`Journal of Applied MathematicsVolumeΒ 2012, Article IDΒ 425890, 7 pageshttp://dx.doi.org/10.1155/2012/425890`
Research Article

## (π,π)-Fuzzy Version of Ideals, Interior Ideals, Quasi-Ideals, and Bi-Ideals

1College of Mathematics and Computer Science, Chongqing Three Gorges University, Wanzhou, Chongqing 404100, China
2Dipartimento di Matematica e Informatica, UniversitΓ  Degli Studi di Udine, Via delle Scienze 206, 33100 Udine, Italy

Received 12 July 2011; Accepted 16 December 2011

Copyright Β© 2012 Yuming Feng and P. Corsini. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We introduced -fuzzy ideals, -fuzzy interior ideals, -fuzzy quasi-ideals, and -fuzzy bi-ideals of an ordered semigroup and studied them. When and , we meet the ordinary fuzzy ones. This paper can be seen as a generalization of Kehayopulu and Tsingelis (2006), Kehayopulu and Tsingelis (2007), and Yao (2009).

#### 1. Introduction and Preliminaries

An ideal of a semigroup is a special subsemigroup satisfying certain conditions. The best way to know an algebraic structure is to begin with a special substructure of it. There are plenty of papers on ideals. After Zadehβ introduction of fuzzy set in 1965 (see [1]), the fuzzy sets have been used in the reconsideration of classical mathematics. Also, fuzzy ideals have been considered by many researchers. For example, Kim [2] studied intuitionistic fuzzy ideals of semigroups, Meng and Guo [3] researched fuzzy ideals of BCK/BCI-algebras, Koguep [4] researched fuzzy ideals of hyperlattices, and Kehayopulu and Tsingelis [5] researched fuzzy interior ideals of ordered semigroups.

Recently, Yuan et al. [6] introduced the concept of fuzzy subfield with thresholds. A fuzzy subfield with thresholds and is also called a -fuzzy subfield. Yao continued to research -fuzzy normal subfields, -fuzzy quotient subfields, -fuzzy subrings, and -fuzzy ideals in [7β10]. Feng et al. researched -fuzzy sublattices and -fuzzy subhyperlattices in [11].

An ordered semigroup is a poset equipped with a binary operation , such that is a semigroup, andif , then

Given an ordered semigroup , a fuzzy subset of (or a fuzzy set in ) is an arbitrary mapping , where is the usual closed interval of real numbers. For any is defined by . For , we define that . For two fuzzy subsets and of , we define the multiplication of and as the fuzzy subset of defined by

In the set of fuzzy subsets of , we define the order relation as follows: if and only if for all . For two fuzzy subsets and of , we define

Note that we use to denote and use to denote .

For any , can be seen as a fuzzy subset of which is defined by , for all .

In the following, we will use or to denote an ordered semigroup and the multiplication of will be instead of .

In the rest of this paper, we will always assume that .

In this paper, we introduced -fuzzy ideals, -fuzzy interior ideals, -fuzzy quasi-ideals and -fuzzy bi-ideals of an ordered semigroup. We obtained the followings:(1)in an ordered semigroup, every -fuzzy ideal is a -fuzzy interior ideal;(2)in an ordered semigroup, every -fuzzy right (resp. left) ideal is a -fuzzy quasi-ideal;(3)in an ordered semigroup, every -fuzzy quasi-ideal is a -fuzzy bi-ideal;(4)in a regular ordered semigroup, the -fuzzy quasi-ideals and the -fuzzy bi-ideals coincide.

#### 2. (π,π)-Fuzzy Ideals and (π,π)-Fuzzy Interior Ideals

Definition 2.1. Let be an ordered semigroup. A fuzzy subset of is called a -fuzzy right ideal (resp. -fuzzy left ideal) of if (resp. ) for all , andif , then for all .
A fuzzy subset of is called a -fuzzy ideal of if it is both a -fuzzy right and a -fuzzy left ideal of .

Example 2.2. Let be an ordered semigroup where and . The multiplication table is defined by the following:
A fuzzy set is defined as follows:
Then, is a -fuzzy ideal of . But it is not a fuzzy ideal of .

Definition 2.3 (see [12]). If is an ordered semigroup, a nonempty subset of is called an interior ideal of if, andif , and , then .

Definition 2.4. If is an ordered semigroup, a fuzzy subset of is called a -fuzzy interior ideal of if for all , andif , then .

In the previous example, is also a -fuzzy interior ideal of . In fact, every fuzzy ideal of an ordered semigroup is a fuzzy interior.

Theorem 2.5. Let be an ordered semigroup and a -fuzzy ideal of , then is a -fuzzy interior ideal of .

Proof. Let . Since is a -fuzzy left ideal of and , we have Since is a -fuzzy right ideal of , we have
From (2.3) and (2.4) we know that .

Theorem 2.6. Let be an ordered semigroup, then is a -fuzzy interior ideal of if and only if is an interior ideal of for all .

Proof. Let be a -fuzzy interior ideal of and .
First of all, we need to show that , for all , .
From and , we conclude that , that is, .
Then, we need to show that for all such that .
From we know that and from we have . Thus, . Notice that , then we conclude that , that is, .
Conversely, let be an interior ideal of for all .
If there are , such that , then and . That is and . This is a contradiction with that is an interior ideal of . Hence holds for all .
If there are such that and , then , and , that is, and . This is a contradiction with that is an interior ideal of . Hence if , then .

#### 3. (π,π)-Fuzzy Quasi-Ideals and (π,π)-Fuzzy Bi-Ideals

Definition 3.1. Let be an ordered semigroup. A subset of is called a quasi-ideal of if, andif and , then .

Definition 3.2. A nonempty subset of an ordered semigroup is called a bi-ideal of if it satisfies, and and , then .

Definition 3.3. Let be an ordered semigroup. A fuzzy subset of is called a -fuzzy quasi-ideal of if, andif , then for all .

Definition 3.4. Let be an ordered semigroup. A fuzzy subset of is called a -fuzzy bi-ideal of if for all ,, andif , then .

Remark 3.5. It is easy to see that a fuzzy quasi-ideal [13] of is a -fuzzy quasi-ideal of , and a fuzzy bi-ideal [13] of is a -fuzzy bi-ideal of .

Theorem 3.6. Let be an ordered semigroup, then is a -fuzzy quasi-ideal of if and only if is a quasi-ideal of for all .

Proof. Let be a -fuzzy quasi-ideal of and .
First of all, we need to show that .
If , then for some and .
From , we conclude that . Thus, , and so . Hence, .
Next, we need to show that for all such that .
From we know that and from we have . Thus, . Notice that , we conclude that , that is, .
Conversely, let be a quasi-ideal of for all . Then, .
If there is , such that , then , and . That is , and .
From and , we obtain that .
From and , we know that there exists at least one pair such that and . Thus, . Hence, .
Similarly, we can prove that .
So . This is a contradiction.
Hence, holds.
If there are such that and , then and , that is, and . This is a contradiction with that is a quasi-ideal of . Hence if , then .

Theorem 3.7. Let be an ordered semigroup, then is a -fuzzy bi-ideal of if and only if is a bi-ideal of for all .

Proof. The proof of this theorem is similar to the proof of the previous theorem.

Theorem 3.8. Let be an ordered semigroup, then the -fuzzy right (resp. left) ideals of are -fuzzy quasi-ideals of .

Proof. Let be a -fuzzy right ideal of and . First we have
If , then we have . So . Thus, .
If , then On the other hand, , for all .
Indeed, if , then , thus .
Hence, we have that . Thus, .
Therefore, is a -fuzzy quasi-ideal of .

Theorem 3.9. Let be an ordered semigroup, then the -fuzzy quasi-ideals of are -fuzzy bi-ideals of .

Proof. Let be a -fuzzy quasi-ideal of and . Then we have that
From , we have that .
From , we have that .
Thus, .
Therefore, is a -fuzzy bi-ideal of .

Definition 3.10 (see [5]). An ordered semigroup is called regular if for all there exists such that .

Theorem 3.11. In a regular ordered semigroup , the -fuzzy quasi-ideals and the -fuzzy bi-ideals coincide.

Proof. Let be a -fuzzy bi-ideal of and . We need to prove that
If , it is easy to verify that condition (3.4) is satisfied.
Let .
If , then we have that . Thus, condition (3.4) is satisfied.
If , then there exists at least one pair such that . That is , and .
We will prove that . Then, , and condition (3.4) is satisfied.
For any , we need to show that .
Let , then for some . Since is regular, there exists such that .
From , and , we obtain that . Since is a -fuzzy bi-ideal of , we have that Note that . Thus, .

#### Acknowledgments

This paper is prepared before the first authorβs visit to UniversitΓ  degli Studi di Udine, the first author wishes to express his gratitude to Professor Corsini, Dr. Paronuzzi, and Professor Russo for their hospitality. The first author is highly grateful to CMEC (KJ091104, KJ111107), CSTC, and CTGU(10QN-27) for financial support.

#### References

1. L. A. Zadeh, βFuzzy sets,β Information and Computation, vol. 8, pp. 338β353, 1965.
2. K. H. Kim, βIntuitionistic fuzzy ideals of semigroups,β Indian Journal of Pure and Applied Mathematics, vol. 33, no. 4, pp. 443β449, 2002.
3. J. Meng and X. Guo, βOn fuzzy ideals of BCK/BCI-algebras,β Fuzzy Sets and Systems, vol. 149, no. 3, pp. 509β525, 2005.
4. B. B. N. Koguep, βOn fuzzy ideals of hyperlattice,β International Journal of Algebra, vol. 2, no. 15, pp. 739β750, 2008.
5. N. Kehayopulu and M. Tsingelis, βFuzzy interior ideals in ordered semigroups,β Lobachevskii Journal of Mathematics, vol. 21, pp. 65β71, 2006.
6. X. Yuan, C. Zhang, and Y. Ren, βGeneralized fuzzy groups and many-valued implications,β Fuzzy Sets and Systems, vol. 138, no. 1, pp. 205β211, 2003.
7. B. Yao, β$\left(\mathrm{\Xi »},\mathrm{\Xi Ό}\right)$-fuzzy normal subgroups and $\left(\mathrm{\Xi »},\mathrm{\Xi Ό}\right)$-fuzzy quotient subgroups,β Journal of Fuzzy Mathematics, vol. 13, no. 3, pp. 695β705, 2005.
8. B. Yao, β$\left(\mathrm{\Xi »},\mathrm{\Xi Ό}\right)$-fuzzy subrings and $\left(\mathrm{\Xi »},\mathrm{\Xi Ό}\right)$-fuzzy ideals,β Journal of Fuzzy Mathematics, vol. 15, no. 4, pp. 981β987, 2007.
9. B. Yao, Fuzzy Theory on Group and Ring, Science and Technology Press, Beijing, China, 2008.
10. B. Yao, β$\left(\mathrm{\Xi »},\mathrm{\Xi Ό}\right)$-fuzzy ideals in semigroups,β Fuzzy Systems and Mathematics, vol. 23, no. 1, pp. 123β127, 2009.
11. Y. Feng, H. Duan, and Q. Zeng, β$\left(\mathrm{\Xi »},\mathrm{\Xi Ό}\right)$-fuzzy sublattices and $\left(\mathrm{\Xi »},\mathrm{\Xi Ό}\right)$-fuzzy sub- hyperlattices,β Fuzzy Information and Engineering, vol. 78, pp. 17β26, 2010.
12. N. Kehayopulu, βNote on interior ideals, ideal elements in ordered semigroups,β Scientiae Mathematicae, vol. 2, no. 3, pp. 407β409, 1999.
13. N. Kehayopulu and M. Tsingelis, βFuzzy ideals in ordered semigroups,β Quasigroups and Related Systems, vol. 15, no. 2, pp. 185β195, 2007.