States and Measures on Hyper BCK-Algebras

Xiao-Long Xin,Pu Wang

doi:10.1155/2014/397265

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∘-compatibledregular congruence relationθand aθ-compatibledinf-Bosbach stateson(H,∘,0,e). By inducing an inf-Bosbach states^on the quotient structureH/[0]θ, we show thatH/[0]θis 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-algebraH/Ker(m)by a reflexive hyper BCK-idealKer(m). Further, we prove thatH/Ker(m)is a bounded commutative BCK-algebra.

Highlights

IntroductionThe theory of hyper structures ( called multialgebras) was introduced in 1934 by Marty [1] at the 8th Congress of Scandinavian Mathematicians
The theory of hyper structures was introduced in 1934 by Marty [1] at the 8th Congress of Scandinavian Mathematicians
We mainly study the state theory on hyper structures and introduce a notion of states on hyper BCKalgebras

Summary

Introduction

The theory of hyper structures ( called multialgebras) was introduced in 1934 by Marty [1] at the 8th Congress of Scandinavian Mathematicians. Jun et al [6] introduced the concept of hyper BCK-algebras which is a generalization of BCKalgebras and studied some properties of them. Corina Ciungu and Dvurecenskij [17] extended the notions of measures and states, which were presented in the paper of Dvurecenskij and Pulmannova [18] to the case of pseudo-BCK-algebras. They studied similar properties and proved that the notion of states in the sense of Dvurecenskij and Pulmannova [18] coincides with the Bosbach state. We mainly introduce and study the state theory on hyper BCK-algebras.

Preliminaries

States on Bounded Hyper BCK-Algebras

States on Quotient Hyper BCK-Algebras

Hyper Measures on Hyper BCK-Algebras

Conclusions