States and Measures on Hyper BCK-Algebras

• Main Authors: Xiao-Long Xin, Pu Wang
• Format: Article
• Language: English
• Published: Hindawi Limited 2014-01-01
• Series: Journal of Applied Mathematics
• Online Access: http://dx.doi.org/10.1155/2014/397265

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.