Generalized state operators on BCI-algebras

Abstract

In this paper, we introduce the notion of a generalized state operator by extending the codomain of a state operator to a more general algebraic structure, that is, from a BCI-algebra X to an arbitrary BCI-algebra Y . Also we give some special types of generalized state operators according to the structures of Y such as GD-states, GK-states and state operators. We get that generalized state operators are the generalization of state-morphisms and state-morphism operators on BCK/BCI-algebras. Then we focus on studying state operators on BCI-algebras and obtain some important results: (1) State BCI-algebras are a genuine generation of state MV-algebras. (2) There is a bijection between all state congruences and all closed state ideals on a state BCI-algebra. (3) If ( X , σ ) is a nontrivial subdirectly irreducible state BCI-algebra with Ker ( σ ) =0, then σ ( X ) is a nontrivial subdirectly irreducible subalgebra of X . Moreover we give some characterizations of several special types of BCI-algebras by use of state operators and differential states.