Fuzzy Set and Its Extension

8Fuzzy Subgroups

8.1 Introduction

In crisp logic, a group is an algebraic structure that is equipped with mathematical operations where two elements a and b are combined to form a third element and satisfies four conditions – closure, identity, associativity, and invertibility. The most familiar example of a group is a set of integers under the operation “addition.”

1. If two integers a, b are added, we get a sum (a + b), which is an integer – closure property.
2. If 0 is added to an integer, we get the same integer – identity.
3. For any three integers, three elements follow the property (a + b) + c = a + (b + c) – associativity.
4. For each integer, a, another number exists b, such that a + b = b + a = 0 implies that the integer b is an inverse element of a which is – a. “−a” is also an integer.

In fuzzy set theory, fuzzy subgroup was first defined by Rosenfeld [1]. Then the definition was generalized by Negoite and Ralescu [2] and Anthony and Sherwood [3]. Here, we give some elementary theory of groups and groupoids.

We know that if X be a set and fuzzy subset, A of X is a function A : X → [0,1].

The definitions of fuzzy subgroup are as follows [1,4]:

Definition 1(a) [1]

Let G be a group. A fuzzy set A in G is said to be a fuzzy subgroup of G if

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Definition 1(b)

Let G be ...