Super BCI / BCK-algebraic States

In order to formalize the intuitive notion of average truth-value of a propo-sition in fuzzy logic, professor Mundici introduced the concept of states on MV-algebras in 1995. In the present study, we concentrate on the fundamental theory of states on hyper BCI/BCK-algebras. The main content we have done is as follows:First of all, states on hyper BCK-algebras have been introduced. Based on the definition of states of a nonempty subset s:s(A)= inf{s(t)|t ∈ A}, we introduce the concept of inf-Bosbach states on a bounded hyper BCK-algebra (H, o,0, e), derive the related properties and propose the equivalence characteri-zation, that is, a function s:H→[0,1], which satisfying the condition s(e)= 1, is an inf-Bosbach state if and only if for all x,y ∈ H, s(x ∧y)= s(y ∧x) and x《 y=(?) s(y o x)= s(y)-s(x) hold.Secondly, we study states on bounded quotient hyper BCK-algebras. We construct the quotient hyper BCK-algebra H/[O]θ by a regular congruence rela-tion θ. Then we induce a θ- compatible inf-Bosbach state s on (H, o,0, e) to H/[O]θ. And then we show that H/[O]θ is a bounded commutative BCK algebra via the definition of o- compatible regular congruence relation θ. Furthermore, we also introduce the notions of hyper measures, hyper state-morphisms and some related notions on hyper BCK-algebras, discuss the basic properties of them, and induce hyper state-morphisms to the quotient hyper BCK-algebra.At last, we research hyper BCI-algebras, quotient hyper BCI-algebras and states on them. Based on the definition of hyperatoms and path, we give the concept of good hyper BCI-algebras. And then we study the quotient hyper BCI-algebras via a regular congruence relation. We introduce the notion of inf- Bosbach states on good hyper BCI-algebras, discuss some basic properties, and then induce it to quotient hyper BCI-algebras.