Research Article | Open Access

# IFP-Intuitionistic Fuzzy Soft -Ideals of Hemirings and Its Decision Making

**Academic Editor:**Tai-Ping Chang

#### Abstract

The aim of this paper is to introduce the concepts of IFP-intuitionistic fuzzy soft *h*-ideals and IFP-equivalent intuitionistic fuzzy soft *h*-ideals of hemirings. Some characterizations and properties of them are given. In particular, some good examples are explored. Finally, we investigate aggregate intuitionistic fuzzy *h*-ideals of hemirings based on decision making.

#### 1. Introduction

Dealing with problems in different areas of applied mathematics and information sciences, we have found that semirings are become more and more useful. They play an important role in studying optimization theory, graph theory, matrices, theory of discrete event dynamical systems, generalized fuzzy computation, and so on. The ideals of semirings play a central role in the structure theory, and the ideals of semirings do not in general coincide with the usual ring ideals; therefore, the usage of ideals in semirings is limited. In order to overcome the difficulty, many researchers mean a special semiring with a zero and with a commutative addition. We call this semiring hemiring. The properties of -ideals in hemirings were thoroughly investigated by LaTorre [1] and by using the -ideals, he established some analogues ring theorems for hemirings. In 2004, Jun et al. [2] introduced the concept of fuzzy -ideals in hemirings and gave some related properties. In particular, Zhan and Dudek [3] studied the -hemiregular hemirings. Some characterizations of -semisimple and -intra-hemirings were further investigated by Yin et al. [4, 5]. In order to continue these papers, Ma et al. [6â€“8] investigated some generalized fuzzy -ideals of hemirings. Zhan and Shum [9] has studied intuitionistic fuzzy -ideals of hemirings. The other important results on semirings (hemirings) were discussed by Dudek et al. [10].

Mathematical modelling and manipulation of some kinds of uncertainties have become an increasingly important issue in solving complicated problems arising in a wide range of areas, such as, economies, engineering, medicine, environmental science, and information science. In 1999, Molodtsov [11] firstly introduced the soft set theory as a general mathematical tool for dealing with uncertainty and vagueness. Since then some researchers studied the operations of soft sets and their various applications [12â€“19]. Recently, some researchers investigated fuzzy soft sets with parameterizations, such as, fuzzy parameterized soft sets (FP-soft sets) [20], fuzzy parameterized fuzzy soft sets (FP-fuzzy soft sets) [20], intuitionistic fuzzy parameterized (IFP) soft set, intuitionistic fuzzy parameterized (IFP) fuzzy soft set, and intuitionistic fuzzy parameterized- (IFP-) intuitionistic fuzzy soft set [21].

In this paper, we introduce the concept of IFP-intuitionistic soft -ideals of hemirings and discuss some related results. Finally, we give some examples to show that the methods can be successfully applied to some uncertain problems.

#### 2. Preliminaries

A semiring is an algebraic system consisting of a nonempty set together with two binary operations on called addition and multiplication (denoted in the usual manner) such that and are semigroups and the following distributive laws are satisfied for all .

By zero of a semiring , we mean an element such that and for all . A semiring with zero and a commutative semigroup is called a hemiring. For the sake of simplicity, we will write for ().

A subhemiring of a hemiring is a subset of closed under addition and multiplication. A subset of is called a left (right) ideal of if is closed under addition and . A subset is called an ideal if it is both a left ideal and a right ideal.

A subhemiring (left ideal, right ideal, and ideal) of is called an -subhemiring (left -ideal, right -ideal, and -ideal) of , respectively, if for any , , and implies .

From now on, is a hemiring, is an initial universe, is a set of parameters, is the power set of , and .

*Definition 1 (see [22]). *Let be an initial universe. A fuzzy set over is a set defined by a function representing a mapping
Here, is called membership function of and the value is called the grade of membership of . The value represents the degree of belonging to fuzzy set . Thus, a fuzzy set over can be represented as follows:
Note that the set of all the fuzzy sets over will be denoted by .

*Definition 2 (see [23]). *An intuitionistic fuzzy set (IFS) in is defined as an object of the following form:
where the functions and define the degree of membership and the degree of nonmembership of the element , respectively, and for every ,
In addition for all , , are intuitionistic fuzzy universal and intuitionistic fuzzy empty set, respectively.

*Definition 3 (see [3]). *(i) A fuzzy set of is said to be a fuzzy left (right) ideal of if the following conditions hold for all , and .

Note that if is a fuzzy left (right) ideal of , then for all .

(ii) A fuzzy left (right) -ideal of is defined to be a fuzzy left (right) ideal of if for any , implies .

A fuzzy set is said to be a fuzzy -ideal of if it is both a fuzzy left -ideal of and a fuzzy right -ideal of .

*Definition 4 (see [9]). *(i) An intuitionistic fuzzy set of is said to be an intuitionistic fuzzy left (right) ideal of if the following conditions hold for all , and , .

Note that if is an intuitionistic fuzzy left (right) ideal of , then and for all .

(ii) An intuitionistic fuzzy left (right) -ideal of is defined to be an intuitionistic fuzzy left (right) ideal of if for any , implies and .

An intuitionistic fuzzy set is said to be an intuitionistic fuzzy -ideal of if it is both an intuitionistic fuzzy left -ideal and an intuitionistic fuzzy right -ideal of .

The following concepts are cited in [21].

*Definition 5. *Let be an initial universe, the set of all parameters, and an intuitionistic fuzzy set over with the membership function and nonmembership function and is an intuitionistic fuzzy set over for all . Then, an -set over is a set defined by a function representing a mapping
Here, is called intuitionistic fuzzy approximation of -set , is an intuitionistic fuzzy set called -element of the -set for all . Thus, an -set over can be represented by the set of ordered pairs
Note that, if , and , we do not display such elements in the set. Also, it must be noted that the sets of all -sets over will be denoted by .

*Definition 6. *Let .(i)If for all , then is called an -empty -set, denoted by .(ii)If , then the -empty -set () is called empty -set, denoted by . Here, means intuitionistic fuzzy empty set.(iii)If , , and for all , then is called -universal -set, denoted by .(iv)If is equal to intuitionistic fuzzy universal set over , then the -universal -set is called universal -set, denoted by . Here, means that intuitionistic fuzzy universal set.

*Definition 7. *Let . Then(i) is an IFP-intuitionistic fuzzy subset of , denoted by , if , , and for all ;(ii) and are IFP-equal, denoted by , if , , and for all .

*Definition 8. *Let . Then the complement of , denoted by , is an IFP-intuitionistic fuzzy soft set defined by

*Definition 9. *Let .

(i) The intersection of and , denoted by , is defined by
Here,
where for all .

(ii) The union of and , denoted by , is defined by
Here,
where for all .

(iii) The -product of and , denoted by , is defined by
Here
where for all .

#### 3. IFP-Intuitionistic Fuzzy Soft -Ideals

*Definition 10. *Let be a hemiring, a set of parameters, and an intuitionistic fuzzy set over , .

Then is said to be an intuitionistic fuzzy parameterized intuitionistic fuzzy soft -ideal (briefly, IFP-intuitionistic fuzzy soft -ideal) over , if for any , is an intuitionistic fuzzy -ideal of .

*Example 11. *Let be a hemiring and be a set of parameters.

If , , , then is an IFP-intuitionistic fuzzy soft -ideal over .

*Example 12. *Let , where is a matrix hemiring, , the meaning of the parameters , , respectively. If is an intuitionistic fuzzy set over defined by

Define by

For all , we can verify that is an intuitionistic fuzzy left -ideal of and, hence, is an IFP-intuitionistic fuzzy soft left -ideal over .

Proposition 13. *If and are IFP-intuitionistic fuzzy soft -ideals over , then their intersection is still an IFP-intuitionistic fuzzy soft -ideal over .*

*Proof. *We can write . For all , , . Now we will prove is a fuzzy -ideal of . For all , ,
Since , and , , then
Thus we see is an intuitionistic fuzzy left ideal of .

Likewise, the proof of right ideal of can be made in a similar way.

For all , , let ; then
Hence is an intuitionistic fuzzy -ideal of ; that is to say, is an intuitionistic fuzzy -ideal of . Therefore, is an IFP-intuitionistic fuzzy soft -ideal over .

We know that the intersection of all IFP-intuitionistic fuzzy soft -ideals over is also an IFP-intuitionistic fuzzy soft -ideal over . Then we would consider whether the union of IFP-intuitionistic fuzzy soft -ideals over is also an IFP-intuitionistic fuzzy soft -ideal over .

*Notation 1. *Let and be IFP-intuitionistic fuzzy soft -ideals over . Then we said the sequence of values is ordered, if for any , : (i) implies , (ii) implies .

Proposition 14. *Let and be IFP-intuitionistic fuzzy soft -ideals over with ordered sequence of values. Then their union is still an IFP-intuitionistic fuzzy soft -ideal over .*

*Proof. *The proof is similar to the proof of Proposition 13.

Proposition 15. *Let and be IFP-intuitionistic fuzzy soft -ideals over . Then is an IFP-intuitionistic fuzzy soft -ideal over .*

*Proof. *
The proof is similar to the proof of Proposition 13.

*Definition 16. *Let be an IFP-intuitionistic fuzzy soft -ideal over and an IFP-intuitionistic fuzzy soft subset over . Then is said to be an IFP-intuitionistic fuzzy soft -ideal of , if is an IFP-intuitionistic fuzzy soft subset of .

*Example 17. *Let be a hemiring and a set of parameters. If , , and and , , and, then is an IFP-intuitionistic fuzzy soft -ideal over , is an IFP-intuitionistic fuzzy soft subset over , and is an IFP-intuitionistic fuzzy soft -ideal of .

Theorem 18. *Let , , and be IFP-intuitionistic fuzzy soft set over . If is an IFP-intuitionistic fuzzy soft -ideal of and is an IFP-intuitionistic fuzzy soft -ideal of , then is an IFP-intuitionistic fuzzy soft -ideal of .*

*Proof. *The proof is obvious; therefore omit it.

#### 4. IFP-Equivalent Intuitionistic Fuzzy Soft -Ideals

*Definition 19. *Let and let be an IFP-intuitionistic fuzzy soft -ideal over hemirings and , respectively. If and are two functions, then is called an IFP-intuitionistic fuzzy soft homomorphism such that is an IFP-intuitionistic fuzzy soft homomorphism over -ideal from to . The latter is written by if the following conditions are satisfied:(1) is an epimorphism from to ,(2) is a surjective mapping,(3), , and for all .

In the above definition, if is an isomorphism from to and is a bijective mapping, then is called an IFP-intuitionistic fuzzy soft isomorphism so that is an IFP-intuitionistic fuzzy soft isomorphism over -ideal from to , denoted by .

*Example 20. *Let and , , and . Define a homomorphism from onto by for , and a mapping from onto by , for .

Let be an intuitionistic fuzzy set over defined by and a fuzzy set over defined by .

Let defined by
defined by

It is clear that and are IFP-intuitionistic fuzzy soft -ideals over and , respectively. We can immediately see that and we can deduce that for all . Hence is an IFP-intuitionistic fuzzy soft homomorphism from to .

Lemma 21 (see [3]). *If is an epimorphism of hemirings and is an intuitionistic fuzzy -ideal of , then is an intuitionistic fuzzy -ideal of .*

Theorem 22. *Let be an IFP-intuitionistic fuzzy soft -ideal over and an IFP-intuitionistic fuzzy soft set over hemiring . If is an IFP-intuitionistic fuzzy soft homomorphic to , then is an IFP-intuitionistic fuzzy soft -ideal over .*

*Proof. *Let be an IFP-intuitionistic fuzzy soft homomorphism from to . Since is an IFP-intuitionistic fuzzy soft -ideal over , and is a fuzzy -ideal of for all . Now, for all , there exists such that , so , . By Lemma 21,â€‰â€‰ is an intuitionistic fuzzy -ideal of the hemiring , so must be an IFP-intuitionistic fuzzy soft -ideal over as well.

*Definition 23. *Let be an IFP-intuitionistic fuzzy soft -ideal over hemiring . Then is said to be an IFP-equivalent intuitionistic fuzzy soft -ideal over if, for any , , and , we have .

By the Definitions 19 and 5, we can easily obtain that is an IFP-equivalent intuitionistic fuzzy soft -ideal over , if is an IFP-equivalent intuitionistic fuzzy soft -ideal over .

*Example 24. *Assume that
is a hemiring, is a set of parameters, and is an intuitionistic fuzzy set over defined by ,
It is clear that is an IFP-equivalent intuitionistic fuzzy soft -ideal over .

*Notation 2. *If and are IFP-equivalent intuitionistic fuzzy soft -ideals over hemiring , then and are not always IFP-equivalent intuitionistic fuzzy soft -ideals over .

*Example 25. *Assume that , . Let be an IFP-intuitionistic fuzzy soft -ideal over defined by

And let be an IFP-intuitionistic fuzzy soft -ideal over defined by

It is clear that and are IFP-equivalent intuitionistic fuzzy soft -ideals over . We can see that
but
Then and are not IFP-equivalent intuitionistic fuzzy soft -ideals over .

Theorem 26. *Let be an IFP-equivalent intuitionistic fuzzy soft -ideal over hemiring and an -intuitionistic fuzzy soft set over hemiring . If is IFP-intuitionistic fuzzy soft homomorphic to , then is an IFP-equivalent intuitionistic fuzzy soft -ideal over .*

*Proof. *Let be an IFP-intuitionistic fuzzy soft homomorphism from to . Since is an IFP-equivalent intuitionistic fuzzy soft -ideal over , we have , if and for any . Now, for all , then there exist with , . Since and , then , and again, we have . And then ; hence must be an IFP-equivalent intuitionistic fuzzy soft -ideal over as well.

*Definition 27. *Let be an IFP-intuitionistic fuzzy soft -ideal over hemiring . Then is said to be an IFP-increasing intuitionistic fuzzy soft -ideal over , if for any , , and , we have , and is said to be an IFP-decreasing intuitionistic fuzzy soft -ideal over , if for any , , and , we have .

*Example 28. *Let , , and be a fuzzy set over defined by
It is clear that is an IFP-increasing intuitionistic fuzzy soft hemiring over .

*Notation 3. *If and are IFP-increasing intuitionistic fuzzy soft -ideals over hemiring , then are not always an IFP-increasing intuitionistic fuzzy soft -ideals over .

*Example 29. *Assume that , . Let be an intuitionistic fuzzy set over defined by
And let be a fuzzy set over defined by
It is clear that and are IFP-increasing intuitionistic fuzzy soft -ideals over . We can see that

but

Then and are not IFP-increasing intuitionistic fuzzy soft -ideal over .

Theorem 30. *Let be an IFP-increasing intuitionistic fuzzy soft -ideal over hemiring and be an IFP-intuitionistic fuzzy soft set over hemiring . If is IFP-intuitionistic fuzzy soft homomorphic to , then is an IFP-increasing intuitionistic fuzzy soft -ideal over .*

*Proof. *Let be an IFP-intuitionistic fuzzy soft homomorphism from to . Since is an IFP-increasing intuitionistic fuzzy soft -ideal over , for all , , and , we have . Now, for all , there exist with , . Since and , then , and again, we have . Hence, and must be an IFP-increasing intuitionistic fuzzy soft -ideal over as well.

#### 5. Decision Making Method on Intuitionistic Fuzzy -Ideals

The approximate function of an -set is an intuitionistic fuzzy set. The on the intuitionistic fuzzy sets is an operation by which several approximate functions of an -set are combined to produce a single intuitionistic fuzzy set that is the aggregate intuitionistic fuzzy set of the -set.

*Definition 31. *Let and be a set of all the intuitionistic fuzzy sets over . Then -aggregation operator, denoted by , is defined by
where
which is an intuitionistic fuzzy set over . The value is called aggregate intuitionistic fuzzy set of . Here the membership degree and nonmembership degree of are defined as follows: