Decidable and undecidable prime theories in infinite-valued logic

In classical propositional logic, a theory T is prime (i.e., for every pair of formulas F,G, either T a2F\u2192G or T a2G\u2192F) iff it is complete. In Lukasiewicz infinite-valued logic the two notions split, completeness being stronger than primeness. Using toric desingularization algorithms and the fine structure of prime ideal spaces of free \u2113-groups, in this paper we shall characterize prime theories in infinite-valued logic. We will show that recursively enumerable (r.e.) prime theories over a finite number of variables are decidable, and we will exhibit an example of an undecidable r.e. prime theory over countably many variables