New types of generalized Bosbach states on non-commutative residuated lattices

Generalized Bosbach states of types I and II and hybrid generalized Bosbach states of types I and II are useful for the development of algebraic theory of probabilistic models for non-commutative fuzzy logics. In this paper, eight types of hybrid generalized Bosbach states of types III-1, III-2, IV-1, IV-2, V-1, V-2, VI-1 and VI-2 (or simply, hybrid types III-1, III-2, IV-1, IV-2, V-1, V-2, VI-1 and VI-2 state) on non-commutative residuated lattices are introduced. The relationships among hybrid generalized Bosbach states and properties of them are studied. Particularly, hybrid type III-1 state (resp. III-2 ) implies type I state (resp. hybrid type I state); hybrid type IV-1 (resp. IV-2) states are a new type of generalized Bosbach state which are different from type I, II and hybrid I, II states; hybrid type V-1 (resp. V-2) states can be equivalently defined by both type I (resp. hybrid type I) states and hybrid type IV-1 (resp. hybrid type IV-2) states; etc. The relationships between new types of generalized Bosbach states and (hybrid) generalized state-morphisms and (hybrid) generalized Riečan states are investigated.