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As we know, intuitionistic fuzzy sets are extensions of the standard fuzzy sets. Now, in this paper, the basic definitions and properties of intuitionistic fuzzy Γ-hyperideals of a Γ-semihyperring are introduced. A few examples are presented. In particular, some characterizations of Artinian and Noetherian Γ-semihyperring are given.

The theory of fuzzy sets proposed by Zadeh [

The concept of a fuzzy ideal of a ring was introduced by Liu [

In 1964, Nobusawa introduced

Algebraic hyperstructures represent a natural extension of classical algebraic structures, and they were introduced by the French mathematician Marty [

In [

The study of fuzzy hyperstructures is an interesting research topic of fuzzy sets. There is a considerable amount of work on the connections between fuzzy sets and hyperstructures, see [

Now, in this work we introduce the notion of an Atanassov’s intuitionistic fuzzy hyperideals of a semihyperrings and investigate some basic properties about it.

Let

A hypergrupoid

A

The multiplication is distributive with respect to hyperoperation +; that is,

The element

A semihyperring

Let

For example,

The concept of

Let

for all

In the above definition, if

A non-empty subset of

Let

Let

Let

Then,

A non-empty subset

Consider the following:

A fuzzy subset

Let

Let

The concept of intuitionistic fuzzy set was introduced and studied by Atanassov [

Consider the universe

One may define an intuitionistic fuzzy set “Very Large” denoted by

Now, we introduce the notion of intuitionistic fuzzy

Let

An intuitionistic fuzzy set

for all

An intuitionistic fuzzy set

for all

An intuitionistic fuzzy set

Let

Let

Let

Note that

Let

Since

We proved the theorem for left intuitionistic fuzzy

If

The proof is straightforward.

An intuitionistic fuzzy subset

Assume that

Also,

Conversely, suppose that

For any fuzzy set

An intuitionistic fuzzy subset

Assume that

Now, let

Conversely, suppose that all non-empty level sets

Let

Now, we give the following theorem about the intuitionistic fuzzy sets

If

Since

Note that if we have a fuzzy

If

The proof is straightforward.

Let

Let

If

The proof is straightforward.

Let

if

the sets

(1) Suppose that

(2) One can easily prove the second part.

Let

Let

the

the

Now, the next theorem is about image and preimage of intuitionistic fuzzy sets.

Let

if

if

The proof is straightforward.

In this section, we give some characterizations of Artinian and Noetherian

Let

Let

Every

If every intuitionistic fuzzy

Suppose that every intuitionistic fuzzy

Let

The set of values of any intuitionistic fuzzy

If a

Suppose that

This research is supported by TUBITAK-BIDEB. The paper was essentially prepared during the second author’s stay at the Department of Mathematics, Yildiz Technical University in 2011. The second author is greatly indebted to Dr. B. A. Ersoy for his hospitality.