Applications of states to BI-algebras

This paper aims is to introduce states, Bosbach states and state-morphism operators on BI-algebras. We define state ideals on BI-algebras and give a characterization of the least state ideal of a BI-algebra. It is proved that the kernel of a Bosbach state on a BI-algebra X is an ideal of X. Further, by these concepts, we introduce the notions of state BI-algebras and state-morphism BI-algebras. The notion of complement pairs of a BI-algebra X is defined, and proves that under suitable conditions, there is a one-to-one correspondence between complement pairs of BI-algebras and state-morphism operators on BI-algebras.