Butler groups with a single \(\tau\)-adic relation (Q1387415)

The following results are proved: Theorem 2. For a torsion-free Abelian group \(B\) of finite rank \(n\) with a \(\tau\)-adic relation, the following conditions are equivalent: 1) \(B\) is a Butler group; 2) the group \(B\) can be defined by a finite system of \(\tau\)-adic relations with rational coefficients; 3) there exists a \(\tau\)-adic relation with piecewise rational coefficients that defines the group \(B\); 4) there exists a \(\tau\)-space of dimension \(n\) that defines the group \(B\) and whose elements are piecewise rational \(\tau\)-adic numbers; 5) any \(\tau\)-space defining \(B\) has the form \(\gamma V\) , where \(\gamma\) is an invertible \(\tau\)-adic number and \(V\) is a space of dimension \(n\) whose elements are piecewise rational numbers. Theorem 3. The system \(\begin{cases} t_1x+s_1y=0, &t_1,s_1\in\mathbb{Q}(\tau_1),\\ \ldots &\ldots\\ t_mx+s_my=0, &t_m,s_m\in\mathbb{Q}(\tau_m),\end{cases}\) defines a Butler group \(B\) of rank 2 with Richman type \(\sigma_1\leq\sigma_2\) where \(\sigma_1=\bigvee_{i\neq j}(\tau_i\wedge\tau_j)\) and \(\sigma_2=\tau_1\vee\dots\vee\tau_m\).