On Mixed Fuzzy Topological Ring

The theory of fuzzy topological ring has wide scope of applicability than order topological ring theory. The reason is fuzzy can provide better result. Therefore, fuzzy topological ring has been found in Robotics, computer, artificial intelligent, etc. In this paper, we induce mixed fuzzy topological ring space and fuzzy neighborhoods system of mixed fuzzy topological ring space. Also, we study fuzzy continuity of mixed fuzzy topological ring.


Definition 1.1[9]
A fuzzy set in  is a map :  →  and, that is, belonging to   (the set of all fuzzy set of ) .Let  ∈   , for every  ∈  , we expressed by () of the degree of membership of  in  .If () be an element of {0, 1} , then  is said a crisp set.

Definition 1.4 [4]
A family  of fuzzy nbhds of   , for 0 <  ≤ 1, is called a fund.system of fuzzy nbhds of   iff for any fuzzy nbhd  of   , there is  ∈  such that   ≤  ≤

Definition 1.5[4]
Let  be a ring and  a FZT on .Let  and  are fuzzy sets in .We define  +  , − and . as follows

bi-fuzzy topological rings and mixed fuzzy topological rings
We study mixed fuzzy topological ring and fuzzy nbhds of mixed fuzzy topological ring

Definition 2.1{5}
Let  be any ring equipped with two fuzzy topological ring space  and .Then the triplet (, , ) is defined as a bi-fuzzy topological ring space.

Example 2.2
Let  be any ring with the indiscrete fuzzy topology  and the discrete fuzzy topology .Then, (, , ) is a bi-fuzzy topological ring

Definition 2.5
Let (, , ) be any bi-fuzzy topological ring.The fuzzy topology ()determined on  by the collection { ∈   : ∃ ∈  .   () ≤ } of all fuzzy open nbhds of 0 such that (, ()) is a fuzzy topological ring, is defined as a mixed fuzzy topology ring

Example 2.6
In the bi-fuzzy topological ring (, , ), let us put  = , the indiscrete fuzzy topology on  and  = , the discrete fuzzy topology on , then in both cases, we have () = .

Example 2.7
In the bi-fuzzy topological ring (, , ), let us put  = , the discrete fuzzy topology and  = , the indiscrete fuzzy topology, then in both cases, we have () =