Generalized state operators on residuated lattices

Abstract

We give a definition of a generalized state operator (g-state operator) $$\sigma $$ on a residuated lattice X and a g-state residuated lattice $$(X,\sigma )$$ , by which the class of all g-state residuated lattices is proved to be a variety, and consider properties of g-state residuated lattices. We prove some fundamental results about them, such as characterizations of $$\sigma $$ -filters, extended $$\sigma $$ -filters, homomorphism theorems for g-state residuated lattices. Moreover, we show that every g-state residuated lattice is a subdirect product of subdirectly irreducible g-state residuated lattices.