Some types of filters and states on hyper NM-algebras

In this paper, we apply the hyper structures theory to NM-algebras and define the notion of hyper NM-algebras, which is a generalization of NM-algebras, and obtain some related results. We also state that any hyper NM-algebra is a hyper residuated lattice and the converse may not hold. Furthermore, we put forth some types of filters and deductive systems of hyper NM-algebras, such as (weak)h-filters, weak h-deductive systems and (positive) implicative weak h-deductive systems. Especially, we focus on discussing relationships of them. Finally, we give the definitions of sup-state, inf-state and hyper state on hyper NM-algebras and obtain there exists a hyper NM-algebra with sup-state and inf-state.