State operators and state-morphism operators on hyper BCK-algebras

In this paper, we first introduce the notion of state operators on a hyper BCK-algebra and investigate some properties of them. Also we introduce the notions of state hyper (weak hyper, strong hyper) BCK-ideals and discuss relations among them. Moreover we study state congruences on state hyper BCK-algebra (X, sigma) and show that there exists a bijection between all state regular strong congruence relations and all state hyper reflexive strong BCK-ideals on (X, sigma). Then, we induce quotient state hyper BCK-algebra on a bounded hyper BCK-algebra by the kernel of regular commutative state operator and prove that it is a bounded commutative BCK-algebra, that is an MV-algebra. Furthermore, we introduce (strong) state-morphism operators on hyper BCK-algebras and discuss the relationship between (strong) state-morphism operators and state operators. We also introduce the adjoint pairs on a hyper BCK-algebra and obtain that there is a one-to-one correspondence between the regular adjoint pairs and the regular state-morphism operators on a hyper BCK-algebra.