States on pseudo EQ-algebras

In this paper, we mainly introduce the concepts of states on pseudo EQ-algebras and investigate their properties and existence as well as their relationships. Firstly, we define some notions and investigate their properties, which will be used in the following sections. Then, we define the concepts of Bosbach states and state-morphisms on pseudo EQ-algebras and investigate their properties and relationships. We prove that each state-morphism is a Bosbach state. Also, we introduce the fantastic filters and pseudo MV-filters and investigate the existence of states by using the two filters. We prove that they coincide on good pseudo EQ-algebras and there exists a Bosbach state if and only if there exists a fantastic filter under special pseudo EQ-algebras. Moreover, we introduce the notion of Riecan state and investigate their properties and connections between the Bosbach states and Riecan states. Also, we prove that any Bosbach state is a Riecan state in the normal pseudo EQ-algebras, but the inverse is not true in general. Finally, we prove that the Bosbach states and Riecan states coincide for a kind of particular pseudo EQ-algebras.