States and Measures on Hyper BCK‐Algebras

Xiao-Long Xin and Pu Wang

Journal of Applied Mathematics, 2014, vol. 2014, issue 1

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.

Date: 2014
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https://doi.org/10.1155/2014/397265

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2014:y:2014:i:1:n:397265

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