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Journal of Applied Mathematics
Volume 2014, Article ID 397265, 7 pages
Research Article

States and Measures on Hyper BCK-Algebras

Department of Mathematics, Northwest University, Xi'an 710069, China

Received 20 November 2013; Accepted 1 February 2014; Published 13 March 2014

Academic Editor: Baolin Wang

Copyright © 2014 Xiao-Long Xin and Pu Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a regular congruence relation and a inf-Bosbach state on . By inducing an inf-Bosbach state on the quotient structure / , we show that / 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 / by a reflexive hyper BCK-ideal . Further, we prove that / is a bounded commutative BCK-algebra.