TYPE-2 FUZZY G-TOLERANCE RELATION AND ITS PROPERTIES

In this short communication we generalize the definition of type-2 fuzzy tolerance relation and consequently the type-2 fuzzy G-tolerance relation in type-2 fuzzy sets. The type-2 fuzzy Gtolerance relation helps in finding the type-2 fuzzy G-equivalence relation. Moreover, we have studied the notion of type-2 fuzzy tolerance relation in abstract algebra.


Introduction
Type-2 fuzzy sets are relatively new in the world of fuzzy sets and systems.Although they were originally introduced in 1975 by L. A. Zadeh [21], type-2 sets did not gain popularity until their reintroduction by J. M. Mendel [12].These newer fuzzy sets were now thought of as an extension of the already popular fuzzy sets (now labelled type-1) to include additional uncertainties in the set.Type-2 fuzzy sets are unique and conceptually appealing, because they are fuzzy extension rather than crisp.Type-2 fuzzy sets have membership functions as type-1 fuzzy sets.The advantage of type-2 fuzzy sets is that they are helpful in some cases where it is difficult to find the exact membership functions for a fuzzy sets.There are wide variety of applications of type-2 fuzzy sets in science and technology like computing with words [14], human resource management [9], forecasting of time-series [11], clustering [1,17], pattern recognition [3], fuzzy logic controller [20], industrial application [4], simulation [18], neural network [2], [19], and transportation problem [13].
The concept of cartesian product of type-2 fuzzy sets was given by Hu et al. [10] as an extension of type-1 fuzzy sets.The properties of membership grades of type-2 fuzzy sets and set-theoretic operations of such sets have been studied by Mizumoto and Tanaka [15], [16].Dubois and Prade [6]- [8] discussed the composition of type-2 fuzzy relations and presented a formulation only for minimum t-norm which is, perhaps, an extension of type-1 sup-star composition.Type-2 fuzzy relations (T2 FRs in short) were introduced in [12].There are different kinds of applications of fuzzy relations in the theory of type-1 fuzzy sets.The other motivation of this research is to investigate T2 FRs and their compositions.A type-2 fuzzy tolerance relation is a type-2 fuzzy reflexive and symmetric relation, but not necessarily transitive relation.
The main objective of this paper is to generalize the definition of type-2 fuzzy tolerance relation and consequently the type-2 fuzzy G-tolerance relation in type-2 fuzzy sets.The type-2 fuzzy G-tolerance relation helps in finding the type-2 fuzzy G-equivalence relation.We have studied the concept of type-2 fuzzy tolerance relation in abstract algebra.
The paper is organized as follows : Section (2) introduces some basic definitions related to the concept.We have discussed type-2 fuzzy G-tolerance relation and type-2 fuzzy G-equivalence relation in Section (3).Section (4) deals with the conversion of type-2 fuzzy tolerance relation.Section (5) describes type-2 fuzzy tolerance relation in algebraic structures.

Preliminaries
Let us define some preliminary concepts in this section.Definition 1. [21] Let X be a non-empty universe.Then a mapping A : X → F ([0, 1]) is called a type-2 fuzzy set (T2 FS, for short) on X, characterized by a membership function µ Ã(x) : X → F ([0, 1]) and is expressed by the following set notation : Ã = { x, μ Ã(x) : x ∈ X}.F ([0, 1]) denotes the set of all type-1 fuzzy sets that can be defined on the set [0, 1].µ Ã(x), itself is a type-1 fuzzy set for value of x ∈ X and is characterized by a secondary membership function f x : J → [0, 1].Therefore, Ã can be represented as : Ã = {< x, {(u, f x (u)) : u ∈ J} >: x ∈ X}, where J ⊆ [0, 1] is the set of all possible primary membership functions corresponding to an element x.
Let µ A (x) and µ B (x) be the secondary membership functions of two type-2 fuzzy sets in the universal set X.The operations of union, intersection and complement of two type-2 fuzzy sets A and B using the Zadeh's Extension Principle [21] are defined below: where is denoted as join, is denoted as meet, ¬ is denoted as negation operator, ∨ is denoted as max and ∧ is denoted as min operator as defined by Mizumoto and Tanaka [16] in type-2 fuzzy sets respectively.
Definition 2. [12] The cartesian product of two type-2 fuzzy sets, say A and B on X is defined as

Definition 3. [12]
Let A, B be two type-2 fuzzy sets on X and Y respectively, then the type-2 fuzzy relation (T2 FR in short) of type-2 fuzzy sets A × B is the type-2 fuzzy subset of X × Y .
Dhiman et al. [5] have extended the definitions of type-2 fuzzy reflexive relation and type-2 fuzzy equivalence relations.
0 and 1 memberships are denoted as 1/0 and 1/1 in type-2 fuzzy sets respectively.0 membership in a type-2 fuzzy set means that it has a secondary membership equal to 1 corresponding to the primary membership of 0, and if it has all other secondary memberships equal to 0. Similarly, the meaning of 1 is same as 0.
Example 1. Suppose that in a biotechnology experiment, three potentially new strains of bacteria have been detected in the area around an anaerobic corrosion pit on a new aluminum-lithium alloy used in the fuel tank of a new experiment aircraft.In a pair wise comparison, the following similarity relation Q is developed .For example, the first strain (column 1) has a strength of similarity to the second strain of 0.1 0.1 + 0.2 0.2 , to the third strain a strength of 0.4 0.4 + 0.5 0.5 .Hence, Therefore, Q is a type-2 fuzzy G-reflexive relation.Again, Q(x, y) = Q(y, x) for all x, y ∈ X.

Type-2 Fuzzy Tolerance Relations in Algebraic Structures
We shall investigate how type-2 fuzzy tolerance relations can be applied in abstract algebra.Let an algebraic structure U = (A, F) be given, where A is the set of elements of this structure and F is the set of operations on this set.If a type-2 fuzzy tolerance τ on A is given, we say that τ , is compatible with U or U is a τ tolerance algebraic structure if and only if the following holds: Let f ∈ F and let f be an n-ary operation.If we have 2n elements x 1 , x 2 , ..., x n ; y 1 , y 2 , ..., y n of A such that (x i , y i ) ∈ τ for i = 1, 2, ....., n, then also (f (x 1 , x 2 , ..., x n ), f (y 1 , y 2 , ..., y n )) ∈ τ .
Theorem 5.1.Let U = (A, F) be an algebra and τ 1 , τ 2 be two type-2 fuzzy tolerance relation on M which are compatible with U. Then the relation τ 1 ∩ τ 2 is a type-2 fuzzy tolerance relation compatible with U.

4 .Example 2 ..
Type-2 fuzzy G-tolerance relation to type-2 fuzzy equivalence relation Type-2 fuzzy tolerance relation can be converted to type-2 fuzzy G-equivalence relation by composition.Let us observe this example below: Consider the earlier Example 1 We conclude that τ is a type-2 fuzzy G-tolerance relation from Example 1Let us now form a new matrix τ with the new elements found from the τ • τ .Suppose,

..Definition 9 .Example 4 .
Consequently, τ • τ τ.Therefore, τ satisfies the transitive property.Therefore, τ is a type-2 fuzzy G-equivalence relation.Definition 8. Let N be an ordinary set and τ be a type-2 fuzzy G-tolerance relation on N .Then a type-2 fuzzy subsetQ of N is called a type-2 fuzzy G pre-class iff Q× Q τ i.e.Q(x)∧ Q(y) τ (x, y) for all x, y ∈ N.Example 3. Let N ={x, y, z}.Then, τ is a type-2 fuzzy G-tolerance relation on M , defined by the following matrix: Let Q be a type-2 fuzzy subset of N defined by Q(x) = 0Clearly, Q × Q τ , and hence is a type-2 fuzzy G pre-class.There can be another R of N larger than Q such that R(x) ∧ R(y) τ (x, y) e.g.R(x) = 0Let τ be a type-2 fuzzy tolerance relation on N .A type-2 fuzzy subset C of N is called a type-2 fuzzy G-tolerance class if C is a type-2 fuzzy G pre-class and there exists no type-2 fuzzy G pre-class D of N s.t.C D. From the Example 3, R is a type-2 fuzzy G-tolerance class while Q is not.Definition 10.A type-2 fuzzy G-tolerance space is a pair (M, τ ), where M is an ordinary set and τ is a type-2 fuzzy tolerance relation defined on M .