Compatibility of Fuzzy Relations

What is it about?

The notion of extensionality, introduced by Hohle and Blanchard, and the notion of compatibility, as coined by Belohlavek, of a fuzzy relation with respect to a fuzzy equality are trivially equivalent. Here, this compatibility property is dissected into left and right compatibility, mimicking the original twofold definition of extensionality, and studied in detail in the context of arbitrary fuzzy relations. Relying on the notions of left and right traces of a fuzzy relation, it is shown that compatibility can be characterized in terms of inclusions, shedding another light on the matter.

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Why is it important?

The notion of compatibility appears, among others, in the study of fuzzy functions by Perfilieva and colleagues, in the study of fuzzy order relations by Bodenhofer and colleagues and has been profoundly used in the lattice-theoretic approach to concept lattices by Belohlavek.