States and measures on hyper BCK-algebras
Summary: 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 \(\circ\)-\textit{compatibled} regular congruence relation \(\theta\) and a \(\theta\)-\textit{compatibled} inf-Bosbach state \(s\) on \((H,\circ,0,e)\). By inducing an inf-Bosbach state \(\hat s\) 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 \(H\)/Ker\((m)\) by a reflexive hyper BCK-ideal Ker\((m)\). Further, we prove that \(H\)/Ker\((m)\) is a bounded commutative BCK-algebra.
scientific article; zbMATH DE number 7010621