Some Properties of Generalized State Operators on Residuated Lattices

We define a generalized state operator σ on a residuated lattice X and a g-state residuated lattice (X,σ), and consider properties of g-state residuated lattices. We show that a characterization theorem of σ-filters and that the class F_σ (X) of all σ-filters of a g-state residuated lattice (X, σ) is a Heyting algebra. Moreover we prove that every g-state residuated lattice (X, σ) is isomprphic to a subdirect product of g-state residuated lattices {(X/P, σ/P)}_P∈Specσ(X), where Spec_σ(X) is the set of all prime σ-filters of (X, σ).