Generalized states on EQ-algebras

In this paper, we introduce a notion of generalized states from an EQ-algebra E۱ to another EQ-algebra E۲, which is a generalization of internal states (or state operators) on an EQ-algebra E. Also we give a type of special generalized state from an EQ-algebra E۱ to E۱, called generalized internal states (or GI-state). Then we give some examples and basic properties of generalized (internal) states on EQ-algebras. Moreover we discuss the relations between generalized states on EQ-algebras and internal states on other algebras, respectively. We obtain the following results: (۱) Every state-morphism on a good EQ-algebra E is a G-state from E to the EQ-algebra E۰ = ([۰,۱],∧۰,⊙۰,∼۰,۱). (۲) Every state operator µ satisfying µ(x)⊙µ(y) ∈ µ(E) on a good EQ-algebra E is a GI-state on E. (۳) Every state operator τ on a residuated lattice (L,∧,∨,⊙,→,۰,۱) can be seen a GI-state on the EQ-algebra (L,∧,⊙,∼,۱), where x ∼ y := (x → y) ∧ (y → x). (۴) Every GI-state σ on a good EQ-algebra (L,∧,⊙,∼,۱) is a internal state on equality algebra (L,∧,∼,۱). (۵) Every GI-state σ on a good EQ-algebra (L,∧,⊙,∼,۱) is a left state operator on BCK-algebra (L,∧,→,۱), where x → y = x ∼ x∧y. 

Keywords:

EQ-algebra , generalized state , internal state , residuated lattice , Equality algebra , BCK-algebra