States on Pseudo-BCI Algebras

Abstract

In this paper, we discuss the structure of pseudo-BCI algebras and get that any pseudo-BCI algebra is a union of it's branches. We introduce the notion of local bounded pseudo-BCI algebras and study some related properties. Moreover we define two operations $\wedge_1$, $\wedge_2$ in a local bounded pseudo-BCI algebra $A$ and two local operations $\vee_1$ and $\vee_2$ in $V(a)$ for $a\in M(A)$. We show that in a local $\wedge_1$($\wedge_2$)-commutative local bounded pseudo-BCI algebra $A$, $(V(A),\wedge_1,\vee_1)$($(V(A),\wedge_2,\vee_2)$) forms a lattice for all $a\in M(a)$. We define a Bosbach state on a local bounded pseudo-BCI algebra. Then we give two examples of local bounded pseudo-BCI algebras to show that there is local bounded pseudo-BCI algebras having a Bosbach state but there is some one having no Bosbach states. Moreover we discuss some basic properties about Bosbach states. If $s$ is a Bosbach state of a local bounded pseudo-BCI algebra $A$, we prove that $A/ker(s)$ is equivalent to an MV-algebra. We also introduce the notion of state-morphisms on local bounded pseudo-BCI algebras and discuss the relations between Bosbach states and state-morphisms. Finally we give some characterization of Bosbach states.