Reciprocal Frank triplets in fuzzy logic

In fuzzy logic, the usual way of choosing interpretations of logic connectives AND, OR, NOT is guided by the De Morgan law, resulting in De Morgan triplets. However, there are some important problem classes where De Morgan triplets do not exist. Instead, a new relationship is required among the connectives, which turn out to be members of the parameterized Frank family of t-norms and t-conorms. Based on such experiments, in this paper we introduce reciprocal Frank triplets in an explicite way. We recall a forgotten approach to their natural introduction and study. Some applications are also mentioned where reciprocal Frank triplets play a key role.