Let \({\mathcal {R}}\) be a 2-torsion-free commutative ring with unity, X a locally finite pre-ordered set with a finite number of connected components and \(I(X,{\mathcal {R}})\) the incidence algebra of X over \({\mathcal {R}}\). In this paper, we give a sufficient and necessary condition for a commuting map on \(I(X,{\mathcal {R}})\) to be proper.

中文翻译：

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在某些入射代数上通勤地图

令\（{\ mathcal {R}} \）是一个无扭转的2交换环，X是一个局部有限的有序集合，具有有限数量的连接分量，并且\（I（X，{\ mathcal { R}}）\）X在\（{\ mathcal {R}} \）上的入射代数。在本文中，我们为\（I（X，{\ mathcal {R}}）\）上的换向图给出适当的充要条件。