CHARACTERIZATION OF FUZZY NEIGHBORHOOD COMMUTATIVE DIVISION RINGS II

We give a characterization of fuzzy neighborhood commutative division ring; and present an alternative formulation of boundedness introduced in fuzzy neighborhood rings. The notion of β-restricted fuzzy set is considered.

Just like our previous work, we consider here the fuzzy neighborhood topology t(E) on D, the one generated by the well-known fuzzy neighborhood system E of R. Lowen [8].The pair (D,t(E)) is termed as a fuzzy neighborhood space.The triplet (D, +,.(or D alone)is considered either a ring, division ring or commutative division ring (whichever we require).D*: D\{0} denotes the multiplicative group of nonzero elements of the commutative division ring D and D + is the additive group of D.
(FNDR2) The mapping r:(D*,t(EiD.))--,(D*,t(Eio.))x -+ x -a is continuous where EID. is the fuzzy neighborhood system on D" induced by D. A commutative division ring structure and a fuzzy neighborhood E on D are said to be compatible if the conditions (FNDR1) and (FNDR2) axe satisfied.
Then by definition it is an additive group and therefore a fuzzy neighborhd group.We show division d:DxD'D,(z,) is continuous.
2=3.If the division d is continuous on D x D*, then certainly the restriction to D*x D* is continuous, i.e., (D*, .,t(EID.))is a fuzzy neighborhood group.3=1.We need to show that m:DxD-,D,(z,y) zy is continuous.But this follows from Theorem 3.3 [4].THEOREM 2.3.Let (D, +, .)be a commutative division ring with characteristic Char(D) 2. Then a fuzzy neighborhood group on the commutative division ring D with respect to which the inversion is continuous is a fuzzy neighborhood commutative division ring.
Now we prove that the right side of (2.8) is less than or equal to # + 6..