State theory on bounded hyper EQ-algebras

Abstract

In a hyper structure $(X,\star )$, $x\star y$ is a non-empty subset of X. For a state s, $s(x\star y)$ need not be well defined. In this paper, by defining $s^*(x\star y)=sup\{s(z)\mid z\in x\star y\}$, we introduce notions of sup-Bosbach states, state-morphisms and sup-Riečan states on a bounded hyper EQ-algebra and discuss the related properties. The states on bounded hyper EQ-algebras are the generalization of states on EQ-algebras. Then we discuss the relations among sup-Bosbach states, state-morphisms and sup-Riečan states on bounded hyper EQ-algebras. By giving a counter example, we show that a sup-Bosbach state may not be a sup-Riečan state on a hyper EQ-algebra. We give conditions in which each sup-Bosbach state becomes a sup-Riečan state on bounded hyper EQ-algebras. Moreover, we introduce several kinds of congruences on bounded hyper EQ-algebras, by which we construct the quotient hyper EQ-algebras. By use of the state s on a bounded hyper EQ-algebra H, we set up a state ${\bar{s}}$ on the quotient hyper EQ-algebra $H/\theta$. We also give the condition, by which a bounded hyper EQ-algebra admits a sup-Bosbach state.