Dualities in the class of extended Boolean functions

We introduce and discuss duality operators on the set of binary extended Boolean functions, i.e., on the set of binary operations on the real interval [0,1] whose restrictions to Boolean inputs yield Boolean functions. These dualities have been divided into seven classes, and the majority of their properties depend on the class they belong to. We study composition of dualities, properties of dual classes, duality of properties of extended Boolean functions and invariantness of extended Boolean functions with respect to particular dualities. Our approach allows to transfer the results known for some studied class of extended Boolean functions into the results for the corresponding dual classes. As typical examples, one can recall the standard duality of the classes of all t-norms and t-conorms and the duality of implication functions and conjunctive (disjunctive) aggregation functions.