Fuzzy implications - A (dis)similarity perspective

Fuzzy implications continue to remain an important class among the fuzzy logic connectives, having found utility in contexts outside of logical inference too. Recently, fuzzy implications have been shown to be a fertile source for obtaining distance functions with very beneficial properties. In this work, we show that fuzzy implications can also be a good wellspring of fuzzy compatibility relations with myriad properties. Quite interestingly, the converse also holds good, i.e., we also show that we can represent any fuzzy implication through a pair (d,E), where d is a distance function - called a pseudo-monometric - and E is a mono-similarity fuzzy relation. This representation helps us to propose an equivalent formulation of the problem of characterisation of QL-implications, that has the potential to offer computational savings in verifying if a QL-operation can be a QL-implication and also throws open further avenues of fruitful theoretical exploration.