The stable marriage problem is a well-known problem of matching n men to n women to achieve a certain type of ``stability''; the \(O(n^ 2)\) time Gale-Shapley algorithm [\textit{D. Gale} and \textit{L.68D30 The author studies the class \({\mathcal K}\) of automata such that an \(\alpha_ 0\)-product of any automata from \({\mathcal K}\) is in \({\mathcal K}\), and for any \(A\in {\mathcal K}\), homomorphic images of A and subautomata of A are in \({\mathcal K}\). The \(\alpha\) \({}_ 0\)-product of automata is a special case of the direct product of automata. }