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The logic FP(Ł,Ł) and two-sorted equational states

The logic FP(Ł,Ł) was introduced by L. Godo and T. Flaminio as an expansion of Łukasiewicz logic with a modality, to reason about the probability of vague events. We prove that FP(Ł,Ł) is complete with respect to a class of two-sorted algebras, called equational states. They are an equational presentation of the well-known theory of states over lattice ordered groups.