Compact propositional Godel logics

Entailment in propositional Godel logics can be defined in a natural way. While all infinite sets of truth values yield the same sets of tautologies, the entailment relations differ. It is shown that there is a rich structure of infinite-valued Godel logics, only one of which is compact. It is also shown that the compact infinite-valued Godel logic is the only one which interpolates, and the only one with an r.e. entailment relation.