Let \(\mathbb {N}\) be the set of all nonnegative integers. Let W be a nonempty subset of \(\mathbb {N}\) and \(F^{*}(W)\) be the set of all finite, nonempty subsets of W. For any integer \(g\ge 2\) , let \(A_{g}(W)\) be the set of all numbers of the form \(\sum \limits _{f\in F} \varepsilon _{f}g^{f}\) , where \(F\in F^{*}(W)\) and \(1\le \varepsilon _{f}\le g-1\) . Let \(\mathbb {N}=W_{1}\cup W_{2} \cup \cdots \cup W_{h}\) be a partition such that set \(W_{i}\) is infinite for \(i=1, 2, \ldots , h\) . Is the asymptotic basis \(A=A_{g}(W_{1})\cup A_{g}(W_{2})\cup \cdots \cup A_{g}(W_{h})\) minimal for all partition \(\mathbb {N}=W_{1}\cup W_{2} \cup \cdots \cup W_{h}\) ? In this paper, we focus on this problem for \(h\ge 3\) and prove that \(A=A_{3}(W_{1})\cup A_{3}(W_{2}) \cup \cdots \cup A_{3}(W_{h})\) is not a minimal asymptotic basis of order h if \(W_{1}\) contains consecutive integers, \(W_{k}-1\subseteq W_{1}\) and \(W_{l}-1\subseteq W_{1}\) for two distinct integers \(k, l\in \{2, \ldots , h\}\) . Under the assumption that \(W_{1}\) contains consecutive integers and \(W_{2}-1\subseteq W_{1}\) , we also prove that \(A=A_{3}(W_{1})\cup A_{3}(W_{2})\cup A_{3}(W_{3})\) is a minimal asymptotic basis of order three if and only if \(W_{3}-1\not \subseteq W_{1}\) .
