States and Measures on Hyper BCK-Algebras

Abstract

We define the notions of Bosbach states and inf-Bosbach states on a bounded hyper BCK-algebra $(H,\circ ,0,e)$ and derive some basic properties of them. We construct a quotient hyper BCK-algebra via a regular congruence relation. We also define a $\circ \text{-}\mathrm{compatibled}$ regular congruence relation $\theta$ and a $\theta \text{-}\mathrm{compatibled}$ inf-Bosbach state $s$ on $(H,\circ ,\mathrm{0,}e)$. By inducing an inf-Bosbach state $\widehat{s}$ on the quotient structure $H$ / ${[0]}_{\theta }$, we show that $H$ / ${[0]}_{\theta }$ 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 $H$ / $\text{Ker}(m)$ by a reflexive hyper BCK-ideal $\text{Ker}(m)$. Further, we prove that $H$ / $\text{Ker}(m)$ is a bounded commutative BCK-algebra.