Elementary equivalence of Chevalley groups (Q2784522)

The so-called elementary properties of an algebraic system are those that can be expressed in the language of restricted predicate calculus, taking as given the basic operations and relations of the system. Two algebraic systems \(G_1\) and \(G_2\) are said to be elementarily equivalent, denoted by \(G_1\equiv G_2\), if they have the same elementary properties. Let \({\mathcal G}_1\) and \({\mathcal G}_2\) be two Chevalley groups constructed from algebraically closed fields \(k_1\) and \(k_2\) of characteristic different from two, from simple Lie algebras \({\mathcal L}_1\) and \({\mathcal L}_2\) and from lattices \(L\) and \(M\), respectively. Assume \(L/L_0\cong\varphi_1\) and \(M/M_0\cong\varphi_2\). The author proves that \({\mathcal G}_1\equiv{\mathcal G}_2\) if and only if \(k_1\equiv k_2\), \({\mathcal L}_1\cong{\mathcal L}_2\) and \(\varphi_1\cong\varphi_2\).