Gödel logics with monotone operators

We consider the extension of Gödel logic by a unary operator interpreted by functions on the unit interval with certain monotonicity properties. We prove that validity of propositional formulas is decidable by giving a sound and complete proof system with finitely many axioms. We show also how to transfer the deduction theorem, the lifting lemma and the agreement of entailment and 1-entailment from Gödel logic to the propositional fragment of our extension. Finally, we prove an enumerability result for a ring-normal prenex fragment.