State theory on bounded hyper EQ-algebras

In a hyper structure (X,⋆)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(X,\star )$$\end{document}, x⋆y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x\star y$$\end{document} is a non-empty subset of X. For a state s, s(x⋆y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s(x\star y)$$\end{document} need not be well defined. In this paper, by defining s∗(x⋆y)=sup{s(z)∣z∈x⋆y}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$s^*(x\star y)=sup\{s(z)\mid z\in x\star y\}$$\end{document}, 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 s¯\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\bar{s}}$$\end{document} on the quotient hyper EQ-algebra H/θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H/\theta $$\end{document}. We also give the condition, by which a bounded hyper EQ-algebra admits a sup-Bosbach state.