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

Authors: Yuming Feng; P. Corsini
Journal: Journal of Applied Mathematics
 (2012)

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## 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).


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## Body

## 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)
ifx,a,b∈S, then
(1.1)a≤b⇒{a∘x≤b∘xx∘a≤x∘b.Given an ordered semigroupS, 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],fα is defined by fα={x∈S|f(x)≥α}. For a∈S, we define that Aα={(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(1.2)(f*g)(a)={sup⁡(y,z)∈Aa(f(y)∧g(z)),ifAa≠∅,0,ifAa=∅.In the set of fuzzy subsets ofS, 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(1.3)(f∩g)(x)=f(x)∧g(x),(f∪g)(x)=f(x)∨g(x).Note that we usea∧b to denote min(a,b) and use a∨b to denote max(a,b).For anyα∈[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 useS 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 that0≤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)
ifx≤y, then f(x)∨λ≥f(y)∧μ for all x,y∈S.
A fuzzy subsetf 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:
(2.1)*eabeeabaabebbea
A fuzzy setf is defined as follows:
(2.2)Seabf0.10.20.3
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)
ifa∈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)
ifx≤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.
Letx,a,y∈S. Since f is a (λ,μ)-fuzzy left ideal of S and x,ay∈S, we have
(2.3)f(x(ay))∨λ≥f(ay)∧μ.
Since f is a (λ,μ)-fuzzy right ideal of S, we have
(2.4)f(ay)∨λ≥f(a)∧μ.
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 fα is an interior ideal of S for all α∈(λ,μ].Proof.
Letf be a (λ,μ)-fuzzy interior ideal of S and α∈(λ,μ].
First of all, we need to show thatxay∈fα, for all a∈fα, x,y∈S.
Fromf(xay)∨λ≥f(a)∧μ≥α∧μ=α and λ<α, we conclude that f(xay)≥α, that is, xay∈fα.
Then, we need to show thatb∈fα for all a∈fα,b∈S such that b≤a.
Fromb≤a we know that f(b)∨λ≥f(a)∧μ and from a∈fα 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∈fα.
Conversely, letfα be an interior ideal of S for all α∈(λ,μ].
If there arex0,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∈fα and x0a0y0∉fα. This is a contradiction with that fα is an interior ideal of S. Hence f(xay)∨λ≥f(a)∧μ holds for all x,a,y∈S.
If there arex0,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∈fα and x0∉fα. This is a contradiction with that fα 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)
ifx∈S and x≤y∈A, then x∈A.Definition 3.2.
A nonempty subsetA 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)
ifx≤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)
ifx≤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 fα is a quasi-ideal of S for all α∈(λ,μ].Proof.
Letf be a (λ,μ)-fuzzy quasi-ideal of S and α∈(λ,μ].
First of all, we need to show thatSfα∩fαS⊆fα.
Ifx∈Sfα∩fαS, then x=st1=t2s for some t1,t2∈fα and s∈S.
Fromf∪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∈fα. Hence, S*fα∩fα*S⊆fα.
Next, we need to show thatb∈fα for all a∈fα,b∈S such that b≤a.
Fromb≤a we know that f(b)∨λ≥f(a)∧μ and from a∈fα 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∈fα.
Conversely, letfα be a quasi-ideal of S for all α∈(λ,μ]. Then, fαS∩Sfα⊆fα.
If there isx0∈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∉fα, supx0≤x1x2f(x1)≥α and supx0≤x1x2f(x2)≥α.
FromfαS∩Sfα⊆fα and x0∉fα, we obtain that x0∉fαS∩Sfα.
Fromsupx0≤x1x2f(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∈fαS. Hence, x0∈fαS.
Similarly, we can prove thatx0∈Sfα.
Sox0∈fαS∩Sfα. This is a contradiction.
Hence,f∪λ⊇(f*1)∩(1*f)∩μ holds.
If there arex0,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∈fα and x0∉fα. This is a contradiction with that fα 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 fα 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.
Letf be a (λ,μ)-fuzzy right ideal of S and x∈S. First we have
(3.1)((f*1)∩(1*f))(x)=(f*1)(x)∧(1*f)(x).
IfAx=∅, 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.
IfAx≠∅, then
(3.2)(f*1)(x)=sup⁡(u,v)∈Ax(f(u)∧1(v)).
On the other hand, f(x)∨λ≥f(u)∧1(v)∧μ, for all (u,v)∈Ax.
Indeed, if(u,v)∈Ax, 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 thatf(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.
Letf be a (λ,μ)-fuzzy quasi-ideal of S and x,y,z∈S. Then we have that
(3.3)f(xyz)∨λ≥(f*1)(xyz)∧(1*f)(xyz)∧μ.
From(x,yz)∈Axyz, we have that (f*1)(xyz)≥f(x)∧1(yz)=f(x).
From(xy,z)∈Axyz, 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 semigroupS, the (λ,μ)-fuzzy quasi-ideals and the (λ,μ)-fuzzy bi-ideals coincide.Proof.
Letf be a (λ,μ)-fuzzy bi-ideal of S and x∈S. We need to prove that
(3.4)f(x)∨λ≥(f*1)(x)∧(1*f)∧μ.
IfAx=∅, it is easy to verify that condition (3.4) is satisfied.
LetAx≠∅.
(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)∈Ax 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)∈Ax, we need to show that 1(u)∧f(v)∧μ≤f(x)∨λ.
Let(u,v)∈Ax, then x≤uv for some u,v∈S. Since S is regular, there exists s∈S such that x≤xsx.
Fromx≤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
(3.5)f(x)∨λ≥(f(zwsuv)∧μ)∨λ=(f(zwsuv)∨λ)∧(μ∨λ)≥f(z)∧f(v)∧μ.
Note that f(z)∧μ>f(x)∨λ. Thus, f(x)∨λ≥f(v)∧μ=1(u)∧f(v)∧μ.



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