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States and Measures on Hyper BCK‐Algebras

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  • Xiao-Long Xin
  • Pu Wang

Abstract

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.

Suggested Citation

  • Xiao-Long Xin & Pu Wang, 2014. "States and Measures on Hyper BCK‐Algebras," Journal of Applied Mathematics, John Wiley & Sons, vol. 2014(1).
    • Handle: RePEc:wly:jnljam:v:2014:y:2014:i:1:n:397265
    DOI: 10.1155/2014/397265
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    References listed on IDEAS

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    1. Jianming Zhan & Dajing Xiang & Zhisong Tan, 2004. "Fuzzy multiply positive implicative hyper BCK-ideals of hyper BCK-algebras," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 2004, pages 1-15, January.
    2. Marc Krasner, 1983. "A class of hyperrings and hyperfields," International Journal of Mathematics and Mathematical Sciences, Hindawi, vol. 6, pages 1-5, January.
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