Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2010, Article ID 781672, 8 pagesdoi:10.1155/2010/781672
Research Article
Quotient of Ideals of an Intuitionistic Fuzzy Lattice
K. V. Thomas1 and Latha S. Nair2
1 Bharata Mata College, Mahatma Gandhi University, Kochi-Kerala, Thrikkakara 682 021, India2 Mar Athanasius College, Mahatma Gandhi University, Kochi, Kothamangalam, Kerala 686 666, India
Correspondence should be addressed to Latha S. Nair, [email protected]
Received 22 September 2010; Revised 10 November 2010; Accepted 2 December 2010
Academic Editor: Jose Luis Verdegay
Copyright © 2010 K. V. Thomas and L. S. Nair. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.
The concept of intuitionistic fuzzy ideal of an intuitionistic fuzzy lattice is introduced, and its certain characterizations areprovided. We defined the quotient (or residual) of ideals of an intuitionistic fuzzy sublattice and studied their properties.
1. Introduction
The concept of intuitionistic fuzzy sets was introduced byAtanassov [1, 2] as a generalization of that of fuzzy sets and itis a very effective tool to study the case of vagueness. Furthermany researches applied this notion in various branches ofmathematics especially in algebra and defined intuitionisticfuzzy subgroups (IFG), intuitionistic fuzzy subrings (IFR),and intuitionistic fuzzy sublattice (IFL), and so forth. Inthe last five years there are so many articles appeared inthis direction. Kim [3], Kim and Jun [4], Kim and Lee [5],introduced different types of IFI’s in Semigroups. Torkzadehand Zahedi [6] defined intuitionistic fuzzy commutativehyper K-ideals, Akram and Dudek [7] defined intuitionisticfuzzy Lie ideals of Lie algebras, and Hur et al. [8] introducedintuitionistic fuzzy prime ideals of a Ring.
The concept of ideal of a fuzzy subring was introducedby Mordeson and Malik in [9]. After that N Ajmal and A.SPrajapathi introduced the concept of residual of ideals ofan L-Ring in [10]. Motivated by this, in this paper we firstdefined the intuitionistic fuzzy ideal of an IFL and certaincharacterizations are given. Lastly we defined quotients(residuals) of ideals of an intuitionistic fuzzy sublattice andstudied their properties.
2. Preliminaries
We recall the following definitions and results which will beused in the sequel. Throughout this paper L stands for alattice (L,∨,∧) with zero element “0” and unit element “1”.
Definition 1 (see [1]). Let X be a nonempty set. Anintuitionistic fuzzy set [IFS] A of X is an object of thefollowing form A = {〈x, μA(x), νA(x)〉 | x ∈ X}, whereμA : X → [01] and νA : X → [01] define the degree ofmembership and the degree of non membership of the elementx ∈ X, respectively, and∀x ∈ X, 0 ≤ μA (x) + νA(x) ≤ 1.
The set of all IFS’s on X is denoted by IFS (X).
Definition 2 (see [1]). If A = {〈x,μA(x), νA(x)〉 | x ∈ S} andB = {〈x,μB(x), νB(x)〉 | x ∈ S} are any two IFS of X then
(i) A ⊆ B ⇔ μA(x) ≤ μB and νA(x) ≥ νB(x)∀x ∈ X;
(ii) A = B⇔ μA(x) = μB(x) and νA(x) = νB(x);
(iii)−A = {〈x, νA(x), μA(x)〉 | x ∈ X};
(iv) [A] = {〈x,μA(x),μAc(x)〉 | x ∈ X}, where μAc(x) =1− μA(x);
(v) 〈A〉 = {〈x, νcA(x), νA(x)〉 | x ∈ X}, where νAc(x) =1− νA(x);
(vi) A∩B= {〈x, min{μA(x),μB(x)}, max{νA(x), νB(x)}〉 |x ∈ X} = {〈x,μA∩B(x), νA∩B(x)〉 | x ∈ X};
(vii) A∪B= {〈x, max{μA(x),μB(x)}, min{νA(x), νB(x)}〉 |x ∈ X} = {〈x,μA∪B(x), νA∪B(x)〉 | x ∈ X}.
Definition 3 (see [11]). Let L be a lattice and A ={〈x,μA(x), νA(x)〉 | x ∈ L} be an IFS of L. Then A is calledan intuitionistic fuzzy sublattice [IFL] of L if the followingconditions are satisfied.
2 Advances in Fuzzy Systems
(i) μA(x ∨ y) ≥ min{μA(x),μA(y)};
(ii) μA(x ∧ y) ≥ min{μA(x),μA(y)};
(iii) νA(x ∨ y) ≤ max{νA(x), νA(y)};
(iv) νA(x ∧ y) ≤ max {νA(x), νA(y)},∀x, y ∈ L.
The set of all intuitionist fuzzy sublattices (IFL’s) of L isdenoted as IFL (L).
Definition 4 (see [11]). An IFSA of L is called an intuitionisticfuzzy ideal (IFI) of L if the following conditions are satisfied.
(i) μA(x ∨ y) ≥ min{μA(x),μA(y)};
(ii) μA(x ∧ y) ≥ max{μA(x),μA(y)};
(iii) νA(x ∨ y) ≤ max{νA(x), νA(y)};
(iv) νA(x ∧ y) ≤ min{νA(x), νA(y)},∀x, y ∈ L.
The set of all IFI’s of L is denoted as IFI (L).
Definition 5 (see [12]). Let A, B ∈ IFS (L). Then we definean IFS (L),
(i) A + B = {〈z,μA+B(z), νA+B(z)〉 | z ∈ L}, where
μA+B(z) = supz=x∨y{min
{μA(x),μB
(y)}}
,
νA+B(z) = inf z=x∨y{
max{νA(x), νB
(y)}}
.(1)
(ii) AB = {〈z,μAB(z), νAB(z)〉 | z ∈ L}, where
μAB(z) = supz=x∧y{
min{μA(x),μB
(y)}}
νAB(z) = inf z=x∧y{
max{νA(x), νB
(y)}}
.(2)
(iii) A · B = {〈z,μA·B(z), νA·B(z)〉 | z ∈ L} where
μA·B(z) = supz=∨ni=1(xi∧yi)
(mini
{min
{μA(xi),μB
(yi)}})
,
νA·B(z) = inf z=∨ni=1(xi∧yi)
(maxi
{max
{νA(xi), νB
(yi)}})
(3)
Lemma 1 (see [13]). Let A, B, and C be IFS (L), then thefollowing assertions hold.
(1) AB = BA, A + B = B + A, A · B = B ·A.
(2) AB ⊆ A · B.
(3) C(A + B) ⊆ CA + CB.
(4) (C + B)A ⊆ CA + BA.
(5) (A∩ B)C ⊆ AC ∩ BC.
(6) A ⊆ B ⇒ AC ⊆ BC and A ·C ⊆ B ·C.
Lemma 2 (see [13]). Let A be an IFL of L, then
(1) A + A = A.
(2) AA = A.
3. Ideal of an Intuitionistic Fuzzy Lattice
In this section we define the ideal of an IFL, and give somecharacterization of these ideals in terms of operations onIFS (L). We also used ∨ and ∧ to represent maximum andminimum, respectively, which is clear from the context.
Definition 6. Let A be an IFL of L and B an IFS of L withB ⊆ A. Then B is called an intuitionistic fuzzy ideal (IFI) of Aif the following conditions are satisfied.
(i) μB(x ∨ y) ≥ μB(x)∧ μB(y).
(ii) μB(x ∧ y) ≥ [μA(x)∧ μB(y)]∨ [μB(x)∧ μA(y)].
(iii) νB(x ∨ y) ≤ νB(x)∨ νB(y).
(iv) νB(x∧y) ≤ [νA(x)∨νB(y)]∧[νB(x)∨νA(y)] ∀x, y ∈L.
If B IFI of A, then we write B � A
Example 1. Consider the lattice L = {1, 2, 5, 10} underdivisibility.
Let A = {〈x,μA(x), νA(x)〉 | x ∈ L} be an IFL ofL defined by 〈1, .5, .1〉, 〈2, .4, .5〉, 〈5, .4, .3〉, 〈10, .7, .3〉 andB = {〈x,μB(x), νB(x)〉 | x ∈ L} be an IFS of L given by〈1, .5, .3〉, 〈2, .4, .5〉, 〈5, .3, .4〉, 〈10, .3, .4〉. Clearly B � A.
Definition 7. Let A be an IFL and B is also an IFL with B ⊆ A.Then B is called an intuitionistic fuzzy sublattice of A.
Lemma 3. The intersection of two IFI’s of A is again an IFI ofA.
Proof. Let B, C be IFI’s of A. Then we can prove that B∩C isalso an IFI of A. Since B ⊆ A and C ⊆ A, we have B ∩ C ⊆ AAlso
μB∩C(x∨ y
) = min{μB(x ∨ y
),μC(x ∨ y
)}
≥ min{μB(x)∧ μB
(y),μC(x)∧ μC
(y)}
,
since B and C are IFI′s of A.
≥ min{μB(x)∧ μC(x),μB
(y)∧ μC
(y)}
≥ min{μB∩C(x),μB∩C
(y)}
≥ μB∩C(x)∧ μB∩C(y),
μB∩C(x∧ y
) = min{μB(x ∧ y
),μC(x ∧ y
)}
≥ min{[μB(x)∧ μA
(y)] ∨ [μB
(y)∧ μA(x)
],
[μC(x)∧ μA
(y)]∨ [μA(x)∧ μC
(y)]}
since B and C are IFI′s of A.
≥ min{μB(x),μC(x)
}∧ μA(y)
∨min{μB(y),μC(y)}∧ μA(x)
≥ [μB∩C(x)∧ μA(y)]∨ [μB∩C
(y)∧ μA(x)
].(4)
Advances in Fuzzy Systems 3
Also
νB∩C(x ∨ y
) = max{νB(x ∨ y
), νC(x ∨ y
)}
≤ max{νB(x)∨ νB
(y), νC(x)∨ νC
(y)}
,
since B and C are IFI′s of A.
≤ max{νB(x)∨ νC(x), νB
(y)∨ νC
(y)}
≤ max{νB∩C(x), νB∩C
(y)}
≤ νB∩C(x)∨ νB∩C(y),
νB∩C(x ∧ y
) = max{νB(x ∧ y
), νC(x ∧ y
)}
≤ max{[
νB(x)∨ νA(y)]∧ [νB
(y)∨ νA(x)
],
[νC(x)∨ νA
(y)]∧ [νA(x)∨ νC
(y)]}
,
since B and C are IFI′s of A.
≤ max{νB(x), νC(x)} ∨ νA(y)
∧max{νB(y), νC(y)}∨ νA(x)
≤ [νB∩C(x)∨ νA(y)]∧ [νB∩C
(y)∨ νA(x)
].(5)
Hence, B ∩ C is an IFI of A.
Theorem 1. Let A an IFL and B an IFS of L with B ⊆ A. ThenB is an IFI of A if and only if
(1) μB(x∨ y) ≥ μB(x)∧ μB(y),
(2) νB(x ∨ y) ≤ νB(x)∨ νB(y),
(3) AB ⊆ B.
Proof. Suppose that conditions (1), (2), and (3) hold. Thenwe prove that B is an IFI of A.
We have
μB(x ∧ y
) ≥ μAB(x ∧ y
), since B ⊇ AB
= supx∧y=xi∧ yi
(μA(xi)∧ μB
(yi))
≥ μA(x)∧ μB(y).
(6)
Similarly
μB(x∧ y
) ≥ μB(x)∧ μA(y), since B ⊇ AB = BA. (7)
Hence
μB(x ∧ y
) ≥ [μA(x)∧ μB(y)]∨ [μB(x)∧ μA
(y)]. (a)
Also
νB(x ∧ y
) ≤ νAB(x ∧ y
), since B ⊇ AB
= infx∧y=xi∧ yi
(νA(xi)∨ νB
(yi))
≤ νA(x)∨ νB(y).
(8)
Similarly
νB(x ∧ y
) ≤ νB(x)∨ νA(y), since B ⊇ AB = BA. (9)
Hence
νB(x ∧ y
) ≤ [νA(x)∨ νB(y)]∧ [νB(x)∨ νA
(y)]. (b)
So from (1), (2), (a), and (b) B is an IFI of A.Conversely suppose B is an IFI of A. Then obviously
conditions (1) and (2) holds. Also we have
μB(x ∧ y
) ≥ μA(x)∧ μB(y),
νB(x ∧ y
) ≤ νA(x)∨ νB(y), ∀x, y ∈ L.
(10)
So∀z ∈ L with z = x ∧ y
μB(z) ≥∨
z=x∧y
[μA(x)∧ μB
(y)] = μAB(z),
νB(z) ≤∧
z=x∧y
[νA(x)∨ νB
(y)] = νAB(z).
(11)
Hence AB ⊆ B.
Theorem 2. Let A be an IFL of L and B an IFS with B ⊆ A.Then B is an IFI of A if and only if
(1) μB(x ∨ y) ≥ μB(x)∧ μB(y),
(2) νB(x ∨ y) ≤ νB(x)∨ νB(y),
(3) A · B ⊆ B.
Proof. Suppose conditions (1), (2), and (3) holds. We proveB is an IFI of A.
We have
μB(x ∧ y
) ≥ μA·B(x∧ y
), since B ⊇ A · B
= supx∧y=∨n
i=1(xi∧ yi)
⎡
⎣n∧
i=1
(μA(xi)∧ μB
(yi))⎤
⎦
≥ μA(x)∧ μB(y).
(12)
Similarly, we can obtain
μB(x ∧ y
) ≥ μB(x)∧ μA(y), since B ⊇ A · B = B · A.
(13)
Hence
μB(x ∧ y
) ≥ [μA(x)∧ μB(y)]∨ [μB(x)∧ μA
(y)]. (a.1)
Also
νB(x ∧ y
) ≤ νA·B(x ∧ y
), since B ⊇ A · B
= infx∧y=∨n
i=1(xi∧ yi)
⎡
⎣n∨
i=1
(νA(xi)∨ νB
(yi))⎤
⎦
≤ νA(x)∨ νB(y).
(14)
4 Advances in Fuzzy Systems
Similarly
νB(x ∧ y
) ≤ νB(x)∨ νA(y), since B ⊇ A · B = B ·A.
(15)
Hence
νB(x ∧ y
) ≤ [νA(x)∨ νB(y)]∧ [νB(x)∨ νA
(y)]. (b.1)
So from (1), (2), (a.1), and (b.1) B is an IFI of A.Conversely suppose that B is an IFI of A. Then obviously
conditions (1) and (2) hold.Let z ∈ L and z = ∨n
i=1(xi ∧ yi), where xi ∈ A, yi ∈ B.We have
μB(z) = μB
⎡
⎣n∨
i=1
(xi ∧ yi
)⎤
⎦ ≥n∧
i=1
μB(xi ∧ yi
)
≥n∧
i=1
{μA(xi)∧ μB
(yi)}
, since B IFI of A.
(16)
Thus
μB(z) ≥∨⎡
⎣n∧
i=1
{μA(xi)∧ μB
(yi)}⎤
⎦ = μA·B(z). (17)
Also
νB(z) = νB
⎡
⎣n∨
i=1
(xi ∧ yi
)⎤
⎦ ≤n∨
i=1
νB(xi ∧ yi
)
≤n∨
i=1
{νA(xi)∨ νB
(yi)}
, since B IFI of A.
(18)
Thus
νB(z) ≤ ∧⎡
⎣n∨
i=1
{νA(xi)∨ νB
(yi)}⎤
⎦ = νA·B(z). (19)
Hence A · B ⊆ B.
Theorem 3. Let A be an IFL of L and B, C are IFI’s of A. ThenB + C is an IFI of A.
Proof. We have
μB+C(x ∨ y
) ≥ μB+C(x)∧ μB+C(y),
νB+C(x ∨ y
) ≤ νB+C(x)∨ νB+C(y) (20)
(by [11, Theorem 5.2]).And
A(B + C) ⊆ AB + AC ⊆ B + C,
(B + C) A ⊆ BA + CA ⊆ B + C.(21)
(by Lemma 1 and Theorem 1).Hence B + C is an IFI of A.
4. Quotient of Ideals
Here first we define the residual of ideals of an IFL and provethat the residual of ideals is again an IFI of the IFL. Moreoverwe establish that it is the largest ideal with respect to someproperty on the operation ·.
Definition 8. Let A be an IFL of L and B, C be IFI’s of A. Thenthe quotient (residual) of B by C denoted as B/C is defined by
B/C = ∪{D/D � A, DC ⊆ B}. (22)
Theorem 4. Let A be an IFL of L and B,C are IFI’s of A. Thenthe quotient B/C is an IFI of A. Also B ⊆ B/C ⊆ A.
Proof. Let η = {D/D � A, DC ⊆ B}. Suppose D, D′ ∈η. Then D and D′ are IFI’s of A such that DC ⊆ B andD′C ⊆ B. Then by Theorem 3 D + D′ is an IFI of A. So byLemmas 1 and 2 (D + D′)C ⊆ DC + D′C ⊆ B + B = B. ThusD + D′ ∈ η. Now
μB/C(x)∧ μB/C(y) =
⎡
⎣∨
D∈ημD(x)
⎤
⎦∧⎡
⎣∨
D′∈ημD′(y)⎤
⎦
=∨{
μD(x)∧ μD′(y)/D, D′ ∈ η
}
≤∨{
μD+D′(x ∨ y
)/D, D′ ∈ η
}
≤ μB/C(x ∨ y
), since D + D′ ∈ η.
(23)
That is,
μB/C(x ∨ y
) ≥ μB/C(x)∧ μB/C(y). (24)
Also
μB/C(x∧ y
) =∨
D∈ημD(x ∧ y
)
≥∨
D∈η
{μD(x)∧ μA
(y)}
, since D � A
=⎡
⎣∨
D∈ημD(x)
⎤
⎦∧ μA(y)
= μB/C(x)∧ μA(y).
(25)
Similarly
μB/C(x ∧ y
) ≥ μB/C(y)∧ μA(x). (26)
Thus
μB/C(x ∧ y
) ≥ [μB/C(x)∧ μA(y)]∨ [μB/C
(y)∧ μA(x)
].
(27)
Advances in Fuzzy Systems 5
Now
νB/C(x)∨ νB/C(y) =
⎡
⎣∧
D∈ηνD(x)
⎤
⎦∨⎡
⎣∧
D′∈ηνD′(y)⎤
⎦
=∧{
νD(x)∨ νD′(y)/D, D′ ∈ η
}
≥∧{
νD+D′(x∨ y
)/D, D′ ∈ η
}
≥ νB/C(x∨ y
), since D + D′ ∈ η.
(28)
That is
νB/C(x ∨ y
) ≤ νB/C(x)∨ νB/C(y). (29)
Also
νB/C(x ∧ y
) =∧
D∈ηνD(x ∧ y
)
≤∧
D∈η
{νD(x)∨ νA
(y)}
, since D � A
=⎡
⎣∧
D∈ηνD
(x)
⎤
⎦∨ νA(y)
= νB/C(x)∨ νA(y).
(30)
Similarly
νB/C(x ∧ y
) ≤ νB/C(y)∨ νA(x). (31)
Thus
νB/C(x ∧ y
) ≤ [νB/C(x)∨ νA(y)]∧ [νB/C
(y)∨ νA(x)
].(32)
From (24), (27), (29), and (32) B/C is an IFI of A.Clearly B/C ⊆ A.Since B is an IFI of A, BA ⊆ B (by Theorem 1).Since C ⊆ A, by Lemma 1 BC ⊆ BA ⊆ B. Hence B ∈ η.
So B ⊆ B/C.Thus we have
B ⊆ B/C ⊆ A. (33)
Theorem 5. Let A be an IFL and B, C be IFI’s of A. Then B/Cis the largest IFI of A with the property (B/C) ·C ⊆ B.
Proof. Let η = {D/D � A and DC ⊆ B}. We have B/C =⋃D∈η D Let x ∈ L such that x = ∨n
i=1(ai ∧ bi).Then
μB(ai ∧ bi) ≥ μDC(ai ∧ bi) ≥ μD(ai)∧ μC(bi), ∀D ∈ η.(34)
So
μB(ai ∧ bi) ≥∨
D∈η
[μD(ai)∧ μC(bi)
]
=⎡
⎣∨
D∈ημD(ai)
⎤
⎦∧ μC(bi)
= μB/C(ai)∧ μC(bi).
(35)
Hence
μB(x) = μB
⎛
⎝n∨
i=1
(ai ∧ bi)
⎞
⎠ ≥n∧
i=1
μB(ai ∧ bi),
since B is an IFI of A
≥n∧
i=1
[μB/C(ai)∧ μC(bi)
].
(36)
Consequently
μB(x) ≥∨⎧⎨
⎩
n∧
i=1
[μB/C(ai)∧ μC(bi)
]/x =
n∨
i=1
(ai ∧ bi)
⎫⎬
⎭
= μ(B/C)·C(x).
(37)
Also
νB(ai ∧ bi) ≤ νDC(ai ∧ bi) ≤ νD(ai)∨ νC(bi), ∀D ∈ η.(38)
So
νB(ai ∧ bi) ≤∧
D∈η[νD(ai)∨ νC(bi)]
=⎡
⎣∧
D∈ηνD(ai)
⎤
⎦∨ νC(bi)
= νB/C(ai)∧ νC(bi).
(39)
Hence
νB(x) = νB
⎛
⎝n∨
i=1
(ai ∧ bi)
⎞
⎠ ≤n∨
i=1
νB(ai ∧ bi),
since B is an IFI of A
≤n∨
i=1
[νB/C(ai)∧ νC(bi)].
(40)
Consequently
νB(x) ≤∧⎧⎨
⎩
n∨
i=1
[νB/C(ai)∧ νC(bi)]/x =n∨
i=1
(ai ∧ bi)
⎫⎬
⎭
= ν(B/C)·C(x).
(41)
Thus from (37) and (41) (B/C) · C ⊆ B.If D is an ideal of A such that D · C ⊆ B then DC ⊆
D ·C ⊆ B. So D ∈ η. Hence D ⊆ B/C. Thus B/C is the largestIFI of A such that (B/C) · C ⊆ B.
Theorem 6. Let A be an IFL and B, C, D be IFI’s of A. Thenthe following holds.
(1) If B ⊆ C then B/D ⊆ C/D and D/C ⊆ D/B.
(2) If B ⊆ C then C/B = A.
(3) B/B = A.
6 Advances in Fuzzy Systems
Proof. (1) Let B ⊆ C. Write η = {E/E � A and ED ⊆ B}and ξ = {E/E � A and ED ⊆ C}. If E ∈ η then E � Aand ED ⊆ B ⊆ C. Thus E ∈ ξ and hence η ⊆ ξ . So B/D =⋃
E∈η E ⊆⋃
E∈ξ E = C/D.Similarly, let η1 = {E/E � A and EC ⊆ D} and ξ1 =
{E/E � A and EB ⊆ D}. If E ∈ η1 then EC ⊆ D. But B ⊆ C.So EB ⊆ EC ⊆ D. Thus E ∈ ξ1 and hence η1 ⊆ ξ1. So D/C =⋃
E∈η1E ⊆ ⋃E∈ξ1
E = D/B.(2) Let η = {E/E � A and EB ⊆ C}. Since B � A, we
have AB ⊆ B ⊆ C, and A � A. Thus A ∈ η and henceA ⊆ ⋃E∈η E = C/B ⊆ A, since C/B is an IFI of A. ThereforeC/B = A.
(3) We have B ⊆ B. So from (2) B/B = A.
Corollary 1. Let A be an IFL of L and B, and C be IFI’s of A.Then
(1) (B/C)/B = A,
(2) (B/B)/C = A,
(3) B/(B ∩ C) = A.
Proof. (1) Since B ⊆ B/C, by Theorem 6 (2), (B/C)/B = A.(2) By Theorem 6 (3) B/B = A. Since C ⊆ A = B/B by
Theorem 6 (2), (B/B)/C = A.(3) Since B � A and C � A. So B∩C � A and B∩C ⊆ B.
Hence by Theorem 6 (2), B/(B ∩ C) = A.
Theorem 7. Let A be an IFL of L and Bi i = 1, 2 . . . . . .m, C,are IFI’s of A. Then
⎛
⎝m⋂
i=1
Bi
⎞
⎠/C =m⋂
i=1
(Bi/C). (42)
Proof. Since⋂m
i=1 Bi ⊆ Bi by Theorem 6 (1) (⋂m
i=1 Bi)/C ⊆Bi/C,∀i.
Hence
⎛
⎝m⋂
i=1
Bi
⎞
⎠/C ⊆m⋂
i=1
(Bi/C). (43)
Let
η1 = {E/E � A, EC ⊆ B1},
η2 = {E/E � A, EC ⊆ B2},
η3 = {E/E � A, EC ⊆ B1 ∩ B2}.(44)
Then ∀x ∈ L
μB1/C∩B2/C(x) = μB1/C(x)∧ μB2/C(x)
=⎛
⎝∨
E∈η1
μE(x)
⎞
⎠∧⎛
⎝∨
E′∈η2
μE′ (x)
⎞
⎠
= ∨{[μE(x)∧ μE′(x)]/E ∈ η1, E′ ∈ η2
}.
(a.2)
Similarly
νB1/C∩B2/C(x) = νB1/C(x)∨ νB2/C(x)
=⎛
⎝∧
E∈η1
νE(x)
⎞
⎠∨⎛
⎝∧
E′∈η2
νE′(x)
⎞
⎠
= ∧{[νE(x)∨ νE′(x)]/E ∈ η1, E′ ∈ η2}.
(b.2)
Now let E ∈ η1 and E′ ∈ η2. Then EC ⊆ B1 and E′C ⊆ B2.Also E∩ E′ IFI of A.
So that
(E∩ E′)C ⊆ EC ∩ E′C ⊆ B1 ∩ B2. (45)
Thus E∩ E′ ∈ η3. So η1 ∩ η2 ⊆ η3.Hence
(B1 ∩ B2)/C =⋃
E∈η3
E ⊇⋃
E∈η1,E′∈η2
(E ∩ E′). (46)
So
μB1∩B2/C(x) ≥∨
μE∩E′ (x)
=∨[
μE(x)∧ μE′(x)]
= μB1/C∩B2/C(x), from (a.2),
νB1∩B2/C(x) ≤∧
νE∩E′ (x)
=∧
[νE(x)∨ νE′(x)]
= νB1/C∩B2/C(x), from (b.2).
(47)
Hence
B1 ∩ B2/C ⊇ B1/C ∩ B2/C. (48)
From (43) and (48) (B1 ∩ B2)/C = B1/C ∩ B2/C. Thiscompletes the proof.
Next, we denote the set of all IFI’s {Bi} i = 1, 2 . . . ,m of an IFL A that satisfies the property μBi(0) = μB j(0)and νBi(0) = νB j(0) ∀i, j by IFI (A∗). Then we have thefollowing results.
Lemma 4. Let A be an IFL of L and B, C ∈ IFI (A∗). Then
(1) B ⊆ B + C and C ⊆ B + C.
(2) B/C = B/B + C.
(3) B + C/B = A and B + C/B ∩C = A.
Advances in Fuzzy Systems 7
Proof. (1) We have
μB+C(x) =∨
x=y∨z
(μB(y)∧ μC(z)
)
≥ μB(x)∧ μC(0) as x = x ∨ 0
= μB(x)∧ μB(0) since μB(0) = μC(0)
= μB(x) since μB(0) ≥ μB(x),
νB+C(x) =∧
x=y∨z
(νB(y)∨ νC(z)
)
≤ νB(x)∨ νC(0) as x = x∨ 0
= νB(x)∨ νB(0) since νB(0) = νC(0)
= νB(x) since νB(0) ≤ νB(x).
(49)
So B ⊆ B + C. Similarly ⊆ B + C.(2) We have B + C � A (by Theorem 3) and C ⊆ B + C
(by (1)). So by Theorem 6 (1),
B/B + C ⊆ B/C. (50)
Write η = {E/E � A and EC ⊆ B} and ξ = {E/E �A and E(B + C) ⊆ B}.
Let E ∈ η, then E � A and E ⊆ A. So EB ⊆ AB (byLemma 1). But AB ⊆ B, since B � A. Hence EB ⊆ B and alsoEC ⊆ B. So By Lemma 1, E(B + C) ⊆ EB + EC ⊆ B + B = B.Therefore E ∈ ξ . So η ⊆ ξ . Thus
B/C =⋃
E∈ηE ⊆
⋃
E∈ξE = B/B + C. (51)
From (50) and (51) B/C = B/B + C.(3) We have B + C � A and B ⊆ B + C. So by Theorem 6
(2), B + C/B = A.Also we have B ∩ C � A and B ∩ C ⊆ B + C. Hence by
Theorem 6 (2) B + C/B ∩ C = A.
Theorem 8. Let A be an IFL of L and {Bi}i = 1, 2 . . .m ∈ IFI(A∗) and C any IFI of A. Then C/
∑mi=1 Bi =
⋂mi=1(C/Bi).
Proof. We have B1 + B2 � A and B1 ⊆ B1 + B2, B2 ⊆ B1 + B2.So by Theorem 6 (1), C/B1 +B2 ⊆ C/B1 and C/B1 +B2 ⊆
C/B2.Therefore
C/B1 + B2 ⊆ C/B1
⋂C/B2. (52)
Let
η1 = {E/E � A, EB1 ⊆ C},η2 = {E/E � A, EB2 ⊆ C},η3 = {E/E � A, E(B1 + B2) ⊆ C}.
(53)
Then ∀x ∈ L
μC/B1∩C/B2 (x) = μC/B1 (x)∧ μC/B2 (x)
=⎛
⎝∨
E∈η1
μE(x)
⎞
⎠∧⎛
⎝∨
E′∈η2
μE′(x)
⎞
⎠
=∨{[
μE(x)∧ μE′(x)]/E ∈ η1, E′ ∈ η2
}.
(a.3)
Similarly
νC/B1∩C/B2 (x) = νC/B1 (x)∨ νC/B2 (x)
=⎛
⎝∧
E∈η1
νE(x)
⎞
⎠∨⎛
⎝∧
E′∈η2
νE′(x)
⎞
⎠
= ∧{[νE(x)∨ νE′(x)]/E ∈ η1, E′ ∈ η2}.
(b.3)
Now let E ∈ η1 and E′ ∈ η2. Then EB1 ⊆ C and E′B2 ⊆ C.Also E∩ E′ IFI of A, so that
(E∩ E′)(B1 + B2) ⊆ (E∩ E′)B1 + (E∩ E′)B2 ⊆ EB1
+ E′B2 ⊆ C + C = C.(54)
So E ∩ E′ ∈ η3. Hence η1 ∩ η2 ⊆ η3. Thus C/B1 + B2 =⋃E∈η3
E ⊇ ⋃E∈η1,E′∈η2(E ∩ E′).
So
μC/B1+B2 (x) ≥ ∨μE∩E′ (x)
= ∨[μE(x)∧ μE′(x)]
= μC/B1∩C/B2 (x) from (a.3),
νC/B1+B2 (x) ≤ ∧νE∩E′ (x)
= ∧[νE(x)∨ νE′(x)]
= νC/B1∩C/B2 (x) from (b.3).
(55)
Therefore
C/B1 + B2 ⊇ C/B1
⋂C/B2. (56)
From (52) and (56) C/B1 + B2 = C/B1 ∩ C/B2, hence theresult.
References
[1] K. T. Atanassov, “Intutionistic fuzzy sets,” Fuzzy Sets andSystems, vol. 20, no. 1, pp. 87–96, 1986.
[2] K. T. Atanassov, “New operations defined over the intuitionis-tic fuzzy sets,” Fuzzy Sets and Systems, vol. 61, no. 2, pp. 137–142, 1994.
[3] K. H. Kim, “On intuitionistic Q-fuzzy semiprime ideals insemigroups,” Advances in Fuzzy Mathematics, vol. 1, no. 1, pp.15–21, 2006.
[4] K. H. Kim and Y. B. Jun, “Intuitionistic fuzzy interior idealsof semigroups,” International Journal of Mathematics andMathematical Sciences, vol. 27, no. 5, pp. 261–267, 2001.
8 Advances in Fuzzy Systems
[5] K. H. Kim and J. G. Lee, “On intuitionistic fuzzy Bi-ideals ofsemigroups,” Turkish Journal of Mathematics, vol. 29, pp. 201–210, 2005.
[6] L. Torkzadeh and M. M. Zahedi, “Intuitionistic fuzzy com-mutative hyper K-ideals,” Journal of Applied Mathematics andComputing, vol. 21, no. 1-2, pp. 451–467, 2006.
[7] M. Akram and W. Dudek, “Interval-valued intuitionistic fuzzyLie ideals of Lie algebras,” World Applied Sciences Journal, vol.7, no. 7, pp. 812–819, 2009.
[8] K. Hur, S. Y. Jang, and H. W. Kang, “Intuitionistic fuzzyideals of a ring,” Journal of the Korea Society of MathematicalEducation. Series B, vol. 12, no. 3, pp. 193–209, 2005.
[9] J. N. Mordeson and D. S. Malik, Fuzzy Commutative Algebra,World Scientific, Singapore, 1998.
[10] A. S. Prajapati, “Residual of ideals of an L-ring,” IranianJournal of Fuzzy Systems, vol. 4, no. 2, pp. 69–82, 2007.
[11] K. V. Thomas and L. S. Nair, “Intuitionistic fuzzy sublatticesand ideals,” Fuzzy Information and Engineering. In press.
[12] L. Atanassov, “On intuitionistic fuzzy versions of L. Zadeh’sextension principle,” Notes on Intuitionistic Fuzzy Sets, vol. 13,no. 3, pp. 33–36, 2006.
[13] K. V. Thomas and L. S. Nair, “Operations on intuitionisticfuzzy ideals of a lattice,” International Journal of FuzzyMathematics. In press.
Submit your manuscripts athttp://www.hindawi.com
Computer Games Technology
International Journal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Distributed Sensor Networks
International Journal of
Advances in
FuzzySystems
Hindawi Publishing Corporationhttp://www.hindawi.com
Volume 2014
International Journal of
ReconfigurableComputing
Hindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Applied Computational Intelligence and Soft Computing
Advances in
Artificial Intelligence
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Advances inSoftware EngineeringHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Electrical and Computer Engineering
Journal of
Journal of
Computer Networks and Communications
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporation
http://www.hindawi.com Volume 2014
Advances in
Multimedia
International Journal of
Biomedical Imaging
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
ArtificialNeural Systems
Advances in
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Computational Intelligence and Neuroscience
Industrial EngineeringJournal of
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Modelling & Simulation in EngineeringHindawi Publishing Corporation http://www.hindawi.com Volume 2014
The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014
Human-ComputerInteraction
Advances in
Computer EngineeringAdvances in
Hindawi Publishing Corporationhttp://www.hindawi.com Volume 2014