Algebraic study to generalized Bosbach states on residuated lattices

Generalized Bosbach states of type I and II, which are also called type I and II states, are useful for the development of algebraic theory of probabilistic models for fuzzy logics. In this paper, a pure algebraic study to the generalization of Bosbach states on residuated lattices is made. By rewriting the equations of Bosbach states, an alternative definition of type II states is given, and five types of generalized Bosbach states of type III, IV, V, VI and VII (or simply, type III, IV, V, VI and VII states) are introduced. The relationships among these generalized Bosbach states and properties of them are investigated by some examples and results. Particularly, type IV states are a new type of generalized Bosbach states which are different from type I, II and III states; type V (resp. VI) states can be equivalently defined by both type I (resp. II) states and type IV states; type I, II and III states are equivalent when the codomain is an MV-algebra as well as type V and type VI states.