D. Spellman, Contemp. Math. 169, 277-281 (1994; see the following review Zbl 0856.20003)] that for every finitely generated \(\exists\)-free group \(G\) there exists a pair \((n,k)\) of natural numbers, and an abelian ordered group \(\Lambda_{nk}\) such that there exists a \(\Lambda_{nk}\)-valued Lyndon length function making \(G\) into a \(\Lambda_{nk}\)-free group in the sense of H. Bass.

A notion of \(\exists\)-free constructible group is introduced, and an example of an \(\exists\)-free constructible non-\(\exists\)-free group is presented. This group is \(\Lambda_{22}\)-free and hence is a counterexample to the question posed by A. M. Gaglione and D. Spellman in the paper mentioned above on the equivalence of the notions of \(\exists\)-free groups and \(\Lambda_{nk}\)-free groups for the class of all finitely generated groups.

For the entire collection see [Zbl 0824.00029].

Reviewer: V.A.Roman’kov (Omsk)

##### MSC:

20A15 | Applications of logic to group theory |

20E05 | Free nonabelian groups |

20F05 | Generators, relations, and presentations of groups |

03C60 | Model-theoretic algebra |