States and measures on hyper BCK-Algebras

We define the notions of bosbach states and inf-bosbach states on a bounded hyper bck-algebra (h,°,0,e) and derive some basic properties of them. we construct a quotient hyper bck-algebra via a regular congruence relation. we also define a ° - compatibled regular congruence relation θ and a θ - compatibled inf-bosbach state s on (h,°,0,e). by inducing an inf-bosbach state s ^ on the quotient structure h / [ 0 ] θ,we show that h / [ 0 ] θ is a bounded commutative bck-algebra which is categorically equivalent to an mv-algebra. in addition,we introduce the notions of hyper measures (states/measure morphisms/state morphisms) on hyper bck-algebras,and present a relation between hyper state-morphisms and bosbach states. then we construct a quotient hyper bck-algebra h / ker (m) by a reflexive hyper bck-ideal ker (m). further,we prove that h / ker (m) is a bounded commutative bck-algebra. © 2014 xiao-long xin and pu wang.