Form equivalence class Howard-Rubin Number: 14
Statement: Let\(\)G_i(x_{i_1}, x_{i_2}, \ldots, x_{i_k}), i\in I\tag* \(\)be a system of equations in the variables \(\{x_j: j\in V\}\)with the following properties:\itemitem{(1)} For each \(j\in V\), the variable \(x_j\), has a finite domain,\(D_j\).\itemitem{(2)} Given any finite number of variables, there is an equationin the system (*) which contains those variable and possibly others.\itemitem{(3)} For each equation in the system (*) there is an equation in(*) with the same, or possibly more, variables which has a solution. \item{}Then, the system (*) has a partial solution. (The family of equations\((x_j=d_j)_{j\in V}\) with \(d_j\in D_j\) is called a partialsolution of the system (*) if for every finite subfamily of equations\(\)x_i = d_i, \ldots, x_k = d_k,\tag**\(\) there isan equation in the system (*) with the variables \(x_i, \ldots, x_k\), andpossibly other variables, such that (**) is part of the completesolution of that equation.) Abian [1973] and Note 149.
Howard-Rubin number: 14 CY
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