Let \(n\) be a positive integer and \(\mathcal{R}\) be a \(2\) and \(n!\)-torsion free commutative ring with unity. Let \(X\) be a locally finite pre-ordered set. If any two directed edges in each connected component of the complete Hasse diagram \((X,\mathfrak{D})\) are contained in one cycle, then every \(n\)-commuting map on the incidence algebra \(I(X,\mathcal{R})\) is proper.

中文翻译：

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关联代数上的线性 $$n$$ n - 通勤图

设\(n\)是一个正整数，而\(\mathcal{R}\)是一个\(2\)和\(n!\) -具有统一性的无扭交换环。令\(X\)是一个局部有限的预排序集合。如果完整 Hasse 图的每个连通分量中的任意两条有向边\((X,\mathfrak{D})\)包含在一个循环中，那么每个\(n\) -关联代数上的通勤映射\(I (X,\mathcal{R})\)是正确的。