#### Abstract

We investigate connections between bialgebras and Atanassov’s intuitionistic fuzzy sets. Firstly we define an intuitionistic fuzzy subbialgebra of a bialgebra with an intuitionistic fuzzy subalgebra structure and also with an intuitionistic fuzzy subcoalgebra structure. Secondly we investigate the related properties of intuitionistic fuzzy subbialgebras. Finally we prove that the dual of an intuitionistic fuzzy strong subbialgebra is an intuitionistic fuzzy strong subbialgebra.

#### 1. Introduction

A bialgebra is, roughly speaking, an algebra on which there exists a dual structure called a coalgebra structure such that the two structures satisfy a compatibility relation. We know that group algebras, tensor algebras, symmetric algebras, enveloping algebras of Lie algebras, and so on are bialgebras [1]. Recently, the theory of bialgebras has many applications in several branches of both pure and applied sciences. So it is natural to ask whether we can extend the concept of bialgebras in uncertainty settings.

Uncertainty is an attribute of information and uncertain data are presented in various domains. The most appropriate theory for dealing with uncertainties is the theory of fuzzy sets developed by Zadeh [2]. Besides several generalizations of fuzzy sets, the intuitionistic fuzzy sets introduced by Atanassov [3, 4] have been found to be highly useful to cope with imperfect and/or imprecise information. Atanassov’s intuitionistic fuzzy sets are an intuitively straightforward extension of Zadeh’s fuzzy sets: while a fuzzy set gives the degree of membership of an element in a given set, an Atanassov’s intuitionistic fuzzy set gives both a degree of membership and a degree of nonmembership. Intuitionistic fuzzy set theory has been successfully applied in decision analysis and pattern recognition, in logic programming, and in medical diagnosis. And recently, many mathematicians have been involved in extending the concepts and results of abstract algebra to the broader framework of the intuitionistic fuzzy settings. For example, Biswas applied the concepts of intuitionistic fuzzy sets to the theory of groups and studied intuitionistic fuzzy subgroups of a group in [5]. Banerjee and Basnet studied intuitionistic fuzzy subrings and ideals of a ring in [6] and Hur et al. also discussed the concepts of the intuitionistic fuzzy ring in [7, 8]. Further, Jun et al. investigated intuitionistic nil radicals of intuitionistic fuzzy ideals in rings in [9]; Akram and Dudek introduced intuitionistic fuzzy left k-ideals of semirings in [10]. Davvaz et al. [11] applied the notion of intuitionistic fuzzy sets to -modules; Davvaz et al. [12–16] had studied the intuitionistic fuzzification of the concept of algebraic hyperstructures and investigated some of their properties; Chen and Zhang [17, 18] discussed Lie superalgebras in intuitionistic fuzzy settings and obtained some results. On the other hand, we had studied fuzzy subcoalgebras and fuzzy subbialgebras in [19–21]. In this paper we will extend the concepts of bialgebras in intuitionistic fuzzy sets.

The structure of this paper is as follows. In Section 2, we recall some basic definitions and notations which will be used in what follows. In Section 3, we give the definition of an intuitionistic fuzzy subbialgebra and discuss the related properties. In Section 4, we define the duality of intuitionistic fuzzy subspaces which generalizes the related results in fuzzy settings. As the application, we prove that the duality of an intuitionistic fuzzy strong subbialgebra is an intuitionistic fuzzy strong subbialgebra. At last, a conclusion is presented.

#### 2. Preliminaries

In this section, we recall some basic definitions and notations for the sake of completeness.

*Definition 1. *Let be a -vector space. An intuitionistic fuzzy set (IFS for short) of is defined as an object having the form , where the functions and denote the degree of membership (namely, ) and the degree of nonmembership (namely, ) of each element to the set , respectively, and for each . For the sake of simplicity, we will use the symbol for the intuitionistic fuzzy set .

*Definition 2. *Let and be intuitionistic fuzzy sets of -vector space . Then one has the following:(1) iff and for all ,(2),(3),(4),(5).

*Definition 3. *Let and be intuitionistic fuzzy sets of -vector space . The intuitionistic sum of and is defined to the intuitionistic fuzzy set of given by

*Definition 4. *Let be an intuitionistic fuzzy set of -vector space . For any , and , , if it satisfies and , then is called an* intuitionistic fuzzy subspace* of .

*Definition 5. *For any and fuzzy subset of , the set (resp., ) is called an upper (resp., lower) -level cut of and (resp., ) is called an upper (resp., lower) strong -level cut of .

Theorem 6. *Let be an intuitionistic fuzzy set of -vector space such that with attained upper bound and with attained lower bound in . Then the following statements are equivalent:*(1)* is an intuitionistic fuzzy subspace of ,*(2)*nonempty sets and are subspaces of for each ,*(3)*nonempty sets and are subspaces of for each ,*(4)*nonempty sets and are subspaces of for each .*

*Definition 7. *Let be an intuitionistic fuzzy set of -algebra . For any , and , , if it satisfies the following conditions:(1) and ,(2) and ,

then is called* an intuitionistic fuzzy subalgebra* of .

*Definition 8. *Let be an intuitionistic fuzzy set of -algebra . For any , and , , if it satisfies the following conditions:(1) and ,(2) and ,

then is called an intuitionistic fuzzy ideal of .

*Definition 9 (see [21]). *Let be an intuitionistic fuzzy set of -coalgebra . For any , , and , , if it satisfies the following conditions:(1) and ,(2) and , for all ,

then is called* intuitionistic fuzzy subcoalgebra* of .

*Definition 10 (see [1]). *A* bialgebra* is a -vector space , endowed with an algebra structure and with a coalgebra structure such that and are morphisms of coalgebras (equivalently, and are morphisms of algebras). In this paper, we always denote as a bialgebra over field .

*Definition 11. *Let be a bialgebra. A -subspace of is called a* subbialgebra* if it is endowed with a subalgebra structure and with a subcoalgebra structure.

*Remark 12. *Let be a bialgebra. Then is a coalgebra. For an element , . In this paper, we require that , are linearly independent, respectively. In this case, the decomposition of is unique.

#### 3. Intuitionistic Fuzzy Subbialgebras of Bialgebras

At first, we give the following.

*Definition 13. *Let be an intuitionistic fuzzy set of bialgebra . Then is called an* intuitionistic fuzzy subbialgebra *of , if, for any , and , , the following conditions are satisfied:(1) is an intuitionistic fuzzy subspace of ,(2) and ,(3) and , for all .

*Remark 14. *If satisfies (1) and (2), then is an intuitionistic fuzzy subalgebra of ; if satisfies (1) and (3), then is an intuitionistic fuzzy subcoalgebra of . So an intuitionistic fuzzy subbialgebra is with an intuitionistic fuzzy subalgebra structure and also with an intuitionistic fuzzy subcoalgebra structure.

*Definition 15. *Let be an intuitionistic fuzzy set of bialgebra . Then is called an* intuitionistic fuzzy strong subbialgebra *of , if, for any , and , , the following conditions are satisfied:(1) is an intuitionistic fuzzy subspace of ,(2) and ,(3) and , for all .

*Remark 16. *If satisfies (1) and (2), then is an intuitionistic fuzzy ideal of . So an intuitionistic fuzzy strong subbialgebra is with an intuitionistic fuzzy ideal structure and with an intuitionistic fuzzy subcoalgebra structure. In fact, it is an intuitionistic fuzzy subbialgebra.

*Example 17. *The field with its algebra structure and with the canonical coalgebra structure ( and ) is a bialgebra (see [1]).

We define by

Then is an intuitionistic fuzzy subbialgebra of . However, is not an intuitionistic fuzzy strong subbialgebra, since for .

*Example 18. *If is a monoid, then the semigroup algebra endowed with a coalgebra structure as and for any is a bialgebra (see [1]).

We define by and for . If , we define and . Then is an intuitionistic fuzzy subbialgebra of .

Lemma 19. *If is an intuitionistic fuzzy subbialgebra of , then so is .*

Lemma 20. *If is an intuitionistic fuzzy subbialgebra of , then so is .*

Combining Lemmas 19 and 20, it is easy to prove the following theorem.

Theorem 21. * is an intuitionistic fuzzy subbialgebra of if and only if and are intuitionistic fuzzy subbialgebras.*

The following theorems explain the relationship between intuitionistic fuzzy subbialgebras and subbialgebras. The proof can be imitated by [19, 22]. Here, we omit.

Theorem 22. *If is an intuitionistic fuzzy subbialgebra of , then the sets and are subbialgebras of for every .*

Theorem 23. *If is an intuitionistic fuzzy set of such that all nonempty level sets and are subbialgebras of , then is an intuitionistic fuzzy subbialgebra of .*

Theorem 24. *Let and be intuitionistic fuzzy subbialgebras of bialgebra such that and . Then is also an intuitionistic fuzzy subbialgebra of .*

*Proof. *Obviously is an intuitionistic fuzzy subspace of . We only show that , and , for , .

Let and . Then . Otherwise, or ; the result can be proved similarly. We have

On the other hand, let . Then

Hence we have
Therefore, is an intuitionistic fuzzy subbialgebra of .

Theorem 25. *Let and be any intuitionistic fuzzy subbialgebras of bialgebra . Then is also an intuitionistic fuzzy subbialgebra of .*

Corollary 26. *Let be a family of intuitionistic fuzzy subbialgebras of bialgebra . Then is also an intuitionistic fuzzy subbialgebra of .*

Let and be two bialgebras. The -linear map is called a* morphism of bialgebras* if it is a morphism of algebras and a morphism of coalgebras between the underlying algebras, respectively, coalgebras of the two bialgebras.

*Definition 27. *Let be a map from bialgebra to bialgebra . If and are intuitionistic fuzzy sets of and , respectively, then the* preimage* of under is defined to be an intuitionistic fuzzy set where and for any and the* image* of under is defined to be an intuitionistic fuzzy set where

Proposition 28. *Let be a homomorphism of bialgebras. If is an intuitionistic fuzzy subbialgebra of , then is an intuitionistic fuzzy subbialgebra of .*

Proposition 29. *Let be a homomorphism of bialgebras. If is an intuitionistic fuzzy subbialgebra of , then is an intuitionistic fuzzy subbialgebra of .*

Proposition 30. *Let and be any intuitionistic fuzzy subbialgebras of bialgebra and let be a homomorphism. If and , then is also an intuitionistic fuzzy subbialgebra of and .*

#### 4. The Duality of Intuitionistic Fuzzy Sub-Bialgebras

The following is well known.

Proposition 31 (see [1]). *Let be a finite dimensional bialgebra. Then , together with the algebra structure which is dual to the coalgebra structure of and with the coalgebra structure which is dual to the algebra structure of , is a bialgebra, which is called the dual bialgebra of .*

Let be an intuitionistic fuzzy subspace of -vector space . Then for any , which implies . We give the definition of the dual of intuitionistic fuzzy subspaces in the following [23].

*Definition 32 (see [23]). *Let be an intuitionistic fuzzy subspace of -vector space .

Define where

Obviously, is an intuitionistic fuzzy set of .

*Remark 33. *(1) If , then .

(2) If , then .

Theorem 34. *The intuitionistic fuzzy set is an intuitionistic fuzzy subspace of .*

*Proof. *Since is the upper bound of and is the lower bound of , by Theorem 6, it suffices to show that the nonempty sets and are subspaces of for all .*Case 1*. and .

First, we show in which and in which , for all .

Indeed, . If , then and . If , then . So there exists , such that and . Then .

Also, .

If , then and .

If , then . So there exists such that and . Then .

Second, if and , then , , so and .

If and , then , so and .

Cases 2–4 can be proved similarly as in Case 1.*Case 2*. and .

We show and , for all . *Case 3. * and .

We show and , for all .

And if , then and .*Case 4. * and .

We show and , for all .

And if , then and .

Hence, is an intuitionistic fuzzy subspace of since and are subspaces of .

Our main theorem is as follows.

Theorem 35. *Let be a finite dimensional bialgebra and let be its dual bialgebra. If is an intuitionistic fuzzy strong subbialgebra of , then is an intuitionistic fuzzy strong subbialgebra of .*

*Proof. * is an intuitionistic fuzzy subspace of by Theorem 34. We need show that, for any , , , and , for all .

Let , . If , the result is obvious.

If and , we have

If and , then, for any , we have . So
which imply and .

If and , the result can be proved similarly.

On the other hand, let . We have

Let . Noting that , then and .

Hence, is an intuitionistic fuzzy strong subbialgebra of .

*Remark 36. *(1) If is an intuitionistic fuzzy subcoalgebra of , then is an intuitionistic fuzzy ideal of , and it is an intuitionistic fuzzy subalgebra of .

(2) If is an intuitionistic fuzzy ideal of , then is an intuitionistic fuzzy subcoalgebra of .

*Question 1. *If is an intuitionistic fuzzy subalgebra of , is the result true?

*Example 37. *See Example 17 and we assume is finite dimension.

We calculate

Since is an intuitionistic fuzzy subbialgebra, we get and following the proof of Theorem 35. If , and , then . However, ; this shows that it is not an intuitionistic fuzzy subcoalgebra of . Hence, Theorem 35 is not true when is an intuitionistic fuzzy subbialgebra.

#### 5. Conclusion

An intuitionistic fuzzy subbialgebra of a bialgebra is defined with an intuitionistic fuzzy subalgebra structure and with an intuitionistic fuzzy subcoalgebra structure. Since the main problem in uncertainty mathematics is how to carry out the ordinary concepts to the uncertainty case and the difficulty lies in how to pick out the rational generalization from the large number of available approaches, the key of studying the dual of intuitionistic fuzzy subbialgebras is to give a rational definition. We define the dual of an intuitionistic fuzzy subspace which generalizes the dual of fuzzy subspaces nontrivially.

Hopf algebra is a bialgebra with antipode which is ubiquity in virtually all fields of mathematics. We will focus on considering Hopf algebras in the intuitionistic fuzzy settings as a continuation of this paper in the future. Also, we will consider bialgebras in rough sets and soft sets. Our obtained results probably can be applied in various fields such as artificial intelligence, signal processing, multiagent systems, pattern recognition, robotics, expert systems, medical diagnosis, and engineering.

#### Conflict of Interests

No conflict of interests exits in the submission of this paper.

#### Acknowledgments

This project is supported by the National Natural Science Foundation of China (Grant no. 11126301) and the Promotive Research Fund for Young and Middle-aged Scientists of Shandong Province (no. BS2011SF002).