Abstract In this paper, we propose a definition of system-consistent (sys-consistent) functions for neighborhood systems and compare it with a previous definition of granule-based consistent (gra-consistent) functions in the literature. We show that sys-consistency achieves the same level of generality as gra-consistency in the sense that the former subsumes all existing definitions of consistent functions that are known to be special cases of the latter. Then, we prove that sys-consistent functions are structure-preserving mappings with respect to interior and closure operators on neighborhood systems, whereas gra-consistent functions are not. In addition, we connect consistent functions with well-known model-theoretic notions of bisimulations and bounded morphisms in modal logic. As a consequence, this implies that properties described by modal formulas remain invariant under consistent mappings. Finally, we show that most (albeit not all) of the above-mentioned results still hold for some variants and extensions of the basic definition.

中文翻译：

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关于邻域系统的一致函数

摘要 在本文中，我们提出了邻域系统的系统一致（sys-consistent）函数的定义，并将其与文献中先前的基于粒度的一致（gra-consistent）函数的定义进行了比较。我们表明 sys-consistency 实现了与 gra-consistency 相同的一般性水平，因为前者包含了已知为后者特例的一致函数的所有现有定义。然后，我们证明 sys-consistent 函数是关于邻域系统上的内部和闭包算子的结构保持映射，而 gra-consistent 函数不是。此外，我们将一致函数与模态逻辑中众所周知的互模拟和有界态射的模型理论概念联系起来。作为结果，这意味着模态公式描述的属性在一致映射下保持不变。最后，我们表明大多数（尽管不是全部）上述结果仍然适用于基本定义的某些变体和扩展。