States on pseudo $L$-algebras

Abstract

This paper introduces states on pseudo $L$-algebras. We first define pseudo $KL$-algebras and pseudo $CL$-algebras, then explore their properties and relationships between pseudo $L$-algebras and pseudo $BCK$-algebras. Next, we define Bosbach states on pseudo $L$-algebras and discuss their properties. At the same time, we also introduce the fantastic ideal and the pseudo $MV$-ideal, also use these ideals to study the existence of states on pseudo $L$-algebras. Moreover, we introduce the state-morphism on pseudo $L$-algebras, discuss the relationship between Bosbach states and state-morphisms, and give an equivalent description of state-morphisms. Further, we introduce the concept of Riecan states and study their properties, as well as the relationship between Bosbach states and Riecan states. We have shown that any Bosbach state can be considered a Riecan state in bounded pseudo $L$-algebras, but not the other way around. Finally, we prove that Riecan states on special pseudo $L$-algebras are Bosbach states.