States on Pseudo BE-Algebras

In the present paper, we study Bosbach and Riecan states on pseudo BE-algebras. Additionally, we establish new properties of pseudo BE-algebras and we define the pseudo BE(A)-algebras. We introduce the notion of a normal Bosbach state proving that any Bosbach state on a pseudo BE(A)-algebra is normal. We prove that the quotient pseudo BE-algebra via the kernel of a normal Bosbach state on a pseudo BE(A)-algebra is a commutative pseudo BE-algebra. In the case of a bounded pseudo BE-algebra, the quotient pseudo BE-algebra via the kernel of a Bosbach state is involutive. The notion of a Riecan state on a good pseudo BE(A)-algebra is defined and it is proved that any Bosbach state on a good pseudo BE(A)-algebra is a Riecan state on it. Some conditions are given for a Riecan state on a good pseudo BE(A)-algebra to be a Bosbach state. Finally, we introduce the state-morphisms on a pseudo BE-algebra and we prove that any state-morphism is a Bosbach state.