(𝝀,𝝁)-Fuzzy Version of Ideals, Interior Ideals, Quasi-Ideals, and Bi-Ideals

Abstract

We introduced (𝜆,𝜇)-fuzzy ideals, (𝜆,𝜇)-fuzzy interior ideals, (𝜆,𝜇)-fuzzy quasi-ideals, and (𝜆,𝜇)-fuzzy bi-ideals of an ordered semigroup and studied them. When 𝜆=0 and 𝜇=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 𝜆 and 𝜇 is also called a (𝜆,𝜇)-fuzzy subfield. Yao continued to research (𝜆,𝜇)-fuzzy normal subfields, (𝜆,𝜇)-fuzzy quotient subfields, (𝜆,𝜇)-fuzzy subrings, and (𝜆,𝜇)-fuzzy ideals in [7–10]. Feng et al. researched (𝜆,𝜇)-fuzzy sublattices and (𝜆,𝜇)-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 𝑥,𝑎,𝑏∈𝑆, then 𝑎≤𝑏⇒𝑎𝑥∘∘𝑥𝑎≤≤𝑏𝑥∘∘𝑥𝑏.(1.1)

Given an ordered semigroup 𝑆, a fuzzy subset of 𝑆 (or a fuzzy set in 𝑆) is an arbitrary mapping 𝑓∶𝑆→[0,1], where [0,1] is the usual closed interval of real numbers. For any 𝛼∈[0,1],𝑓_𝛼 is defined by 𝑓_𝛼={𝑥∈𝑆|𝑓(𝑥)≥𝛼}. For 𝑎∈𝑆, we define that 𝐴_𝛼={(𝑦,𝑧)∈𝑆×𝑆|𝑎≤𝑦𝑧}. For two fuzzy subsets 𝑓 and 𝑔 of 𝑆, we define the multiplication of 𝑓 and 𝑔 as the fuzzy subset of 𝑆 defined by(𝑓∗𝑔)(𝑎)=sup^{(𝑦,𝑧)∈𝐴𝑎}0,(𝑓(𝑦)∧𝑔(𝑧)),if𝐴_𝑎≠∅,if𝐴_𝑎=∅.(1.2)

In the set of fuzzy subsets of 𝑆, we define the order relation as follows: 𝑓⊆𝑔 if and only if 𝑓(𝑥)≤𝑔(𝑥) for all 𝑥∈𝑆. For two fuzzy subsets 𝑓 and 𝑔 of 𝑆, we define(𝑓∩𝑔)(𝑥)=𝑓(𝑥)∧𝑔(𝑥),(𝑓∪𝑔)(𝑥)=𝑓(𝑥)∨𝑔(𝑥).(1.3)

Note that we use 𝑎∧𝑏 to denote min(𝑎,𝑏) and use 𝑎∨𝑏 to denote max(𝑎,𝑏).

For any 𝛼∈[0,1], 𝛼 can be seen as a fuzzy subset of 𝑆 which is defined by 𝛼(𝑥)=𝛼, for all 𝑥∈𝑆.

In the following, we will use 𝑆 or (𝑆,∘,≤) to denote an ordered semigroup and the multiplication of 𝑥,𝑦 will be 𝑥𝑦 instead of 𝑥∘𝑦.

In the rest of this paper, we will always assume that 0≤𝜆<𝜇≤1.

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

2. (𝜆,𝜇)-Fuzzy Ideals and (𝜆,𝜇)-Fuzzy Interior Ideals

Definition 2.1. Let (𝑆,⋅,≤) be an ordered semigroup. A fuzzy subset 𝑓 of 𝑆 is called a (𝜆,𝜇)-fuzzy right ideal (resp. (𝜆,𝜇)-fuzzy left ideal) of 𝑆 if(1)𝑓(𝑥𝑦)∨𝜆≥𝑓(𝑥)∧𝜇 (resp. 𝑓(𝑥𝑦)∨𝜆≥𝑓(𝑦)∧𝜇 ) for all 𝑥,𝑦∈𝑆, and(2)if 𝑥≤𝑦, then 𝑓(𝑥)∨𝜆≥𝑓(𝑦)∧𝜇 for all 𝑥,𝑦∈𝑆.
A fuzzy subset 𝑓 of 𝑆 is called a (𝜆,𝜇)-fuzzy ideal of 𝑆 if it is both a (𝜆,𝜇)-fuzzy right and a (𝜆,𝜇)-fuzzy left ideal of 𝑆.

Example 2.2. Let (𝑆,∗,≤) be an ordered semigroup where 𝑆={𝑒,𝑎,𝑏} and 𝑒≤𝑎≤𝑏. The multiplication table is defined by the following: ∗𝑒𝑎𝑏𝑎𝑒𝑒𝑎𝑏𝑏𝑎𝑏𝑒𝑏𝑒𝑎(2.1)
A fuzzy set 𝑓 is defined as follows: 𝑆𝑒𝑎𝑏𝑓0.10.20.3(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)𝑆𝐴𝑆⊆𝐴, and(2)if 𝑎∈𝐴,𝑏∈𝑆, and 𝑏≤𝑎, then 𝑏∈𝐴.

Definition 2.4. If (𝑆,∘,≤) is an ordered semigroup, a fuzzy subset 𝑓 of 𝑆 is called a (𝜆,𝜇)-fuzzy interior ideal of 𝑆 if(1)𝑓(𝑥𝑎𝑦)∨𝜆≥𝑓(𝑎)∧𝜇 for all 𝑥,𝑎,𝑦∈𝑆, and(2)if 𝑥≤𝑦, then 𝑓(𝑥)∨𝜆≥𝑓(𝑦)∧𝜇.

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 (𝜆,𝜇)-fuzzy ideal of 𝑆, then 𝑓 is a (𝜆,𝜇)-fuzzy interior ideal of 𝑆.

Proof. Let 𝑥,𝑎,𝑦∈𝑆. Since 𝑓 is a (𝜆,𝜇)-fuzzy left ideal of 𝑆 and 𝑥,𝑎𝑦∈𝑆, we have 𝑓(𝑥(𝑎𝑦))∨𝜆≥𝑓(𝑎𝑦)∧𝜇.(2.3) Since 𝑓 is a (𝜆,𝜇)-fuzzy right ideal of 𝑆, we have 𝑓(𝑎𝑦)∨𝜆≥𝑓(𝑎)∧𝜇.(2.4)
From (2.3) and (2.4) we know that 𝑓(𝑥𝑎𝑦)∨𝜆=(𝑓(𝑥(𝑎𝑦))∨𝜆)∨𝜆≥(𝑓(𝑎𝑦)∧𝜇)∨𝜆=(𝑓(𝑎𝑦)∨𝜆)∧(𝜇∨𝜆)≥𝑓(𝑎)∧𝜇.

Theorem 2.6. Let (𝑆,∘,≤) be an ordered semigroup, then 𝑓 is a (𝜆,𝜇)-fuzzy interior ideal of 𝑆 if and only if 𝑓_𝛼 is an interior ideal of 𝑆 for all 𝛼∈(𝜆,𝜇].

Proof. Let 𝑓 be a (𝜆,𝜇)-fuzzy interior ideal of 𝑆 and 𝛼∈(𝜆,𝜇].
First of all, we need to show that 𝑥𝑎𝑦∈𝑓_𝛼, for all 𝑎∈𝑓_𝛼, 𝑥,𝑦∈𝑆.
From 𝑓(𝑥𝑎𝑦)∨𝜆≥𝑓(𝑎)∧𝜇≥𝛼∧𝜇=𝛼 and 𝜆<𝛼, we conclude that 𝑓(𝑥𝑎𝑦)≥𝛼, that is, 𝑥𝑎𝑦∈𝑓_𝛼.
Then, we need to show that 𝑏∈𝑓_𝛼 for all 𝑎∈𝑓_𝛼,𝑏∈𝑆 such that 𝑏≤𝑎.
From 𝑏≤𝑎 we know that 𝑓(𝑏)∨𝜆≥𝑓(𝑎)∧𝜇 and from 𝑎∈𝑓_𝛼 we have 𝑓(𝑎)≥𝛼. Thus, 𝑓(𝑏)∨𝜆≥𝛼∧𝜇=𝛼. Notice that 𝜆<𝛼, then we conclude that 𝑓(𝑏)≥𝛼, that is, 𝑏∈𝑓_𝛼.
Conversely, let 𝑓_𝛼 be an interior ideal of 𝑆 for all 𝛼∈(𝜆,𝜇].
If there are 𝑥₀,𝑎₀,𝑦₀∈𝑆, such that 𝑓(𝑥₀𝑎₀𝑦₀)∨𝜆<𝛼=𝑓(𝑎₀)∧𝜇, then 𝛼∈(𝜆,𝜇],𝑓(𝑎₀)≥𝛼 and 𝑓(𝑥₀𝑎₀𝑦₀)<𝛼. That is 𝑎₀∈𝑓_𝛼 and 𝑥₀𝑎₀𝑦₀∉𝑓_𝛼. This is a contradiction with that 𝑓_𝛼 is an interior ideal of 𝑆. Hence 𝑓(𝑥𝑎𝑦)∨𝜆≥𝑓(𝑎)∧𝜇 holds for all 𝑥,𝑎,𝑦∈𝑆.
If there are 𝑥₀,𝑦₀∈𝑆 such that 𝑥₀≤𝑦₀ and 𝑓(𝑥₀)∨𝜆<𝛼=𝑓(𝑦₀)∧𝜇, then 𝛼∈(𝜆,𝜇],𝑓(𝑦₀)≥𝛼, and 𝑓(𝑥₀)<𝛼, that is, 𝑦₀∈𝑓_𝛼 and 𝑥₀∉𝑓_𝛼. This is a contradiction with that 𝑓_𝛼 is an interior ideal of 𝑆. Hence if 𝑥≤𝑦, then 𝑓(𝑥)∨𝜆≥𝑓(𝑦)∧𝜇.

3. (𝜆,𝜇)-Fuzzy Quasi-Ideals and (𝜆,𝜇)-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)𝐴𝑆∩𝑆𝐴⊆𝑆, and(2)if 𝑥∈𝑆 and 𝑥≤𝑦∈𝐴, then 𝑥∈𝐴.

Definition 3.2. A nonempty subset 𝐴 of an ordered semigroup 𝑆 is called a bi-ideal of 𝑆 if it satisfies(1)𝐴𝑆𝐴⊆𝐴, and(2)𝑥∈𝑆 and 𝑥≤𝑦∈𝐴, then 𝑥∈𝐴.

Definition 3.3. Let (𝑆,∘,≤) be an ordered semigroup. A fuzzy subset 𝑓 of 𝑆 is called a (𝜆,𝜇)-fuzzy quasi-ideal of 𝑆 if(1)𝑓∪𝜆⊇(𝑓∗1)∩(1∗𝑓)∩𝜇, and(2)if 𝑥≤𝑦, then 𝑓(𝑥)∨𝜆≥𝑓(𝑦)∧𝜇 for all 𝑥,𝑦∈𝑆.

Definition 3.4. Let (𝑆,∘,≤) be an ordered semigroup. A fuzzy subset 𝑓 of 𝑆 is called a (𝜆,𝜇)-fuzzy bi-ideal of 𝑆 if for all 𝑥,𝑦,𝑧∈𝑆,(1)𝑓(𝑥𝑦𝑧)∨𝜆≥(𝑓(𝑥)∧𝑓(𝑧))∧𝜇, and(2)if 𝑥≤𝑦, then 𝑓(𝑥)∨𝜆≥𝑓(𝑦)∧𝜇.

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 (𝜆,𝜇)-fuzzy quasi-ideal of 𝑆 if and only if 𝑓_𝛼 is a quasi-ideal of 𝑆 for all 𝛼∈(𝜆,𝜇].

Proof. Let 𝑓 be a (𝜆,𝜇)-fuzzy quasi-ideal of 𝑆 and 𝛼∈(𝜆,𝜇].
First of all, we need to show that 𝑆𝑓_𝛼∩𝑓_𝛼𝑆⊆𝑓_𝛼.
If 𝑥∈𝑆𝑓_𝛼∩𝑓_𝛼𝑆, then 𝑥=𝑠𝑡₁=𝑡₂𝑠 for some 𝑡₁,𝑡₂∈𝑓_𝛼 and 𝑠∈𝑆.
From 𝑓∪𝜆⊇(𝑓∗1)∩(1∗𝑓)∩𝜇, we conclude that 𝑓(𝑥)∨𝜆≥(𝑓∗1)(𝑥)∧(𝑓∗1)(𝑥)∧𝜇≥𝑓(𝑡₁)∧𝑓(𝑡₂)∧𝜇≥𝛼∧𝜇=𝛼. Thus, 𝑓(𝑥)≥𝛼, and so 𝑥∈𝑓_𝛼. Hence, 𝑆∗𝑓_𝛼∩𝑓_𝛼∗𝑆⊆𝑓_𝛼.
Next, we need to show that 𝑏∈𝑓_𝛼 for all 𝑎∈𝑓_𝛼,𝑏∈𝑆 such that 𝑏≤𝑎.
From 𝑏≤𝑎 we know that 𝑓(𝑏)∨𝜆≥𝑓(𝑎)∧𝜇 and from 𝑎∈𝑓_𝛼 we have 𝑓(𝑎)≥𝛼. Thus, 𝑓(𝑏)∨𝜆≥𝛼∧𝜇=𝛼. Notice that 𝜆<𝛼, we conclude that 𝑓(𝑏)≥𝛼, that is, 𝑏∈𝑓_𝛼.
Conversely, let 𝑓_𝛼 be a quasi-ideal of 𝑆 for all 𝛼∈(𝜆,𝜇]. Then, 𝑓_𝛼𝑆∩𝑆𝑓_𝛼⊆𝑓_𝛼.
If there is 𝑥₀∈𝑆, such that 𝑓(𝑥₀)∨𝜆<𝛼=(𝑓∗1)(𝑥)∧(1∗𝑓)(𝑥)∧𝜇, then 𝛼∈(𝜆,𝜇],𝑓(𝑥₀)<𝛼, (𝑓∗1)(𝑥₀)≥𝛼 and (1∗𝑓)(𝑥₀)≥𝛼. That is 𝑥₀∉𝑓_𝛼, sup_{𝑥0≤𝑥1𝑥2}𝑓(𝑥₁)≥𝛼 and sup_{𝑥0≤𝑥1𝑥2}𝑓(𝑥₂)≥𝛼.
From 𝑓_𝛼𝑆∩𝑆𝑓_𝛼⊆𝑓_𝛼 and 𝑥₀∉𝑓_𝛼, we obtain that 𝑥₀∉𝑓_𝛼𝑆∩𝑆𝑓_𝛼.
From sup_{𝑥0≤𝑥1𝑥2}𝑓(𝑥₁)≥𝛼 and 𝛼≠0, we know that there exists at least one pair (𝑥₁,𝑥₂)∈𝑆×𝑆 such that 𝑥₀≤𝑥₁𝑥₂ and 𝑓(𝑥₁)≥𝛼. Thus, 𝑥₀≤𝑥₁𝑥₂∈𝑓_𝛼𝑆. Hence, 𝑥₀∈𝑓_𝛼S.
Similarly, we can prove that 𝑥₀∈𝑆𝑓_𝛼.
So 𝑥₀∈𝑓_𝛼𝑆∩𝑆𝑓_𝛼. This is a contradiction.
Hence, 𝑓∪𝜆⊇(𝑓∗1)∩(1∗𝑓)∩𝜇 holds.
If there are 𝑥₀,𝑦₀∈𝑆 such that 𝑥₀≤𝑦₀ and 𝑓(𝑥₀)∨𝜆<𝛼=𝑓(𝑦₀)∧𝜇, then 𝛼∈(𝜆,𝜇],𝑓(𝑦₀)≥𝛼 and 𝑓(𝑥₀)<𝛼, that is, 𝑦₀∈𝑓_𝛼 and 𝑥₀∉𝑓_𝛼. This is a contradiction with that 𝑓_𝛼 is a quasi-ideal of 𝑆. Hence if 𝑥≤𝑦, then 𝑓(𝑥)∨𝜆≥𝑓(𝑦)∧𝜇.

Theorem 3.7. Let (𝑆,∘,≤) be an ordered semigroup, then 𝑓 is a (𝜆,𝜇)-fuzzy bi-ideal of 𝑆 if and only if 𝑓_𝛼 is a bi-ideal of 𝑆 for all 𝛼∈(𝜆,𝜇].

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 (𝜆,𝜇)-fuzzy right (resp. left) ideals of 𝑆 are (𝜆,𝜇)-fuzzy quasi-ideals of 𝑆.

Proof. Let 𝑓 be a (𝜆,𝜇)-fuzzy right ideal of 𝑆 and 𝑥∈𝑆. First we have ((𝑓∗1)∩(1∗𝑓))(𝑥)=(𝑓∗1)(𝑥)∧(1∗𝑓)(𝑥).(3.1)
If 𝐴_𝑥=∅, then we have (𝑓∗1)(𝑥)=0=(1∗𝑓)(𝑥). So 𝑓(𝑥)∨𝜆≥0=(𝑓∗1)(𝑥)∧(1∗𝑓)(𝑥)∧𝜇. Thus, 𝑓∪𝜆⊇(𝑓∗1)∩(1∗𝑓)∩𝜇.
If 𝐴_𝑥≠∅, then (𝑓∗1)(𝑥)=sup_{(𝑢,𝑣)∈𝐴𝑥}(𝑓(𝑢)∧1(𝑣)).(3.2) On the other hand, 𝑓(𝑥)∨𝜆≥𝑓(𝑢)∧1(𝑣)∧𝜇, for all (𝑢,𝑣)∈𝐴_𝑥.
Indeed, if (𝑢,𝑣)∈𝐴_𝑥, then 𝑥≤𝑢𝑣, thus 𝑓(𝑥)∨𝜆=𝑓(𝑥)∨𝜆∨𝜆≥(𝑓(𝑢𝑣)∧𝜇)∨𝜆=(𝑓(𝑢𝑣)∨𝜆)∧(𝜆∨𝜇)≥(𝑓(𝑢)∧𝜇)∧𝜇=𝑓(𝑢)∧𝜇=𝑓(𝑢)∧1(𝑣)∧𝜇.
Hence, we have that 𝑓(𝑥)∨𝜆≥(sup_{(𝑢,𝑣)∈𝐴𝑥}(𝑓(𝑢)∧1(𝑣)))∧𝜇=(𝑓∗1)(𝑥)∧𝜇≥(𝑓∗1)(𝑥)∧(1∗𝑓)(𝑥)∧𝜇. Thus, 𝑓∪𝜆⊇(𝑓∗1)∩(1∗𝑓)∩𝜇.
Therefore, 𝑓 is a (𝜆,𝜇)-fuzzy quasi-ideal of 𝑆.

Theorem 3.9. Let (𝑆,∘,≤) be an ordered semigroup, then the (𝜆,𝜇)-fuzzy quasi-ideals of 𝑆 are (𝜆,𝜇)-fuzzy bi-ideals of 𝑆.

Proof. Let 𝑓 be a (𝜆,𝜇)-fuzzy quasi-ideal of 𝑆 and 𝑥,𝑦,𝑧∈𝑆. Then we have that 𝑓(𝑥𝑦𝑧)∨𝜆≥(𝑓∗1)(𝑥𝑦𝑧)∧(1∗𝑓)(𝑥𝑦𝑧)∧𝜇.(3.3)
From (𝑥,𝑦𝑧)∈𝐴_𝑥𝑦𝑧, we have that (𝑓∗1)(𝑥𝑦𝑧)≥𝑓(𝑥)∧1(𝑦𝑧)=𝑓(𝑥).
From (𝑥𝑦,𝑧)∈𝐴_𝑥𝑦𝑧, we have that (1∗𝑓)(𝑥𝑦𝑧)≥1(𝑥𝑦)∧𝑓(𝑧)=𝑓(𝑧).
Thus, 𝑓(𝑥𝑦𝑧)∨𝜆≥𝑓(𝑥)∧𝑓(𝑧)∧𝜇.
Therefore, 𝑓 is a (𝜆,𝜇)-fuzzy bi-ideal of 𝑆.

Definition 3.10 (see [5]). An ordered semigroup (𝑆,∘,≤) is called regular if for all 𝑎∈𝑆 there exists 𝑥∈𝑆 such that 𝑎≤𝑎𝑥𝑎.

Theorem 3.11. In a regular ordered semigroup 𝑆, the (𝜆,𝜇)-fuzzy quasi-ideals and the (𝜆,𝜇)-fuzzy bi-ideals coincide.

Proof. Let 𝑓 be a (𝜆,𝜇)-fuzzy bi-ideal of 𝑆 and 𝑥∈𝑆. We need to prove that 𝑓(𝑥)∨𝜆≥(𝑓∗1)(𝑥)∧(1∗𝑓)∧𝜇.(3.4)
If 𝐴_𝑥=∅, it is easy to verify that condition (3.4) is satisfied.
Let 𝐴_𝑥≠∅.
(1) If (𝑓∗1)(𝑥)∧𝜇≤𝑓(𝑥)∨𝜆, then we have that 𝑓(𝑥)∨𝜆≥(𝑓∗1)(𝑥)∧𝜇≥(𝑓∗1)(𝑥)∧(1∗𝑓)(𝑥)∧𝜇. Thus, condition (3.4) is satisfied.
(2) If (𝑓∗1)(𝑥)∧𝜇>𝑓(𝑥)∨𝜆, then there exists at least one pair (𝑧,𝑤)∈𝐴_𝑥 such that 𝑓(𝑧)∧1(𝑤)∧𝜇>𝑓(𝑥)∨𝜆. That is 𝑧,𝑤∈𝑆, 𝑥≤𝑧𝑤 and 𝑓(𝑧)∧𝜇>𝑓(𝑥)∨𝜆.
We will prove that (1∗𝑓)(𝑥)∧𝜇≤𝑓(𝑥)∨𝜆. Then, 𝑓(𝑥)∨𝜆≥(1∗𝑓)(𝑥)∧𝜇≥(𝑓∗1)(𝑥)∧(1∗𝑓)(𝑥)∧𝜇, and condition (3.4) is satisfied.
For any (𝑢,𝑣)∈𝐴_𝑥, we need to show that 1(𝑢)∧𝑓(𝑣)∧𝜇≤𝑓(𝑥)∨𝜆.
Let (𝑢,𝑣)∈𝐴_𝑥, then 𝑥≤𝑢𝑣 for some 𝑢,𝑣∈𝑆. Since 𝑆 is regular, there exists 𝑠∈𝑆 such that 𝑥≤𝑥𝑠𝑥.
From 𝑥≤𝑥𝑠𝑥, 𝑥≤𝑧𝑤 and 𝑥≤𝑢𝑣, we obtain that 𝑥≤𝑧𝑤𝑠𝑢𝑣. Since 𝑓 is a (𝜆,𝜇)-fuzzy bi-ideal of 𝑆, we have that 𝑓(𝑥)∨𝜆≥(𝑓(𝑧𝑤𝑠𝑢𝑣)∧𝜇)∨𝜆=(𝑓(𝑧𝑤𝑠𝑢𝑣)∨𝜆)∧(𝜇∨𝜆)≥𝑓(𝑧)∧𝑓(𝑣)∧𝜇.(3.5) Note that 𝑓(𝑧)∧𝜇>𝑓(𝑥)∨𝜆. Thus, 𝑓(𝑥)∨𝜆≥𝑓(𝑣)∧𝜇=1(𝑢)∧𝑓(𝑣)∧𝜇.

Acknowledgments

This paper is prepared before the first author’s visit to Università 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.