On consistent functions for neighborhood systems

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.