(λ, μ)-fuzzy interior ideals of ordered Г-semigroups
For all λ, μ ∈ [0, 1] such that λ < μ, we first introduced the definitions of (λ, μ)-fuzzy ideals and (λ, μ)-fuzzy interior ideals of an ordered Γ-semigroup. Then we proved that in regular and in intra-regular ordered semigroups the (λ, μ)-fuzzy ideals and the (λ, μ)-fuzzy interior ideals coincid...
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Інститут прикладної математики і механіки НАН України
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| Цитувати: | (λ, μ)-fuzzy interior ideals of ordered Г-semigroups / Yu. Feng, P. Corsini // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 61–70. — Бібліогр.: 24 назв. — англ. |
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irk-123456789-1523082019-06-10T01:25:39Z (λ, μ)-fuzzy interior ideals of ordered Г-semigroups Feng, Yu. Corsini, P. For all λ, μ ∈ [0, 1] such that λ < μ, we first introduced the definitions of (λ, μ)-fuzzy ideals and (λ, μ)-fuzzy interior ideals of an ordered Γ-semigroup. Then we proved that in regular and in intra-regular ordered semigroups the (λ, μ)-fuzzy ideals and the (λ, μ)-fuzzy interior ideals coincide. Lastly, we introduced the concept of a (λ, μ)-fuzzy simple ordered Γ-semigroup and characterized the simple ordered Γ-semigroups in terms of (λ, μ)-fuzzy interior ideals. 2013 Article (λ, μ)-fuzzy interior ideals of ordered Г-semigroups / Yu. Feng, P. Corsini // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 61–70. — Бібліогр.: 24 назв. — англ. 1726-3255 http://dspace.nbuv.gov.ua/handle/123456789/152308 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України |
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For all λ, μ ∈ [0, 1] such that λ < μ, we first introduced the definitions of (λ, μ)-fuzzy ideals and (λ, μ)-fuzzy interior ideals of an ordered Γ-semigroup. Then we proved that in regular and in intra-regular ordered semigroups the (λ, μ)-fuzzy ideals and the (λ, μ)-fuzzy interior ideals coincide. Lastly, we introduced the concept of a (λ, μ)-fuzzy simple ordered Γ-semigroup and characterized the simple ordered Γ-semigroups in terms of (λ, μ)-fuzzy interior ideals. |
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Article |
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Feng, Yu. Corsini, P. |
| spellingShingle |
Feng, Yu. Corsini, P. (λ, μ)-fuzzy interior ideals of ordered Г-semigroups Algebra and Discrete Mathematics |
| author_facet |
Feng, Yu. Corsini, P. |
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Feng, Yu. |
| title |
(λ, μ)-fuzzy interior ideals of ordered Г-semigroups |
| title_short |
(λ, μ)-fuzzy interior ideals of ordered Г-semigroups |
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(λ, μ)-fuzzy interior ideals of ordered Г-semigroups |
| title_fullStr |
(λ, μ)-fuzzy interior ideals of ordered Г-semigroups |
| title_full_unstemmed |
(λ, μ)-fuzzy interior ideals of ordered Г-semigroups |
| title_sort |
(λ, μ)-fuzzy interior ideals of ordered г-semigroups |
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Інститут прикладної математики і механіки НАН України |
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2013 |
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http://dspace.nbuv.gov.ua/handle/123456789/152308 |
| citation_txt |
(λ, μ)-fuzzy interior ideals of ordered Г-semigroups / Yu. Feng, P. Corsini // Algebra and Discrete Mathematics. — 2013. — Vol. 16, № 1. — С. 61–70. — Бібліогр.: 24 назв. — англ. |
| series |
Algebra and Discrete Mathematics |
| work_keys_str_mv |
AT fengyu lmfuzzyinterioridealsoforderedgsemigroups AT corsinip lmfuzzyinterioridealsoforderedgsemigroups |
| first_indexed |
2025-07-13T02:47:51Z |
| last_indexed |
2025-07-13T02:47:51Z |
| _version_ |
1837498237591748608 |
| fulltext |
Algebra and Discrete Mathematics RESEARCH ARTICLE
Volume 16 (2013). Number 1. pp. 61 – 70
© Journal “Algebra and Discrete Mathematics”
(λ, µ)-fuzzy interior ideals
of ordered Γ-semigroups
Yuming Feng and P. Corsini
Communicated by V. Mazorchuk
Abstract. For all λ, µ ∈ [0, 1] such that λ < µ, we first intro-
duced the definitions of (λ, µ)-fuzzy ideals and (λ, µ)-fuzzy interior
ideals of an ordered Γ-semigroup. Then we proved that in regular
and in intra-regular ordered semigroups the (λ, µ)-fuzzy ideals and
the (λ, µ)-fuzzy interior ideals coincide. Lastly, we introduced the
concept of a (λ, µ)-fuzzy simple ordered Γ-semigroup and charac-
terized the simple ordered Γ-semigroups in terms of (λ, µ)-fuzzy
interior ideals.
1. Introduction and preliminaries
The formal study of semigroups began in the early 20th century. Semi-
groups are important in many areas of mathematics, for example, coding
and language theory, automata theory, combinatorics and mathematical
analysis.
Γ-semigroups were first defined by Sen and Saha [14] as a generalization
of semigroups and studied by many researchers, for example [1, 2, 5, 6, 8,
9, 3, 12, 15, 16, 17, 18].
The concept of fuzzy sets was first introduced by Zadeh [24] in 1965
and then the fuzzy sets have been used in the reconsideration of classical
mathematics. Recently, Yuan [23] 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
Key words and phrases: Γ-semigroup; (λ, µ)-fuzzy interior ideal; (λ, µ)-fuzzy
simple; regular ordered Γ-semigroup; intra-regular ordered Γ-semigroup.
62 (λ, µ)-fuzzy interior ideals of ordered Γ-semigroups
subfields, (λ, µ)-fuzzy quotient subfields , (λ, µ)-fuzzy subrings and (λ, µ)-
fuzzy ideals in [19, 20, 21, 22].
In this paper, we studied (λ, µ)-fuzzy ideals in ordered Γ-semigroups.
This can be seen as an application of [22] and as a generalization of
[7, 11, 13].
Let S = {x, y, z, ...} and Γ = {α, β, γ, ...} be two non-empty sets.
An ordered Γ-semigroup SΓ = (S, Γ, ≤) is a poset (S, ≤) such that
there exists a mapping S × Γ × S → S (images of (a, α, b) to be denoted
by aαb), such that, for all x, y, z ∈ S, α, β, γ ∈ Γ, we have
(1) (xβy)γz = xβ(yγz).
(2) x ≤ y ⇒
{
xαz ≤ yαz
zαx ≤ zαy.
Note that an ordered semigroup is a special ordered Γ-semigroup with
Γ = {◦}, i.e., Γ is a set with one element.
Let (S, ◦, ≤) be an ordered semigroup. A nonempty subset A of S is
called a left (respectively, right) ideal of S if (1) S ◦ A ⊆ A (respectively,
A ◦ S ⊆ A); (2) a ∈ A, b ∈ S, b ≤ a ⇒ b ∈ A. A is called an ideal of S if
it is both a left and a right ideal of S .
If (S, Γ, ≤) is an ordered Γ-semigroup, and A is a subset of S, we
denote by (A] the subset of S defined as follows:
(A] = {t ∈ S|t ≤ a for some a ∈ A}.
Given an ordered Γ-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 t ∈ [0, 1], ft is defined by
ft = {x ∈ S|f(x) ≥ t}.
For each subset A of S, the characteristic function fA is a fuzzy subset
of S defined by
fA(x) =
{
1, if x ∈ A
0, if x 6∈ A.
In the following, we will use S, SΓ or (S, Γ, ≤) to denote an ordered
Γ-semigroup.
In the rest of this paper, we will always assume that 0 ≤ λ < µ ≤ 1.
We will use a ∨ b to denote max{a, b} and a ∧ b to stand for min{a, b}.
Note that ([0, 1], ∧, ∨) is a distributive lattice.
Yu. Feng, P. Corsini 63
2. (λ, µ)-fuzzy ideals and (λ, µ)-fuzzy interior ideals
In this section, we first introduce the concepts of (λ, µ)-fuzzy ideals
and (λ, µ)-fuzzy interior ideals of an ordered Γ-semigroup. Then we show
that every (λ, µ)-fuzzy ideal is a (λ, µ)-fuzzy interior ideal.
Definition 1. A fuzzy subset f of an ordered Γ-semigroup S is called a
(λ, µ)-fuzzy right ideal of S if
(1) f(xαy) ∨ λ ≥ f(x) ∧ µ for all x, y ∈ S, α ∈ Γ 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 left ideal of S if
(1) f(xαy) ∨ λ ≥ f(y) ∧ µ for all x, y ∈ S, α ∈ Γ 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.
Definition 2. If (S, Γ, ≤) is an ordered Γ-semigroup, a nonempty subset
A of S is called an interior ideal of S if
(1) SΓAΓS ⊆ A and
(2)If a ∈ A, b ∈ S and b ≤ a, then b ∈ A.
Definition 3. If (S, Γ, ≤) is an ordered Γ-semigroup, a fuzzy subset f of
S is called a (λ, µ)-fuzzy interior ideal of S if the following assertions are
satisfied:
(1) f(xβaγy) ∨ λ ≥ f(a) ∧ µ for all x, a, y ∈ S, β, γ ∈ Γ and
(2) If x ≤ y, then f(x) ∨ λ ≥ f(y) ∧ µ.
Theorem 1. Let (S, Γ, ≤) be an ordered Γ-semigroup, Then f is a (λ, µ)-
fuzzy interior ideal of S if and only if ft is an interior ideal of S for all
t ∈ (λ, µ].
Proof. Let f be a (λ, µ)-fuzzy interior ideal of S, ∀t ∈ (λ, µ] and ∀β, γ ∈ Γ.
First of all, we need to show that xβaγy ∈ ft, for all a ∈ ft, x, y ∈ S.
From f(xβaγy) ∨ λ ≥ f(a) ∧ µ ≥ t ∧ µ = t and λ < t we conclude
that f(xβaγy) ≥ t, that is xβaγy ∈ ft.
Then, we need to show that b ∈ ft for all a ∈ ft, b ∈ S such that b ≤ a.
From b ≤ a we know that f(b) ∨ λ ≥ f(a) ∧ µ and from a ∈ ft we
have f(a) ≥ t. Thus f(b) ∨ λ ≥ t ∧ µ = t. Notice that λ < t, we conclude
that f(b) ≥ t, that is b ∈ ft.
Conversely, let ft be an interior ideal of S for all t ∈ (λ, µ].
If there are x0, a0, y0 ∈ S, such that f(x0βa0γy0) ∨ λ < t = f(a0) ∧ µ,
then t ∈ (λ, µ], f(a0) ≥ t and f(x0βa0γy0) < t. That is a0 ∈ ft and
64 (λ, µ)-fuzzy interior ideals of ordered Γ-semigroups
x0βa0γy0 6∈ ft. This is a contradiction with that ft is an interior ideal of
S. Hence f(xβaγy) ∨ 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 < t = f(y0)∧µ,
then t ∈ (λ, µ], f(y0) ≥ t and f(x0) < t, that is y0 ∈ ft and x0 6∈ ft. This
is a contradiction with that ft is an interior ideal of S. Hence if x ≤ y,
then f(x) ∨ λ ≥ f(y) ∧ µ.
Theorem 2. Let (S, Γ, ≤) be an ordered Γ-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, β, γ ∈ Γ, Since f is a (λ, µ)-fuzzy left ideal of S
and x, aγy ∈ S, we have that
f(xβ(aγy)) ∨ λ ≥ f(aγy) ∧ µ (1)
Since f is a (λ, µ)-fuzzy right ideal of S, we have that
f(aγy) ∨ λ ≥ f(a) ∧ µ (2)
From (1) and (2) we know that f(xβaγy)∨λ = (f(xβ(aγy))∨λ)∨λ ≥
(f(aγy) ∧ µ) ∨ λ = (f(aγy) ∨ λ) ∧ (µ ∨ λ) ≥ f(a) ∧ µ.
3. (λ, µ)-fuzzy interior ideals of regular/intra-regular or-
dered Γ-semigroups
We prove here that in regular and in intra-regular ordered Γ-semigroups
the (λ, µ)-fuzzy ideals and the (λ, µ)-fuzzy interior ideals coincide.
Definition 4. An ordered Γ-semigroup (S, Γ, ≤) is called regular if for
all a ∈ S there exists x ∈ S such that a ≤ aβxγa, for all β, γ ∈ Γ.
Definition 5. An ordered Γ-semigroup (S, Γ, ≤) is called intra-regular
if for all a ∈ S there exists x, y ∈ S such that a ≤ xβaγaδy, for all
β, γ, δ ∈ Γ.
Theorem 3. Let (S, Γ, ≤) be a regular ordered Γ-semigroup and f a
(λ, µ)-fuzzy interior ideal of S, then f is a (λ, µ)-fuzzy ideal of S.
Proof. Let x, y ∈ S, then f(xβy) ∨ λ ≥ f(x) ∧ µ, for all β ∈ Γ.
Indeed, since S is regular and x ∈ S, there exist z ∈ S such that
x ≤ xβzγx, for all β, γ ∈ Γ. Thus we have that xβy ≤ (xβzγx)βy =
(xβz)γxβy. So
f(xβy) ∨ λ ≥ f((xβz)γxβy) ∧ µ (3)
Yu. Feng, P. Corsini 65
for f is a (λ, µ)-fuzzy interior ideal. Again since f is a (λ, µ)-fuzzy interior
ideal of S, we have
f((xβz)γxβy) ∨ λ ≥ f(x) ∧ µ. (4)
From (3) and (4) we have that f(xβy) ∨ λ = (f(xβy) ∨ λ) ∨ λ ≥
(f((xβz)γxβy) ∧ µ) ∨ λ = (f((xβz)γxβy) ∨ λ) ∧ (µ ∨ λ) ≥ f(x) ∧ µ,
and f is a (λ, µ)-fuzzy right ideal of S.
In a similar way, we can prove that f is a (λ, µ)-fuzzy left ideal of S.
Thus f is a (λ, µ)-fuzzy ideal of S.
Theorem 4. Let (S, Γ, ≤) be an intra-regular ordered semigroup and f a
(λ, µ)-fuzzy interior ideal of S, then f is a (λ, µ)-fuzzy ideal of S.
Proof. Let a, b ∈ S, then f(aβb) ∨ λ ≥ f(a) ∧ µ, for all β ∈ Γ.
Indeed, since S is intra-regular and a ∈ S, there exist x, y ∈ S such
that a ≤ xβaγaδy. Then aβb ≤ (xβaγaδy)βb.
Since f is a (λ, µ)-fuzzy interior ideal of S, we have that f(aβb) ∨ λ =
(f(aβb)∨λ)∨λ ≥ (f(xβaγaδyβb)∧µ)∨λ = (f(xβaγaδyβb)∨λ)∧(µ∨λ).
Again since f is a (λ, µ)-fuzzy interior ideal of S, we have f(xβaγaδyβb)∨
λ = f((xβa)γaδ(yβb)) ∨ λ ≥ f(a) ∧ µ.
Thus we have that f(aβb) ∨ λ ≥ f(a) ∧ µ, and f is a (λ, µ)-fuzzy right
ideal of S.
In a similar way we can prove that f is a (λ, µ)-fuzzy left ideal of S.
Therefore, f is a (λ, µ)-fuzzy ideal of S.
Remark 1. From previous theorems we know that in regular or intra-
regular ordered Γ-semigroups the concepts of (λ, µ)-fuzzy ideals and
(λ, µ)-fuzzy interior ideals coincide.
4. (λ, µ)-fuzzy simple ordered Γ-semigroups
In this section, we introduce the concept of (λ, µ)-fuzzy simple ordered
Γ-semigroups and characterize this type of ordered Γ-semigroups in terms
of (λ, µ)-fuzzy interior ideals.
Definition 6. An ordered Γ-semigroup S is called simple if it does not
contain proper ideals, that is, for any ideal A 6= ∅ of S, we have A = S.
Definition 7. An ordered Γ-semigroup S is called (λ, µ)-fuzzy simple
if for any (λ, µ)-fuzzy ideal f of S, we have f(a) ∨ λ ≥ f(b) ∧ µ, for all
a, b ∈ S.
66 (λ, µ)-fuzzy interior ideals of ordered Γ-semigroups
Remark 2. In [11], Kehayopulu and Tsingelis studied (0, 1)-fuzzy simple
ordered semigroup (They called it fuzzy simple ordered semigroup. see
Definition 3.1 of [11]).
Sardar, Davvaz and Majumder researched (0, 1)-fuzzy simple ordered
Γ-semigroup , which was called fuzzy simple ordered Γ-semigroup in their
paper [13].
Theorem 5. Let S be an ordered Γ-semigroup, then S is (λ, µ)-fuzzy
simple if and only if for any (λ, µ)-fuzzy ideal f of S, if ft 6= ∅, then
ft = S, for all t ∈ (λ, µ].
Proof. Assume that S is is (λ, µ)-fuzzy simple. For any (λ, µ)-fuzzy ideal
f of S, suppose that ft 6= ∅. We need to prove that x ∈ ft for all x ∈ S,
where t ∈ (λ, µ].
Since ft 6= ∅, we can suppose that there exists y ∈ ft, that is f(y) ≥ t.
So f(x) ∨ λ ≥ f(y) ∧ µ ≥ t ∧ µ = t.
Notice that λ < t, we have that f(x) ≥ t, that is x ∈ ft.
Conversely, for any (λ, µ)-fuzzy ideal f of S, suppose that ft = S, for
all t ∈ (λ, µ]. We need to prove that f(a) ∨ λ ≥ f(b) ∧ µ, for all a, b ∈ S.
If there exist a0, b0 ∈ S, such that f(a0) ∨ λ < t = f(b0) ∧ µ, then
t ∈ (λ, µ], f(a0) < t and f(b0) ≥ t. Thus a0 6∈ ft = S. This is a
contradiction.
So f(a) ∨ λ ≥ f(b) ∧ µ holds, for all a, b ∈ S.
Proposition 1. Let S be an ordered Γ-semigroup and f a (λ, µ)-fuzzy
right ideal of S, then Ia = {b ∈ S|f(b) ∨ λ ≥ f(a) ∧ µ} is a right ideal of
S for every a ∈ S.
Proof. Let a ∈ S, then Ia 6= ∅ since a ∈ Ia.
(1) Let b ∈ Ia and s ∈ S, then bβs ∈ Ia for any β ∈ Γ. Indeed, since
f is a (λ, µ)-fuzzy right ideal of S and b, s ∈ S, we have
f(bβs) ∨ λ ≥ f(b) ∧ µ. (5)
Since b ∈ Ia, we have that
f(b) ∨ λ ≥ f(a) ∧ µ. (6)
From (5) and (6) we conclude that f(bβs) ∨ λ = (f(bβs) ∨ λ) ∨ λ ≥
(f(b) ∧ µ) ∨ λ = (f(b) ∨ λ) ∧ (µ ∨ λ) ≥ f(a) ∧ µ. So bβs ∈ Ia.
(2) Let b ∈ Ia and S ∋ s ≤ b, then s ∈ Ia. Indeed, since f is a
(λ, µ)-fuzzy right ideal of S , s, b ∈ S and s ≤ b, we have
f(s) ∨ λ ≥ f(b) ∧ µ. (7)
Yu. Feng, P. Corsini 67
Since b ∈ Ia, we have
f(b) ∨ λ ≥ f(a) ∧ µ. (8)
From (7) and (8) we obtain that f(s) ∨ λ = (f(s) ∨ λ) ∨ λ ≥ (f(b) ∧
µ) ∨ λ = (f(b) ∨ λ) ∧ (µ ∨ λ) ≥ f(a) ∧ µ. So s ∈ Ia.
Similarly, we have
Proposition 2. Let S be an ordered Γ-semigroup and f a (λ, µ)-fuzzy
left ideal of S, then Ia = {b ∈ S|f(b) ∨ λ ≥ f(a) ∧ µ} is a left ideal of S
for every a ∈ S.
By the previous two propositions we have
Proposition 3. Let S be an ordered Γ-semigroup and f a (λ, µ)-fuzzy
ideal of S, then Ia = {b ∈ S|f(b) ∨ λ ≥ f(a) ∧ µ} is an ideal of S for
every a ∈ S.
Lemma 1. Let S be an ordered Γ-semigroup and ∅ 6= I ⊆ S, then I is
an ideal of S if and only if the characteristic function fI is a (λ, µ)-fuzzy
ideal of S.
Proof. “⇒”
Suppose I is an ideal of S. For any x ∈ S, two cases are possible:
(1) x ∈ I. In this case, xγy ∈ I for any γ ∈ Γ and y ∈ S. This is
because I is an ideal of S.
Thus fI(xγy) = fI(x) = 1 and so fI(xγy) ∨ λ ≥ fI(x) ∧ µ.
Similarly, we have fI(yγx) ∨ λ ≥ fI(x) ∧ µ.
So fI is a (λ, µ)-fuzzy ideal of S.
(2) x 6∈ I. In this case, fI(x) = 0. So fI(xγy) ∨ λ ≥ fI(x) ∧ µ and
fI(yγx) ∨ λ ≥ fI(x) ∧ µ hold. Thus fI is a (λ, µ)-fuzzy ideal of S.
“⇐”
Conversely, suppose that fI is a (λ, µ)-fuzzy ideal of S. Then fI(xγy)∨
λ ≥ fI(x) ∧ µ and fI(yγx) ∨ λ ≥ fI(x) ∧ µ.
Set x ∈ I, we need to show that xγy ∈ I and yγx ∈ I for any γ ∈ Γ
and y ∈ S.
Since x ∈ I, we have that fI(x) = 1, so fI(xγy)∨λ ≥ µ and fI(yγx)∨
λ ≥ µ. Note that λ < µ, we have that fI(xγy) ≥ µ and fI(yγx) ≥ µ.
Thus fI(xγy) = 1 and fI(yγx) = 1. That is xγy ∈ I and yγx ∈ I for any
γ ∈ Γ and y ∈ S.
Theorem 6. An ordered Γ-semigroup S is simple if and only if it is
(λ, µ)-fuzzy simple.
68 (λ, µ)-fuzzy interior ideals of ordered Γ-semigroups
Proof. Suppose S is simple, let f be a (λ, µ)-fuzzy ideal of S and a, b ∈ S.
By previous propostion, the set Ia is an ideal of S. Since S is simple, we
have Ia = S. Then b ∈ Ia, from which we have that f(b) ∨ λ ≥ f(a) ∧ µ.
Thus S is (λ, µ)-fuzzy simple.
Conversely, suppose S contains proper ideals and let I be such ideal
of S. By the previous lemma, we know that fI is a (λ, µ)-fuzzy ideal
of S. We have that S ⊆ I. Indeed, let x ∈ S. Since S is (λ, µ)-fuzzy
simple, fI(x) ∨ λ ≥ fI(b) ∧ µ for all b ∈ S. Now let a ∈ I. Then we have
fI(x) ∨ λ ≥ fI(a) ∧ µ = 1 ∧ µ = µ. Notice that λ < µ, we conclude that
fI(x) ≥ µ, which implies that fI(x) = 1, that is x ∈ I. Thus we have that
S ⊆ I, and so S = I. We get a contradiction.
Lemma 2. An ordered Γ-semigroup S is simple if and only if for every
a ∈ S, we have S = (SΓaΓS].
Proof. It is easy from Lemma 1.2 of [10] or from Theorem 1.1 of [4].
Theorem 7. Let S be an ordered Γ-semigroup, then S is simple if and only
if for every (λ, µ)-fuzzy interior ideal f of S, we have f(a) ∨ λ ≥ f(b) ∧ µ,
for all a, b ∈ S.
Proof. Suppose S is simple. Let f be a (λ, µ)-fuzzy interior ideal of S and
a, b ∈ S. Since S is simple and b ∈ S, by the previous lemma, we have that
S = (SΓbΓS]. Since a ∈ S, we have that a ∈ (SΓbΓS]. Then there exist
x, y ∈ S and β, γ ∈ Γ such that a ≤ xβbγy. Since a, xβbγy ∈ S, a ≤ xβbγy
and f is a (λ, µ)-fuzzy interior ideal of S, we have that
f(a) ∨ λ ≥ f(xβbγy) ∧ µ. (9)
Since x, b, y ∈ S and f is a (λ, µ)-fuzzy interior ideal of S, we have that
f(xβbγy) ∨ λ ≥ f(b) ∧ µ. (10)
From (9) and (10) we conclude that f(a)∨λ = (f(a)∨λ)∨λ ≥ (f(xβbγy)∧
µ) ∨ λ = (f(xβbγy) ∨ λ) ∧ (µ ∨ λ) ≥ f(b) ∧ µ.
Conversely, Suppose that for every (λ, µ)-fuzzy interior ideal f of S,
we have f(a) ∨ λ ≥ f(b) ∧ µ, for all a, b ∈ S.
Now let f be any (λ, µ)-fuzzy ideal f of S, then it is a (λ, µ)-fuzzy
interior ideal of S. So we have f(a) ∨ λ ≥ f(b) ∧ µ, for all a, b ∈ S. Thus
S is (λ, µ)-fuzzy simple by its definition. And from the previous theorem,
we conclude that S is simple.
As a consequence we have
Yu. Feng, P. Corsini 69
Theorem 8. For an ordered Γ-semigroup S, the following are equivalent:
(1) S is simple.
(2) S = (SΓaΓS] for every a ∈ S.
(3) S is (λ, µ)-fuzzy simple.
(4) For every (λ, µ)-fuzzy interior ideal f of S, we have f(a) ∨ λ ≥
f(b) ∧ µ, for all a, b ∈ S.
5. Conclusion and further research
In this paper, we generalized results of [11, 13]. We introduced (λ, µ)-
fuzzy ideals and (λ, µ)-fuzzy interior ideals of an ordered Γ-semigroup and
studied them. When λ = 0 and µ = 1, we meet ordinary fuzzy ideals and
fuzzy interior ideals. So we can say that (λ, µ)-fuzzy ideals and (λ, µ)-fuzzy
interior ideals are more general concepts than fuzzy ones.
In [22], Yao gave the definition of (λ, µ)-fuzzy bi-ideals in semigroups.
One can study (λ, µ)-fuzzy bi-ideals in ordered Γ-semigroups. We would
like to explore this in next papers.
Acknowledgment
This paper is finished during the first author’s visiting in Universita’
degli Studi di Udine. The first author wishes to give his hearted thanks
to Mr. Zuddas, Mr. Sciarretta, Mr. Gessel and Mr. Li Lei for the kind
help during his stay in Udine.
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Contact information
Yuming Feng1,2,
P. Corsini2
1School of Mathematics and Statistics,
Chongqing Three Gorges University, Wanzhou,
Chongqing, 404100, P.R. China
yumingfeng25928@163.com
2Dipartimento di Ingegneria Civile e Architet-
tura dell’Universita’ degli Studi di Udine, Via
delle Scienze, 206, 33100 Udine, Italy
Received by the editors: 13.02.2011
and in final form 01.03.2012.
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