Rotation-invariant t-norm solutions of a system of functional equations

The generalization of the disjunctive and conjunctive Boolean normal forms to fuzzy set theory, obtained by interpreting ∧ as a t-norm, ∨ as a t-conorm and ′ as a negator, often provides a kind of standard fuzzification procedure. A system of functional equations turns up if some functional independence of the difference between both generalizations is demanded. In this paper, we explore which De Morgan triplets, based on a left-continuous t-norm T, solve this system. Imposing some extra continuity conditions on T, these t-norm solutions can be obtained by the rotation construction of Jenei. The Cauchy equation plays a key role in the reasoning process.