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

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 hideals in hemirings were thoroughly investigated by LaTorre [1] and by using the h-ideals, he established some analogues ring theorems for hemirings. In 2004, Jun et al. [2] introduced the concept of fuzzy h-ideals in hemirings and gave some related properties. In particular, Zhan and Dudek [3] studied the h-hemiregular hemirings. Some characterizations of hsemisimple and h-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 h-ideals of hemirings. Zhan and Shum [9] has studied intuitionistic fuzzy h-ideals of hemirings. The other important results on semirings (hemirings) were discussed by Dudek et al. [10]. Mathematicalmodelling andmanipulation 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 (FPsoft 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 IFPintuitionistic soft h-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.


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 ℎideals in hemirings were thoroughly investigated by LaTorre [1] and by using the ℎ-ideals, he established some analogues ring theorems for hemirings.In 2004, Jun et al. [2] introduced the concept of fuzzy ℎ-ideals in hemirings and gave some related properties.In particular, Zhan and Dudek [3] studied the ℎ-hemiregular hemirings.Some characterizations of ℎsemisimple and ℎ-intra-hemirings were further investigated by Yin et al. [4,5].In order to continue these papers, Ma et al. [6][7][8] investigated some generalized fuzzy ℎ-ideals of hemirings.Zhan and Shum [9] has studied intuitionistic fuzzy ℎ-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][13][14][15][16][17][18][19].Recently, some researchers investigated fuzzy soft sets with parameterizations, such as, fuzzy parameterized soft sets (FPsoft 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 IFPintuitionistic soft ℎ-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.

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 , ,  ∈ .
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.
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 ().
Note that if  is a fuzzy left (right) ideal of , then (0) ≥ () for all  ∈ .
A fuzzy set  is said to be a fuzzy ℎ-ideal of  if it is both a fuzzy left ℎ-ideal of  and a fuzzy right ℎ-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 ,  ∈  : ( + ) ≥ min{(), ()}, ]( + ) ≤ max{](), ]()} and () ≥
An intuitionistic fuzzy set  is said to be an intuitionistic fuzzy ℎ-ideal of  if it is both an intuitionistic fuzzy left ℎ-ideal and an intuitionistic fuzzy right ℎ-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   :  → [0, 1] and nonmembership function ]  :  → [0, 1] and   is an intuitionistic fuzzy set over  for all  ∈ .Then, an Ω-set Ω  over IF() is a set defined by a function   () representing a mapping Here,   is called intuitionistic fuzzy approximation of Ω-set Ω  ,   () is an intuitionistic fuzzy set called -element of the Ω-set for all  ∈ .Thus, an Ω-set Ω  over  can be represented by the set of ordered pairs Note that, if   () = 0, ]  () = 1 and   () = 0, we do not display such elements in the set.Also, it must be noted that the sets of all Ω-sets over IF() will be denoted by Ω().

IFP-Intuitionistic Fuzzy Soft ℎ-Ideals
Definition 10.Let  be a hemiring,  a set of parameters, and  an intuitionistic fuzzy set over , Then Ω  is said to be an intuitionistic fuzzy parameterized intuitionistic fuzzy soft ℎ-ideal (briefly, IFP-intuitionistic fuzzy soft ℎ-ideal) over , if for any  ∈ ,   () is an intuitionistic fuzzy ℎ-ideal of .
Define   by () () = { 0,  is not upper triangular matrix, 1,  is upper triangular matrix, For all  ∈ , we can verify that   () is an intuitionistic fuzzy left ℎ-ideal of  and, hence, Ω  is an IFP-intuitionistic fuzzy soft left ℎ-ideal over .
Likewise, the proof of right ideal of  can be made in a similar way.
We know that the intersection of all IFP-intuitionistic fuzzy soft ℎ-ideals over  is also an IFP-intuitionistic fuzzy soft ℎ-ideal over .Then we would consider whether the union of IFP-intuitionistic fuzzy soft ℎ-ideals over  is also an IFP-intuitionistic fuzzy soft ℎ-ideal over .Notation 1.Let Ω  and Ω  be IFP-intuitionistic fuzzy soft ℎideals over .Then we said the sequence of values is ordered, if for any  ∈ ,  1 ,  2 ∈ : Proposition 14.Let Ω  and Ω  be IFP-intuitionistic fuzzy soft ℎ-ideals over  with ordered sequence of values.Then their union Ω  ⋃Ω  is still an IFP-intuitionistic fuzzy soft ℎ-ideal over .
Proof.The proof is similar to the proof of Proposition 13.
Proof.The proof is similar to the proof of Proposition 13.Definition 16.Let Ω  be an IFP-intuitionistic fuzzy soft ℎideal over  and Ω  an IFP-intuitionistic fuzzy soft subset over .Then Ω  is said to be an IFP-intuitionistic fuzzy soft ℎ-ideal of Ω  , if   is an IFP-intuitionistic fuzzy soft subset of   .
Proof.The proof is obvious; therefore omit it.

IFP-Equivalent Intuitionistic
Fuzzy Soft ℎ-Ideals In the above definition, if  is an isomorphism from  to  and  is a bijective mapping, then (, ) is called an IFPintuitionistic fuzzy soft isomorphism so that (, ) is an IFPintuitionistic fuzzy soft isomorphism over ℎ-ideal from Ω  to Ω  , denoted by Ω  ≃ Ω  .
In the above Example, the decision-making method only gives one-digit number from 0 to 4. By the similar way, we can arrange many digits from 0 to 9.

Conclusion
In the present paper, we showed the basic results of IFPintuitionistic fuzzy soft ℎ-ideal.And then we defined IFPintuitionistic fuzzy soft homomorphisms (isomorphisms), IFP-equivalent intuitionistic fuzzy soft ℎ-ideals, and IFPincreasing intuitionistic fuzzy soft ℎ-ideals and discussed some properties of them.Finally, we investigate aggregate intuitionistic fuzzy ℎ-ideals of hemirings based on decision making.
As an extension of this work, one could apply the method to solve the related problems in applied mathematics and information sciences.