On Hyperrings Associated with -fuzzy Relations

The association between hyperstructures and binary relations had been intensively studied 5,6,10,11,2,4,20 . Chvalina 5,6 and Hort 2 use ordered structures for the construction of semigroups and hypergroups. Rosenberg 4 associated with any binary relation of full domain, a hypergroupoid and found conditions on , such that is a hypergroup. Corsini and Leoreanu 16 study hypergroups and binary relations. Connections between algebraic hyperstructures and fuzzy sets have been considered for the first time by Corsini 12 . They were studied afterwards by Corsini 13,14 Corsini and Leoreanu 17,18 , Cristea 3 , Davvaz 1 and many others. The interest in fuzzy logic has been rapidly growing recently. Several new algebras playing the role of structures of true values have been introduced and axiomatized. The most general structure considered here is that of a residuated lattice. In this paper we will use complete residuated lattice ( ) as the structure of truth values. An -fuzzy relation on a set is any function from into . -fuzzy relations and fuzzy homomorphisms of algebras have been studied by Ignjatović, Ćirić and Bogdanović 7,8 and many others. In section 3 of this paper we associate a hyperring ( -ring) to every semigroup ( ) equipped with reflexive and transitive -fuzzy relation which is compatible with operation . Then we show that under certain condition there exists (inclusion) monomorphism between the constructed hyperring ( ) and the hyperring of multiendomorphisms of hypergroup ( ). We investigate morphisms of hyperrings associated with -fuzzy relations, too. Also, we show that if the hyperring ( ) is associated with fuzzy congruence on the semigroup ( ) then there exists a class of functions which are not necessary homomorphisms of semigroups ( ) and ( ) such that is a strong homomorphism of hyperrings ( ) and ( ) for the responsive fuzzy – congruence and it is valid for the responsive classes of associated -rings, too. Finally, we show that if ( ) is a hyperring associated with -fuzzy relation on a semigroup ( ), then there exists a class of congruences on that hyperring,