8. Generalized States on Residuated Structures

We introduce two kinds of generalized states; namely we define generalized states of type I and II, we study their properties and we prove that every strong type II state is an order-preserving type I state. We prove that any perfect FL _w -algebra admits a strong type I and type II state. Some conditions are given for a generalized state of type I on a linearly ordered bounded Rℓ-monoid to be a state operator.

We introduce the notion of generalized state-morphism and we prove that any generalized state morphism is an order-preserving type I state and, in some particular conditions, an order-preserving type I state is a generalized state-morphism. The notion of a strong perfect FL _w -algebra is introduced and it is proved that any strong perfect FL _w -algebra admits a generalized state-morphism. The notion of generalized Riečan state is also given, and the main results are proved based on the notion of Glivenko property defined for the non-commutative case. The main results consist of proving that any order-preserving type I state is a generalized Riečan state and in some particular conditions the two states coincide. We introduce the notion of a generalized local state on a perfect pseudo-MTL algebra A and we prove that, if A is relative free of zero divisors, then every generalized local state can be extended to a generalized Riečan state. The notions of extension property and Horn-Tarski property are introduced. Finally, we outline how the generalized states give an approach of a theory of probabilistic models for non-commutative fuzzy logics associated to a pseudo t-norm.