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. [6–8] 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 [12–19]. 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 .