(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].
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(1) (S, ∘) is a semigroup, and
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(2) if x, a, b ∈ S, then
()
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.
- (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
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(1) f(xy)∨λ ≥ f(x)∧μ (resp. f(xy)∨λ ≥ f(y)∧μ ) for all x, y ∈ S, and
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(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:
A fuzzy set f is defined as follows:
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
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(1) SAS⊆A, and
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(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
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(1) f(xay)∨λ ≥ f(a)∧μ for all x, a, y ∈ S, and
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(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
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. Let f be a (λ, μ)-fuzzy interior ideal of S and α ∈ (λ, μ].
First of all, we need to show that xay ∈ fα, for all a ∈ fα, 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 ∈ fα.
Then, we need to show that b ∈ fα for all a ∈ fα, b ∈ S such that b ≤ a.
From b ≤ 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, let fα 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 ∈ 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 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 ∈ 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
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(1) AS∩SA⊆S, and
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(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
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(1) ASA⊆A, and
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(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
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(1) f ∪ λ⊇(f*1)∩(1*f)∩μ, and
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(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,
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(1) f(xyz)∨λ ≥ (f(x)∧f(z))∧μ, and
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(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 fα 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 Sfα∩fαS⊆fα.
If x ∈ Sfα∩fαS, then x = st1 = t2s for some t1, t2 ∈ fα 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 ∈ fα. Hence, S*fα∩fα*S⊆fα.
Next, we need to show that b ∈ fα for all a ∈ fα, b ∈ S such that b ≤ a.
From b ≤ 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, let fα be a quasi-ideal of S for all α ∈ (λ, μ]. Then, fαS∩Sfα⊆fα.
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 ∉ fα, and .
From fαS∩Sfα⊆fα and x0 ∉ fα, we obtain that x0 ∉ fαS∩Sfα.
From 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 that x0 ∈ Sfα.
So x0 ∈ fαS∩Sfα. 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 ∈ 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. Let f be a (λ, μ)-fuzzy right ideal of S and x ∈ S. First we have
If Ax = ∅, 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 Ax ≠ ∅, then
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 that . 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
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 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
If Ax = ∅, it is easy to verify that condition (3.4) is satisfied.
Let Ax ≠ ∅.
(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.
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
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.
