On monotony of fuzzy closure

Closure operators and systems play a significant role in a variety of areas of fuzzy logic. While the condition of monotony for ordinary closure operators has a straightforward form, two basic conditions of monotony naturally arise from the existing examples in a fuzzy setting. To unify these two conditions, two approaches can be found in the literature, one based on the notion of a filter of truth degrees and the other on the notion of a linguistic hedge. We present results connecting these approaches, explore their variants, and provide a notion of monotony that subsumes both the filter-based and hedge-based approaches. We study properties of this general concept of monotony and discuss open problems along with topics for future research.