On state residuated lattices

Abstract

In the paper, we introduce the notion of state operators on residuated lattices and investigate some related properties of such operators. Also, we give characterizations of Rl-monoids and Heyting algebras, and discuss relations between state operators and states on residuated lattices. Using state filters on state residuated lattices, we characterize two kinds of state residuated lattices, which are state simple and state local. Moreover, we focus on algebraic structures of the set $$SF[L]$$SF[L] of all state filters on state residuated lattices. We obtain that $$SF[L]$$SF[L] forms a coherent frame and a pseudocomplemented lattice. Then, we introduce the co-annihilator of a nonempty set $$X$$X with respect to a state filter $$F$$F and study some properties of them. As applications, we show that if $$F$$F and $$G$$G are filters in a residuated lattice, then the co-annihilator of $$G$$G with respect to $$F$$F is the relative pseudocomplement of $$G$$G with respect to $$F$$F in the lattice $$F[L]$$F[L] of all filters of residuated lattices.