Interval-valued fuzzy coimplications and related dual interval-valued conjugate functions

• The notions of representability, conjugacy and duality of fuzzy connectives are discussed.
• The canonical representation of fuzzy coimplications are conceived as the dual construction of interval-valued fuzzy implications.
• The conjugate class of such representable coimplications preserves the corresponding main properties of interval-valued fuzzy implication class.
• Interval automorphisms acting on interval classes of border, model and S-coimplications are studied and summarized in commutative diagrams.

The aim of this paper is to introduce the dual notion of interval conjugate implications, the interval coimplications, as interval representations of corresponding conjugate fuzzy coimplications. Using the canonical representation, this paper considers both the correctness and the optimality criteria, in order to provide interpretation for fuzzy coimplications as the non-truth degree of conditional rule in expert systems and study the action of interval automorphisms on such interval fuzzy connectives. It is proved that interval automorphisms acting on NN-dual interval coimplications preserve the main properties of interval implications discussed in the literature including the duality principle. Lastly, the action of interval automorphisms on interval classes of border, model and S-coimplications are considered, summarized in commutative diagrams.