Gödel Homomorphisms as Gödel Modal Operators

We extend propositional Gödel logic by a unary modal operator, which we interpret as Gödel homomorphisms, i.e. functions [0, 1] → [0, 1] that distribute over the interpretations of the binary connectives of Gödel logic. We show weak completeness of the propositional fragment w.r.t. a simple superintuitionistic Hilbert-type proof system, and we prove that validity does not change if we use the function class of continuous, strictly increasing functions. We also give proof systems for restrictions to sub- and superdiagonal functions.