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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)]}


≥ 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)}

,


≤ 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)]}

,


≤ 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)


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)


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.


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.


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.


[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.

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