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Journal of Applied Mathematics
VolumeΒ 2012Β (2012), Article IDΒ 425890, 7 pages
http://dx.doi.org/10.1155/2012/425890
Research Article

(𝝀,𝝁)-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

Academic Editor: VuΒ Phat

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 (πœ†,πœ‡)-fuzzy ideals, (πœ†,πœ‡)-fuzzy interior ideals, (πœ†,πœ‡)-fuzzy quasi-ideals, and (πœ†,πœ‡)-fuzzy bi-ideals of an ordered semigroup and studied them. When πœ†=0 and πœ‡=1, 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 Zadeh’ 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 πœ† and πœ‡ is also called a (πœ†,πœ‡)-fuzzy subfield. Yao continued to research (πœ†,πœ‡)-fuzzy normal subfields, (πœ†,πœ‡)-fuzzy quotient subfields, (πœ†,πœ‡)-fuzzy subrings, and (πœ†,πœ‡)-fuzzy ideals in [7–10]. Feng et al. researched (πœ†,πœ‡)-fuzzy sublattices and (πœ†,πœ‡)-fuzzy subhyperlattices in [11].

An ordered semigroup (𝑆,∘,≀) is a poset (𝑆,≀) equipped with a binary operation ∘, such that(1)(𝑆,∘) is a semigroup, and(2)if π‘₯,π‘Ž,π‘βˆˆπ‘†, then ξ‚»π‘Žβ‰€π‘β‡’π‘Žβˆ˜π‘₯β‰€π‘βˆ˜π‘₯π‘₯βˆ˜π‘Žβ‰€π‘₯βˆ˜π‘.(1.1)

Given an ordered semigroup 𝑆, a fuzzy subset of 𝑆 (or a fuzzy set in 𝑆) is an arbitrary mapping π‘“βˆΆπ‘†β†’[0,1], where [0,1] is the usual closed interval of real numbers. For any π›Όβˆˆ[0,1],𝑓𝛼 is defined by 𝑓𝛼={π‘₯βˆˆπ‘†|𝑓(π‘₯)β‰₯𝛼}. For π‘Žβˆˆπ‘†, we define that 𝐴𝛼={(𝑦,𝑧)βˆˆπ‘†Γ—π‘†|π‘Žβ‰€π‘¦π‘§}. For two fuzzy subsets 𝑓 and 𝑔 of 𝑆, we define the multiplication of 𝑓 and 𝑔 as the fuzzy subset of 𝑆 defined byξƒ―(π‘“βˆ—π‘”)(π‘Ž)=sup(𝑦,𝑧)βˆˆπ΄π‘Ž(𝑓(𝑦)βˆ§π‘”(𝑧)),ifπ΄π‘Žβ‰ βˆ…,0,ifπ΄π‘Ž=βˆ….(1.2)

In the set of fuzzy subsets of 𝑆, we define the order relation as follows: π‘“βŠ†π‘” if and only if 𝑓(π‘₯)≀𝑔(π‘₯) for all π‘₯βˆˆπ‘†. For two fuzzy subsets 𝑓 and 𝑔 of 𝑆, we define(π‘“βˆ©π‘”)(π‘₯)=𝑓(π‘₯)βˆ§π‘”(π‘₯),(𝑓βˆͺ𝑔)(π‘₯)=𝑓(π‘₯)βˆ¨π‘”(π‘₯).(1.3)

Note that we use π‘Žβˆ§π‘ to denote min(π‘Ž,𝑏) and use π‘Žβˆ¨π‘ to denote max(π‘Ž,𝑏).

For any π›Όβˆˆ[0,1], 𝛼 can be seen as a fuzzy subset of 𝑆 which is defined by 𝛼(π‘₯)=𝛼, for all π‘₯βˆˆπ‘†.

In the following, we will use 𝑆 or (𝑆,∘,≀) to denote an ordered semigroup and the multiplication of π‘₯,𝑦 will be π‘₯𝑦 instead of π‘₯βˆ˜π‘¦.

In the rest of this paper, we will always assume that 0β‰€πœ†<πœ‡β‰€1.

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

2. (πœ†,πœ‡)-Fuzzy Ideals and (πœ†,πœ‡)-Fuzzy Interior Ideals

Definition 2.1. Let (𝑆,β‹…,≀) be an ordered semigroup. A fuzzy subset 𝑓 of 𝑆 is called a (πœ†,πœ‡)-fuzzy right ideal (resp. (πœ†,πœ‡)-fuzzy left ideal) of 𝑆 if(1)𝑓(π‘₯𝑦)βˆ¨πœ†β‰₯𝑓(π‘₯)βˆ§πœ‡ (resp. 𝑓(π‘₯𝑦)βˆ¨πœ†β‰₯𝑓(𝑦)βˆ§πœ‡ ) for all π‘₯,π‘¦βˆˆπ‘†, and(2)if π‘₯≀𝑦, then 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑦)βˆ§πœ‡ for all π‘₯,π‘¦βˆˆπ‘†.
A fuzzy subset 𝑓 of 𝑆 is called a (πœ†,πœ‡)-fuzzy ideal of 𝑆 if it is both a (πœ†,πœ‡)-fuzzy right and a (πœ†,πœ‡)-fuzzy left ideal of 𝑆.

Example 2.2. Let (𝑆,βˆ—,≀) be an ordered semigroup where 𝑆={𝑒,π‘Ž,𝑏} and π‘’β‰€π‘Žβ‰€π‘. The multiplication table is defined by the following: βˆ—π‘’π‘Žπ‘π‘’π‘Žπ‘’π‘Žπ‘π‘π‘Žπ‘π‘’π‘π‘’π‘Ž(2.1)
A fuzzy set 𝑓 is defined as follows: π‘†π‘’π‘Žπ‘π‘“0.10.20.3(2.2)
Then, 𝑓 is a (0.3,0.7)-fuzzy ideal of 𝑆. But it is not a fuzzy ideal of 𝑆.

Definition 2.3 (see [12]). If (𝑆,∘,≀) is an ordered semigroup, a nonempty subset 𝐴 of 𝑆 is called an interior ideal of 𝑆 if(1)π‘†π΄π‘†βŠ†π΄, and(2)if π‘Žβˆˆπ΄,π‘βˆˆπ‘†, and π‘β‰€π‘Ž, then π‘βˆˆπ΄.

Definition 2.4. If (𝑆,∘,≀) is an ordered semigroup, a fuzzy subset 𝑓 of 𝑆 is called a (πœ†,πœ‡)-fuzzy interior ideal of 𝑆 if(1)𝑓(π‘₯π‘Žπ‘¦)βˆ¨πœ†β‰₯𝑓(π‘Ž)βˆ§πœ‡ for all π‘₯,π‘Ž,π‘¦βˆˆπ‘†, and(2)if π‘₯≀𝑦, then 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑦)βˆ§πœ‡.

In the previous example, 𝑓 is also a (0.3,0.7)-fuzzy interior ideal of 𝑆. In fact, every fuzzy ideal of an ordered semigroup is a fuzzy interior.

Theorem 2.5. Let (𝑆,∘,≀) be an ordered semigroup and 𝑓 a (πœ†,πœ‡)-fuzzy ideal of 𝑆, then 𝑓 is a (πœ†,πœ‡)-fuzzy interior ideal of 𝑆.

Proof. Let π‘₯,π‘Ž,π‘¦βˆˆπ‘†. Since 𝑓 is a (πœ†,πœ‡)-fuzzy left ideal of 𝑆 and π‘₯,π‘Žπ‘¦βˆˆπ‘†, we have 𝑓(π‘₯(π‘Žπ‘¦))βˆ¨πœ†β‰₯𝑓(π‘Žπ‘¦)βˆ§πœ‡.(2.3) Since 𝑓 is a (πœ†,πœ‡)-fuzzy right ideal of 𝑆, we have .𝑓(π‘Žπ‘¦)βˆ¨πœ†β‰₯𝑓(π‘Ž)βˆ§πœ‡(2.4)
From (2.3) and (2.4) we know that 𝑓(π‘₯π‘Žπ‘¦)βˆ¨πœ†=(𝑓(π‘₯(π‘Žπ‘¦))βˆ¨πœ†)βˆ¨πœ†β‰₯(𝑓(π‘Žπ‘¦)βˆ§πœ‡)βˆ¨πœ†=(𝑓(π‘Žπ‘¦)βˆ¨πœ†)∧(πœ‡βˆ¨πœ†)β‰₯𝑓(π‘Ž)βˆ§πœ‡.

Theorem 2.6. Let (𝑆,∘,≀) be an ordered semigroup, then 𝑓 is a (πœ†,πœ‡)-fuzzy interior ideal of 𝑆 if and only if 𝑓𝛼 is an interior ideal of 𝑆 for all π›Όβˆˆ(πœ†,πœ‡].

Proof. Let 𝑓 be a (πœ†,πœ‡)-fuzzy interior ideal of 𝑆 and π›Όβˆˆ(πœ†,πœ‡].
First of all, we need to show that π‘₯π‘Žπ‘¦βˆˆπ‘“π›Ό, for all π‘Žβˆˆπ‘“π›Ό, π‘₯,π‘¦βˆˆπ‘†.
From 𝑓(π‘₯π‘Žπ‘¦)βˆ¨πœ†β‰₯𝑓(π‘Ž)βˆ§πœ‡β‰₯π›Όβˆ§πœ‡=𝛼 and πœ†<𝛼, we conclude that 𝑓(π‘₯π‘Žπ‘¦)β‰₯𝛼, that is, π‘₯π‘Žπ‘¦βˆˆπ‘“π›Ό.
Then, we need to show that π‘βˆˆπ‘“π›Ό for all π‘Žβˆˆπ‘“π›Ό,π‘βˆˆπ‘† such that π‘β‰€π‘Ž.
From π‘β‰€π‘Ž we know that 𝑓(𝑏)βˆ¨πœ†β‰₯𝑓(π‘Ž)βˆ§πœ‡ and from π‘Žβˆˆπ‘“π›Ό we have 𝑓(π‘Ž)β‰₯𝛼. Thus, 𝑓(𝑏)βˆ¨πœ†β‰₯π›Όβˆ§πœ‡=𝛼. Notice that πœ†<𝛼, then we conclude that 𝑓(𝑏)β‰₯𝛼, that is, π‘βˆˆπ‘“π›Ό.
Conversely, let 𝑓𝛼 be an interior ideal of 𝑆 for all π›Όβˆˆ(πœ†,πœ‡].
If there are π‘₯0,π‘Ž0,𝑦0βˆˆπ‘†, such that 𝑓(π‘₯0π‘Ž0𝑦0)βˆ¨πœ†<𝛼=𝑓(π‘Ž0)βˆ§πœ‡, then π›Όβˆˆ(πœ†,πœ‡],𝑓(π‘Ž0)β‰₯𝛼 and 𝑓(π‘₯0π‘Ž0𝑦0)<𝛼. That is π‘Ž0βˆˆπ‘“π›Ό and π‘₯0π‘Ž0𝑦0βˆ‰π‘“π›Ό. This is a contradiction with that 𝑓𝛼 is an interior ideal of 𝑆. Hence 𝑓(π‘₯π‘Žπ‘¦)βˆ¨πœ†β‰₯𝑓(π‘Ž)βˆ§πœ‡ holds for all π‘₯,π‘Ž,π‘¦βˆˆπ‘†.
If there are π‘₯0,𝑦0βˆˆπ‘† such that π‘₯0≀𝑦0 and 𝑓(π‘₯0)βˆ¨πœ†<𝛼=𝑓(𝑦0)βˆ§πœ‡, then π›Όβˆˆ(πœ†,πœ‡],𝑓(𝑦0)β‰₯𝛼, and 𝑓(π‘₯0)<𝛼, that is, 𝑦0βˆˆπ‘“π›Ό and π‘₯0βˆ‰π‘“π›Ό. This is a contradiction with that 𝑓𝛼 is an interior ideal of 𝑆. Hence if π‘₯≀𝑦, then 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑦)βˆ§πœ‡.

3. (πœ†,πœ‡)-Fuzzy Quasi-Ideals and (πœ†,πœ‡)-Fuzzy Bi-Ideals

Definition 3.1. Let (𝑆,∘,≀) be an ordered semigroup. A subset 𝐴 of 𝑆 is called a quasi-ideal of 𝑆 if(1)π΄π‘†βˆ©π‘†π΄βŠ†π‘†, and(2)if π‘₯βˆˆπ‘† and π‘₯β‰€π‘¦βˆˆπ΄, then π‘₯∈𝐴.

Definition 3.2. A nonempty subset 𝐴 of an ordered semigroup 𝑆 is called a bi-ideal of 𝑆 if it satisfies(1)π΄π‘†π΄βŠ†π΄, and(2)π‘₯βˆˆπ‘† and π‘₯β‰€π‘¦βˆˆπ΄, then π‘₯∈𝐴.

Definition 3.3. Let (𝑆,∘,≀) be an ordered semigroup. A fuzzy subset 𝑓 of 𝑆 is called a (πœ†,πœ‡)-fuzzy quasi-ideal of 𝑆 if(1)𝑓βˆͺπœ†βŠ‡(π‘“βˆ—1)∩(1βˆ—π‘“)βˆ©πœ‡, and(2)if π‘₯≀𝑦, then 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑦)βˆ§πœ‡ for all π‘₯,π‘¦βˆˆπ‘†.

Definition 3.4. Let (𝑆,∘,≀) be an ordered semigroup. A fuzzy subset 𝑓 of 𝑆 is called a (πœ†,πœ‡)-fuzzy bi-ideal of 𝑆 if for all π‘₯,𝑦,π‘§βˆˆπ‘†,(1)𝑓(π‘₯𝑦𝑧)βˆ¨πœ†β‰₯(𝑓(π‘₯)βˆ§π‘“(𝑧))βˆ§πœ‡, and(2)if π‘₯≀𝑦, then 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑦)βˆ§πœ‡.

Remark 3.5. It is easy to see that a fuzzy quasi-ideal [13] of 𝑆 is a (0,1)-fuzzy quasi-ideal of 𝑆, and a fuzzy bi-ideal [13] of 𝑆 is a (0,1)-fuzzy bi-ideal of 𝑆.

Theorem 3.6. Let (𝑆,∘,≀) be an ordered semigroup, then 𝑓 is a (πœ†,πœ‡)-fuzzy quasi-ideal of 𝑆 if and only if 𝑓𝛼 is a quasi-ideal of 𝑆 for all π›Όβˆˆ(πœ†,πœ‡].

Proof. Let 𝑓 be a (πœ†,πœ‡)-fuzzy quasi-ideal of 𝑆 and π›Όβˆˆ(πœ†,πœ‡].
First of all, we need to show that π‘†π‘“π›Όβˆ©π‘“π›Όπ‘†βŠ†π‘“π›Ό.
If π‘₯βˆˆπ‘†π‘“π›Όβˆ©π‘“π›Όπ‘†, then π‘₯=𝑠𝑑1=𝑑2𝑠 for some 𝑑1,𝑑2βˆˆπ‘“π›Ό and π‘ βˆˆπ‘†.
From 𝑓βˆͺπœ†βŠ‡(π‘“βˆ—1)∩(1βˆ—π‘“)βˆ©πœ‡, we conclude that 𝑓(π‘₯)βˆ¨πœ†β‰₯(π‘“βˆ—1)(π‘₯)∧(π‘“βˆ—1)(π‘₯)βˆ§πœ‡β‰₯𝑓(𝑑1)βˆ§π‘“(𝑑2)βˆ§πœ‡β‰₯π›Όβˆ§πœ‡=𝛼. Thus, 𝑓(π‘₯)β‰₯𝛼, and so π‘₯βˆˆπ‘“π›Ό. Hence, π‘†βˆ—π‘“π›Όβˆ©π‘“π›Όβˆ—π‘†βŠ†π‘“π›Ό.
Next, we need to show that π‘βˆˆπ‘“π›Ό for all π‘Žβˆˆπ‘“π›Ό,π‘βˆˆπ‘† such that π‘β‰€π‘Ž.
From π‘β‰€π‘Ž we know that 𝑓(𝑏)βˆ¨πœ†β‰₯𝑓(π‘Ž)βˆ§πœ‡ and from π‘Žβˆˆπ‘“π›Ό we have 𝑓(π‘Ž)β‰₯𝛼. Thus, 𝑓(𝑏)βˆ¨πœ†β‰₯π›Όβˆ§πœ‡=𝛼. Notice that πœ†<𝛼, we conclude that 𝑓(𝑏)β‰₯𝛼, that is, π‘βˆˆπ‘“π›Ό.
Conversely, let 𝑓𝛼 be a quasi-ideal of 𝑆 for all π›Όβˆˆ(πœ†,πœ‡]. Then, π‘“π›Όπ‘†βˆ©π‘†π‘“π›ΌβŠ†π‘“π›Ό.
If there is π‘₯0βˆˆπ‘†, such that 𝑓(π‘₯0)βˆ¨πœ†<𝛼=(π‘“βˆ—1)(π‘₯)∧(1βˆ—π‘“)(π‘₯)βˆ§πœ‡, then π›Όβˆˆ(πœ†,πœ‡],𝑓(π‘₯0)<𝛼, (π‘“βˆ—1)(π‘₯0)β‰₯𝛼 and (1βˆ—π‘“)(π‘₯0)β‰₯𝛼. That is π‘₯0βˆ‰π‘“π›Ό, supπ‘₯0≀π‘₯1π‘₯2𝑓(π‘₯1)β‰₯𝛼 and supπ‘₯0≀π‘₯1π‘₯2𝑓(π‘₯2)β‰₯𝛼.
From π‘“π›Όπ‘†βˆ©π‘†π‘“π›ΌβŠ†π‘“π›Ό and π‘₯0βˆ‰π‘“π›Ό, we obtain that π‘₯0βˆ‰π‘“π›Όπ‘†βˆ©π‘†π‘“π›Ό.
From supπ‘₯0≀π‘₯1π‘₯2𝑓(π‘₯1)β‰₯𝛼 and 𝛼≠0, we know that there exists at least one pair (π‘₯1,π‘₯2)βˆˆπ‘†Γ—π‘† such that π‘₯0≀π‘₯1π‘₯2 and 𝑓(π‘₯1)β‰₯𝛼. Thus, π‘₯0≀π‘₯1π‘₯2βˆˆπ‘“π›Όπ‘†. Hence, π‘₯0βˆˆπ‘“π›ΌS.
Similarly, we can prove that π‘₯0βˆˆπ‘†π‘“π›Ό.
So π‘₯0βˆˆπ‘“π›Όπ‘†βˆ©π‘†π‘“π›Ό. This is a contradiction.
Hence, 𝑓βˆͺπœ†βŠ‡(π‘“βˆ—1)∩(1βˆ—π‘“)βˆ©πœ‡ holds.
If there are π‘₯0,𝑦0βˆˆπ‘† such that π‘₯0≀𝑦0 and 𝑓(π‘₯0)βˆ¨πœ†<𝛼=𝑓(𝑦0)βˆ§πœ‡, then π›Όβˆˆ(πœ†,πœ‡],𝑓(𝑦0)β‰₯𝛼 and 𝑓(π‘₯0)<𝛼, that is, 𝑦0βˆˆπ‘“π›Ό and π‘₯0βˆ‰π‘“π›Ό. This is a contradiction with that 𝑓𝛼 is a quasi-ideal of 𝑆. Hence if π‘₯≀𝑦, then 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑦)βˆ§πœ‡.

Theorem 3.7. Let (𝑆,∘,≀) be an ordered semigroup, then 𝑓 is a (πœ†,πœ‡)-fuzzy bi-ideal of 𝑆 if and only if 𝑓𝛼 is a bi-ideal of 𝑆 for all π›Όβˆˆ(πœ†,πœ‡].

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

Theorem 3.8. Let (𝑆,∘,≀) be an ordered semigroup, then the (πœ†,πœ‡)-fuzzy right (resp. left) ideals of 𝑆 are (πœ†,πœ‡)-fuzzy quasi-ideals of 𝑆.

Proof. Let 𝑓 be a (πœ†,πœ‡)-fuzzy right ideal of 𝑆 and π‘₯βˆˆπ‘†. First we have ((π‘“βˆ—1)∩(1βˆ—π‘“))(π‘₯)=(π‘“βˆ—1)(π‘₯)∧(1βˆ—π‘“)(π‘₯).(3.1)
If 𝐴π‘₯=βˆ…, then we have (π‘“βˆ—1)(π‘₯)=0=(1βˆ—π‘“)(π‘₯). So 𝑓(π‘₯)βˆ¨πœ†β‰₯0=(π‘“βˆ—1)(π‘₯)∧(1βˆ—π‘“)(π‘₯)βˆ§πœ‡. Thus, 𝑓βˆͺπœ†βŠ‡(π‘“βˆ—1)∩(1βˆ—π‘“)βˆ©πœ‡.
If 𝐴π‘₯β‰ βˆ…, then (π‘“βˆ—1)(π‘₯)=sup(𝑒,𝑣)∈𝐴π‘₯(𝑓(𝑒)∧1(𝑣)).(3.2) On the other hand, 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑒)∧1(𝑣)βˆ§πœ‡, for all (𝑒,𝑣)∈𝐴π‘₯.
Indeed, if (𝑒,𝑣)∈𝐴π‘₯, then π‘₯≀𝑒𝑣, thus 𝑓(π‘₯)βˆ¨πœ†=𝑓(π‘₯)βˆ¨πœ†βˆ¨πœ†β‰₯(𝑓(𝑒𝑣)βˆ§πœ‡)βˆ¨πœ†=(𝑓(𝑒𝑣)βˆ¨πœ†)∧(πœ†βˆ¨πœ‡)β‰₯(𝑓(𝑒)βˆ§πœ‡)βˆ§πœ‡=𝑓(𝑒)βˆ§πœ‡=𝑓(𝑒)∧1(𝑣)βˆ§πœ‡.
Hence, we have that 𝑓(π‘₯)βˆ¨πœ†β‰₯(sup(𝑒,𝑣)∈𝐴π‘₯(𝑓(𝑒)∧1(𝑣)))βˆ§πœ‡=(π‘“βˆ—1)(π‘₯)βˆ§πœ‡β‰₯(π‘“βˆ—1)(π‘₯)∧(1βˆ—π‘“)(π‘₯)βˆ§πœ‡. Thus, 𝑓βˆͺπœ†βŠ‡(π‘“βˆ—1)∩(1βˆ—π‘“)βˆ©πœ‡.
Therefore, 𝑓 is a (πœ†,πœ‡)-fuzzy quasi-ideal of 𝑆.

Theorem 3.9. Let (𝑆,∘,≀) be an ordered semigroup, then the (πœ†,πœ‡)-fuzzy quasi-ideals of 𝑆 are (πœ†,πœ‡)-fuzzy bi-ideals of 𝑆.

Proof. Let 𝑓 be a (πœ†,πœ‡)-fuzzy quasi-ideal of 𝑆 and π‘₯,𝑦,π‘§βˆˆπ‘†. Then we have that 𝑓(π‘₯𝑦𝑧)βˆ¨πœ†β‰₯(π‘“βˆ—1)(π‘₯𝑦𝑧)∧(1βˆ—π‘“)(π‘₯𝑦𝑧)βˆ§πœ‡.(3.3)
From (π‘₯,𝑦𝑧)∈𝐴π‘₯𝑦𝑧, we have that (π‘“βˆ—1)(π‘₯𝑦𝑧)β‰₯𝑓(π‘₯)∧1(𝑦𝑧)=𝑓(π‘₯).
From (π‘₯𝑦,𝑧)∈𝐴π‘₯𝑦𝑧, we have that (1βˆ—π‘“)(π‘₯𝑦𝑧)β‰₯1(π‘₯𝑦)βˆ§π‘“(𝑧)=𝑓(𝑧).
Thus, 𝑓(π‘₯𝑦𝑧)βˆ¨πœ†β‰₯𝑓(π‘₯)βˆ§π‘“(𝑧)βˆ§πœ‡.
Therefore, 𝑓 is a (πœ†,πœ‡)-fuzzy bi-ideal of 𝑆.

Definition 3.10 (see [5]). An ordered semigroup (𝑆,∘,≀) is called regular if for all π‘Žβˆˆπ‘† there exists π‘₯βˆˆπ‘† such that π‘Žβ‰€π‘Žπ‘₯π‘Ž.

Theorem 3.11. In a regular ordered semigroup 𝑆, the (πœ†,πœ‡)-fuzzy quasi-ideals and the (πœ†,πœ‡)-fuzzy bi-ideals coincide.

Proof. Let 𝑓 be a (πœ†,πœ‡)-fuzzy bi-ideal of 𝑆 and π‘₯βˆˆπ‘†. We need to prove that 𝑓(π‘₯)βˆ¨πœ†β‰₯(π‘“βˆ—1)(π‘₯)∧(1βˆ—π‘“)βˆ§πœ‡.(3.4)
If 𝐴π‘₯=βˆ…, it is easy to verify that condition (3.4) is satisfied.
Let 𝐴π‘₯β‰ βˆ….
(1) If (π‘“βˆ—1)(π‘₯)βˆ§πœ‡β‰€π‘“(π‘₯)βˆ¨πœ†, then we have that 𝑓(π‘₯)βˆ¨πœ†β‰₯(π‘“βˆ—1)(π‘₯)βˆ§πœ‡β‰₯(π‘“βˆ—1)(π‘₯)∧(1βˆ—π‘“)(π‘₯)βˆ§πœ‡. Thus, condition (3.4) is satisfied.
(2) If (π‘“βˆ—1)(π‘₯)βˆ§πœ‡>𝑓(π‘₯)βˆ¨πœ†, then there exists at least one pair (𝑧,𝑀)∈𝐴π‘₯ such that 𝑓(𝑧)∧1(𝑀)βˆ§πœ‡>𝑓(π‘₯)βˆ¨πœ†. That is 𝑧,π‘€βˆˆπ‘†, π‘₯≀𝑧𝑀 and 𝑓(𝑧)βˆ§πœ‡>𝑓(π‘₯)βˆ¨πœ†.
We will prove that (1βˆ—π‘“)(π‘₯)βˆ§πœ‡β‰€π‘“(π‘₯)βˆ¨πœ†. Then, 𝑓(π‘₯)βˆ¨πœ†β‰₯(1βˆ—π‘“)(π‘₯)βˆ§πœ‡β‰₯(π‘“βˆ—1)(π‘₯)∧(1βˆ—π‘“)(π‘₯)βˆ§πœ‡, and condition (3.4) is satisfied.
For any (𝑒,𝑣)∈𝐴π‘₯, we need to show that 1(𝑒)βˆ§π‘“(𝑣)βˆ§πœ‡β‰€π‘“(π‘₯)βˆ¨πœ†.
Let (𝑒,𝑣)∈𝐴π‘₯, then π‘₯≀𝑒𝑣 for some 𝑒,π‘£βˆˆπ‘†. Since 𝑆 is regular, there exists π‘ βˆˆπ‘† such that π‘₯≀π‘₯𝑠π‘₯.
From π‘₯≀π‘₯𝑠π‘₯, π‘₯≀𝑧𝑀 and π‘₯≀𝑒𝑣, we obtain that π‘₯≀𝑧𝑀𝑠𝑒𝑣. Since 𝑓 is a (πœ†,πœ‡)-fuzzy bi-ideal of 𝑆, we have that 𝑓(π‘₯)βˆ¨πœ†β‰₯(𝑓(𝑧𝑀𝑠𝑒𝑣)βˆ§πœ‡)βˆ¨πœ†=(𝑓(𝑧𝑀𝑠𝑒𝑣)βˆ¨πœ†)∧(πœ‡βˆ¨πœ†)β‰₯𝑓(𝑧)βˆ§π‘“(𝑣)βˆ§πœ‡.(3.5) Note that 𝑓(𝑧)βˆ§πœ‡>𝑓(π‘₯)βˆ¨πœ†. Thus, 𝑓(π‘₯)βˆ¨πœ†β‰₯𝑓(𝑣)βˆ§πœ‡=1(𝑒)βˆ§π‘“(𝑣)βˆ§πœ‡.

Acknowledgments

This paper is prepared before the first author’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€œFuzzy sets,” Information and Computation, vol. 8, pp. 338–353, 1965. View at Google Scholar Β· View at Zentralblatt MATH
  2. K. H. Kim, β€œIntuitionistic fuzzy ideals of semigroups,” Indian Journal of Pure and Applied Mathematics, vol. 33, no. 4, pp. 443–449, 2002. View at Google Scholar Β· View at Zentralblatt MATH
  3. J. Meng and X. Guo, β€œOn fuzzy ideals of BCK/BCI-algebras,” Fuzzy Sets and Systems, vol. 149, no. 3, pp. 509–525, 2005. View at Publisher Β· View at Google Scholar
  4. B. B. N. Koguep, β€œOn fuzzy ideals of hyperlattice,” International Journal of Algebra, vol. 2, no. 15, pp. 739–750, 2008. View at Google Scholar Β· View at Zentralblatt MATH
  5. N. Kehayopulu and M. Tsingelis, β€œFuzzy interior ideals in ordered semigroups,” Lobachevskii Journal of Mathematics, vol. 21, pp. 65–71, 2006. View at Google Scholar Β· View at Zentralblatt MATH
  6. X. Yuan, C. Zhang, and Y. Ren, β€œGeneralized fuzzy groups and many-valued implications,” Fuzzy Sets and Systems, vol. 138, no. 1, pp. 205–211, 2003. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  7. B. Yao, β€œ(Ξ»,ΞΌ)-fuzzy normal subgroups and (Ξ»,ΞΌ)-fuzzy quotient subgroups,” Journal of Fuzzy Mathematics, vol. 13, no. 3, pp. 695–705, 2005. View at Google Scholar
  8. B. Yao, β€œ(Ξ»,ΞΌ)-fuzzy subrings and (Ξ»,ΞΌ)-fuzzy ideals,” Journal of Fuzzy Mathematics, vol. 15, no. 4, pp. 981–987, 2007. View at Google Scholar
  9. B. Yao, Fuzzy Theory on Group and Ring, Science and Technology Press, Beijing, China, 2008.
  10. B. Yao, β€œ(Ξ»,ΞΌ)-fuzzy ideals in semigroups,” Fuzzy Systems and Mathematics, vol. 23, no. 1, pp. 123–127, 2009. View at Google Scholar
  11. Y. Feng, H. Duan, and Q. Zeng, β€œ(Ξ»,ΞΌ)-fuzzy sublattices and (Ξ»,ΞΌ)-fuzzy sub- hyperlattices,” Fuzzy Information and Engineering, vol. 78, pp. 17–26, 2010. View at Publisher Β· View at Google Scholar
  12. N. Kehayopulu, β€œNote on interior ideals, ideal elements in ordered semigroups,” Scientiae Mathematicae, vol. 2, no. 3, pp. 407–409, 1999. View at Google Scholar Β· View at Zentralblatt MATH
  13. N. Kehayopulu and M. Tsingelis, β€œFuzzy ideals in ordered semigroups,” Quasigroups and Related Systems, vol. 15, no. 2, pp. 185–195, 2007. View at Google Scholar Β· View at Zentralblatt MATH