(untitled) (European Journal of Pure and Applied Mathematics, 2017)
In this paper, we discuss the structure of pseudo-BCI algebras and getÂ that any pseudo-BCI algebra is a union of it’s branches. We introduce the notion of local boundedÂ pseudo-BCI algebras and study some related properties. Moreover we define two operations _1, _2 in a local bounded pseudo-BCI algebra A and two local operations _1 and _2 in V(a) for aM(A). We show that in a local _1(_2)-commutative local boundedÂ pseudo-BCI algebra A, (V(A),_1,_1)((V(A),_2,_2)) forms a lattice for all aM(a).Â We define a Bosbach state on a local bounded pseudo-BCI algebra. Then we give two examples of local boundedÂ 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 BosbachÂ 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 boundedÂ pseudo-BCI algebras and discuss the relations between Bosbach states and state-morphisms.Â Finally we give some characterization of Bosbach states.