Probabilistic averaging in bounded commutative residuated ℓ-monoids

AbstractBounded commutative Rℓ-monoids generalize BL-algebras (and consequently MV-algebras). Nevertheless that such monoids in contrast to MV-algebras or Boolean algebras do not admit an analogue of the addition, in general, we are able to introduce states, which generalize states on MV-algebras. States are analogues of probability measures. We exhibit the state space of the monoids proving that the set of extremal states is a nonempty compact Hausdorff topological space homeomorphic with the set of maximal filters endowed with the hull-kernel topology