Abstract

We deal with topics regarding -fuzzy subgroups, mainly -fuzzy cosets and -fuzzy normal subgroups. We give basic properties of -fuzzy subgroups and present some results related to -fuzzy cosets and -fuzzy normal subgroups.

1. Introduction

Since Zadeh [1] introduced the concept of a fuzzy set in 1965, various algebraic structures have been fuzzified. As a result, the theory of fuzzy group was developed. In 1971, Rosenfeld [2] introduced the notion of a fuzzy subgroup and thus initiated the study of fuzzy groups.

In recent years, some variants and extensions of fuzzy groups emerged. In 1996, Bhakat and Das proposed the concept of an -fuzzy subgroup in [3] and investigated their fundamental properties. They showed that is an -fuzzy subgroup if and only if is a crisp group for any provided . A question arises naturally: can we define a type of fuzzy subgroups such that all of their nonempty -level sets are crisp subgroups for any in an interval ? In 2003, Yuan et al. [4] answered this question by defining a so-called -fuzzy subgroups, which is an extension of -fuzzy subgroup. As in the case of fuzzy group, some counterparts of classic concepts can be found for -fuzzy subgroups. For instance, -fuzzy normal subgroups and -fuzzy quotient groups are defined and their elementary properties are investigated, and an equivalent characterization of -fuzzy normal subgroups was presented in [5]. However, there is much more research on -fuzzy subgroups if we consider rich results both in the classic group theory and the fuzzy group theory in the sense of Rosenfeld.

In this paper, we conduct a detailed investigation on -fuzzy subgroups. The research includes further properties of -fuzzy subgroups, -fuzzy normal subgroups, and -fuzzy left and right cosets.

The rest of this paper is organized as follows. In Section 2, we give properties of -fuzzy subgroups. In Section 3, we present properties of -fuzzy normal subgroups. In Section 4, we define -fuzzy left and right cosets and discuss their properties.

2. Properties of -Fuzzy Subgroups

In this paper, stands for a group with identity , and . By a fuzzy subset of , we mean a mapping from to the closed unit interval . In this section, we present some basic properties of -fuzzy subgroups.

Definition 1. Let be a fuzzy subset of . is called a fuzzy subgroup of if, for all , (i),(ii).

Definition 1 was introduced by Rosenfeld [2] in 1971.

Definition 2 (see [4]). Let be a fuzzy subset of . is called a -fuzzy subgroup of if, for all , (i),(ii).

Clearly, a -fuzzy subgroup is just a fuzzy subgroup, and thus a -fuzzy subgroup is a generalization of fuzzy subgroup.

Definition 3. For a fuzzy subset of and , let . Then is called a level subset of .

Proposition 4 (see [2]). A fuzzy subset of is a fuzzy subgroup if and only if is a crisp subgroup of for every .

Proposition 5 (see [5]). If is a -fuzzy subgroup of , then for all .

Corollary 6. Let be a -fuzzy subgroup of . Then (i)If   for some , then .(ii)If   for all and , then .(iii)If , then for all .

Proof. (i) If for some , then by Proposition 5; that is, .
(ii) When , by Proposition 5, ; that is, . Thus .
When , due to . by Proposition 5. Therefore, for all , ; that is, .
(iii) If , then for all , which implies due to .

Corollary 7. Let be a -fuzzy subgroup of and . Then holds for all .

Proof. If for some , then by Corollary 6(i), which is a contradiction. Hence for all . Since , holds for all by Corollary 6(ii).

Corollary 8. Let be a -fuzzy subgroup of and . Then for all .

Proof. For every , . By Corollary 7, . Hence .

Proposition 9 (see [4]). Let be a -fuzzy subset of . Then is a -fuzzy subgroup of if and only if is a subgroup of for all .

Proposition 10. Let be a -fuzzy subgroup of and . Assume(i)if  ,  then  .(ii)If  ,  then  .(iii)If  ,  then  .

Proof. Since is a -fuzzy subgroup of , is a subgroup of for all by Proposition 9.(i)If , then . Since is a subgroup of , ; that is, .(ii)If , then . If , then ; that is, , which is contradictory to that . Hence . In addition, since and is a subgroup of , we have . Therefore, . In summary, .(iii)Suppose . Let .
Then , and . Thus . By Proposition 9, is a subgroup of , and thus . As a result, , which is a contradiction to that . Therefore, .

Proposition 11. Let be a -fuzzy subgroup of and . (i)If    and  ,  then  .(ii)If  ,  ,  then  .(iii)If  ,  ,  then    and  .

Proof. Since is a -fuzzy subgroup of , is a subgroup of for all by Proposition 9.(i)From and , it follows that . Since is a subgroup of , we have ; that is, .(ii)Let and , . Then and . Now we have the following implications successively:  If , let . Then and . By Proposition 9, is a subgroup of , thus . Hence , which is a contradiction. Consequently, ; that is, . Similarly, .(iii)Suppose . Let .
Then and . By Proposition 9, is a subgroup of , thus . It follows that , which is a contradiction. Hence . Similarly, .

Proposition 12. Let be a cyclic group with generator . If is a -fuzzy subgroup of and , then .

Proof. For any , there must be a positive integer such that . By the definition of -fuzzy subgroup, which implies . Similarly, and thus . Continuing in this way, we have ; that is, .

Proposition 13. If is a cyclic group and , then .

Proof. For any , there must be a positive integer such that . By the definition of -fuzzy subgroup, which implies . Similarly, which implies . Continuing in this way, we have . Hence ; that is, . Therefore .

Corollary 14. Let be a cyclic group with generators and , a -fuzzy subgroup of . If , then .

Proof. By Proposition 12, . If , then by Proposition 13. Particularly, , which is a contradiction to that . Hence ; thus . By Proposition 12, . Consequently, .

Corollary 15. Let be a cyclic group of a prime order with generator , a -fuzzy subgroup of . If , then for .

Proof. When the order of is a prime, every in is a generator of . By Corollary 14, for .

3. Properties of -Fuzzy Normal Subgroups

The notion of fuzzy normal subgroup was first proposed and investigated by Wu [6] in 1981.

Definition 16 (see [6]). Let be a fuzzy subgroup of . is called a fuzzy normal subgroup of if for all ,

Proposition 17 (see [6]). Let be a fuzzy subgroup of . is a fuzzy normal subgroup if and only if for all ,

Proposition 18 (see [6]). Let be a fuzzy subset of . Then is a fuzzy normal subgroup of if and only if is a normal subgroup of for all .

In 2005, the following notion of -fuzzy normal subgroup was put forward by Yao [5].

Definition 19. Let be a -fuzzy subgroup of . is called a -fuzzy normal subgroup of if, for all , Clearly, a -fuzzy normal subgroup is just a fuzzy normal subgroup, and thus a -fuzzy normal subgroup is a generalization of fuzzy normal subgroup.

Proposition 20 (see [5]). Let be a -fuzzy subgroup of . is a -fuzzy normal subgroup if and only if, for all ,

Proposition 21 (see [5]). Let be a -fuzzy subset of . Then is a -fuzzy normal subgroup of if and only if is a normal subgroup of for all .

Proposition 22. Let be a -fuzzy normal subgroup of and . (i)If  ,  then    for all  .(ii)If  ,  then    for all  .(iii)If    and  ,  then  .(iv)If    and  ,  then  .(v)If    and  ,  then  .

Proof. (i) If , then . By Proposition 21, is a normal subgroup of and thus . Hence .
(ii) Let . Then . By Proposition 21, is a normal subgroup of . Hence ; that is, .
Suppose . Set . Then . By Proposition 21, is a normal subgroup of , and thus . Therefore, ; that is, , which is a contradiction to that . Consequently, .
(iii) If , then by (ii); that is, .
(iv) If , then . Since is a normal subgroup of by Proposition 21, ; that is, .
(v) Suppose on the contrary.
If , then, by (i), , which is contradictory to that . If , then, by (iii), , which is contradictory to that . Hence .

Proposition 23. Let be a -fuzzy subgroup of . Then is a -fuzzy normal subgroup of if and only if for all , where is a commutator in .

Proof. For any , Since is a -fuzzy normal subgroup of , and . Therefore, Conversely, if , then Hence is a -fuzzy normal subgroup of .

Proposition 24. If is an abelian group and is a -fuzzy subgroup of , then is a -fuzzy normal subgroup of .

Proof. Since is an abelian group, we have ; hence for all by Proposition 5. By Proposition 23, is a -fuzzy normal subgroup of .

Since a cyclic group is an abelian group, the following result is immediate by Proposition 24.

Corollary 25. If is a cyclic group and is a -fuzzy subgroup of , then is a -fuzzy normal subgroup of .

4. Properties of Left Cosets and Right Cosets of

Definition 26 (see [5]). Let be a -fuzzy subgroup of and . Define fuzzy subsets and of respectively by
and will be called a left coset and a right coset of , respectively.

Clearly, we have , and which are valid for all .

The following conclusions can be found in [5].

Proposition 27 (see [5]). Let and be -fuzzy subgroups of and . Then (i).(ii).(iii).(iv).

In particular, we have the following corollary when .

Corollary 28. Let be a -fuzzy subgroup of and . Then (i);(ii).

Firstly, we present some basic properties of .

Proposition 29. Let be a -fuzzy subgroup of and .(i)If  ,  then  .(ii)If  ,  then  .(iii)If  ,  then  .

Proof. (i) If , then . (ii) and (iii) can be similarly proved.

Proposition 30. Let be a -fuzzy subgroup of . Then is a -fuzzy subgroup of and a fuzzy subgroup of in the sense of Rosenfeld as well.

Proof. Firstly, we prove that for all .
If , then . Hence, which implies . So .
Conversely, if , then , whence ; that is, . Thus, . In summary, .
Since is a -fuzzy subgroup of , is the subgroup of by Proposition 9. It follows from that is a subgroup of . Hence is a -fuzzy subgroup of by Proposition 9.
Since , for , and clearly it is a subgroup of . Considering that for , is subgroup of for all . As a result, is a fuzzy subgroup of in the sense of Rosenfeld by Proposition 4.

Proposition 31. Let be a -fuzzy normal subgroup of . Then is a -fuzzy normal subgroup of and a fuzzy normal subgroup of as well.

Proof. By Proposition 30, is a -fuzzy subgroup of and a fuzzy subgroup of . For any , Therefore, is a -fuzzy normal subgroup of by Definition 19 and a fuzzy normal subgroup of by Definition 16.

Furthermore, we have the following results.

Proposition 32. If is a -fuzzy subgroup of and , then if and only if .

Proof. Firstly, we assume . Since is a subgroup of by Proposition 9, we have ; that is, by Proposition 10. Let . Consider the following three cases.
Case  1. If , then , which implies ; that is, by Proposition 11(i). Therefore,
Case  2. If , then by Proposition 11(ii). Hence
Case  3. If , then by Proposition 11(iii). Hence In summary, .
Conversely, assume . Then we have the following implications successively:

Similarly, if is a -fuzzy subgroup of and , then if and only if .

Corollary 33. If is a -fuzzy subgroup of and , then

Proof. Since , we have by Corollary 6, and hence and . It follows from that .

Corollary 34. Let be a -fuzzy subgroup of and . Then if and only if provided .

Proof. The desired result follows from the following equivalences:

Similarly, under the conditions stated in Corollary 34, we have the equivalence if and only if .

Proposition 35. If is a -fuzzy subgroup of and , then if and only if .

Proof. Firstly, we assume ; that is, . By Proposition 9, is a subgroup of . Hence we have ; that is, , so by Corollary 6. Let . We have by Corollary 6. Consider the following three cases.
Case  1  .  In this case, by Proposition 11(iii). So Case  2  .  In this case, by Proposition 11(ii). So Case  3  .  Since is a subgroup of by Proposition 9, we have . Hence ; that is, . Thus by Corollary 6. As a result, . Therefore, In summary, .
Conversely, assume . Then we have the following implications successively:

Similarly, if is a -fuzzy subgroup of and , then if and only if .

Corollary 36. Let be a -fuzzy subgroup of and . Then if and only if provided .

Proof. The desired result follows from the following equivalencies:

Similarly, under the conditions stated in Corollary 36, we have the equivalence if and only if .

Proposition 37. Let be a -fuzzy subgroup of . Then is a -fuzzy subgroup of if and only if .

Proof. Suppose that is a -fuzzy subgroup of .
If , then , which implies , and thus . Since is a subgroup of by Proposition 9, we have . Hence , which implies ; that is, . Hence we have by Proposition 10(i). Therefore, by Proposition 32.
If , then by Corollary 6 and , which implies . Since is a -fuzzy subgroup of , is a subgroup of by Proposition 9, which implies . Hence and thus ; that is, , which implies since is a subgroup of . Therefore, , and hence by Corollary 6. It follows from Proposition 35.
If , then for all by Corollary 6. Consider Hence .
Conversely, if , then is a -fuzzy subgroup of by Proposition 30.

Proposition 38. Let be a -fuzzy subgroup of , , and . If , then .

Proof. Since , is a subgroup of by Proposition 9, and thus . Hence, from , we know that ; that is, If , then ; that is, . Hence .
If , then ; that is, . By , thus . Hence .

Similarly, we have the following conclusion.

Let be a -fuzzy subgroup of , , and . If , then .

Proposition 39. Let be a -fuzzy subgroup of and . If and , then .

Proof. Since , by Corollary 28.
Case  1. If , then by Proposition 32, ; that is, . By Proposition 11 and , .
Case  2. If and , then and then by Proposition 35.
If , then by Proposition 11, .
If , then . Since is a -fuzzy subgroup of , is a subgroup of by Proposition 9. Hence ; that is, ; hence by Corollary 7.

It is noteworthy that the converse of Proposition 39 is not true as shown by the following example.

Example 40. Let be a cyclic group of order 3 with generator . Define by: , . Then is a -fuzzy subgroup of . In this example, however, , . Hence .

5. Concluding Remarks

In this paper, we present a further investigation into properties of -fuzzy subgroups, -fuzzy normal subgroups, and -fuzzy left and right cosets. It should be noted that we concentrate on the case in studying fuzzy left and right cosets. The reason is that is valid for all when , which is of little mathematical significance. Certainly, some other topics of -fuzzy subgroups such as the characterization of -fuzzy subgroups and operations of -fuzzy subgroups are still open for the research in the future.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this article.

Acknowledgments

The research project is supported by Shanxi Scholarship Council of China 2013-052, the Natural Science foundation of Shanxi 2013011004-1 and Shanxi Provincial Teaching Reform Project J2013134.