We give a description of elementary subgroups (in the sense of first-order logic) of finitely generated virtually free groups. In particular, we recover the fact that elementary subgroups of finitely generated free groups are free factors. Moreover, one gives an algorithm that takes as input a finite presentation of a virtually free group $G$ and a finite subset $X$ of $G$, and decides if the subgroup of $G$ generated by $X$ is $\forall\exists$-elementary. We also prove that every elementary embedding of an equationally noetherian group into itself is an automorphism.

中文翻译：

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几乎自由群的基本子群

我们给出了有限生成的虚拟自由群的基本子群（在一阶逻辑意义上）的描述。特别地，我们恢复了有限生成自由群的基本子群是自由因子这一事实。此外，有人给出了一种算法，该算法将虚拟自由群 $G$ 的有限表示和 $G$ 的有限子集 $X$ 作为输入，并确定 $X$ 生成的 $G$ 的子群是否为 $\ forall\exists$-基本的。我们还证明了等式诺特群的每一个基本嵌入到自身都是自同构。