On <i>α</i>-state filters in state residuated lattices

Abstract The purpose of this paper is to investigate $\alpha $-state filters in a state residuated lattice. First, the notion of $\alpha $-state filters in a state residuated lattice is introduced. It is proved that the set of all $\alpha $-state filters in a state residuated lattice forms a complete Heyting algebra. Furthermore, the concept of quasicomplemented state residuated lattices is presented. Some equivalent conditions are derived for quasicomplemented state residuated lattices. Finally, the hull-kernel topology on the set of all prime $\alpha $-state filters in a state residuated lattice is investigated. It is demonstrated that the set of all prime $\alpha $-state filters under the hull-kernel topology is a spectral space and some certain conditions are given for the space to be either a $T_{1}$-space or a Hausdorff space. In addition, some topological characterizations are summarized for a state residuated lattice to be quasicomplemented.