DOUBT INTUITIONISTIC FUZZY IDEALS IN BCK/BCI-ALGEBRAS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

DOUBT INTUITIONISTIC FUZZY IDEALS IN BCK/BCI-ALGEBRAS Tripti Bej1 and Madhumangal Pal2 Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore – 721 102, India.

ABSTRACT In this paper, we introduce the concept of doubt intuitionistic fuzzy subalgebras and doubt intuitionistic fuzzy ideals in BCK/BCI-algebras. We show that an intuitionistic fuzzy subset of BCK/BCI-algebras is an intuitionistic fuzzy subalgebra and an intuitionistic fuzzy ideal if and only if the complement of this intuitionistic fuzzy subset is a doubt intuitionistic fuzzy subalgebra and a doubt intuitionistic fuzzy ideal. And at the same time we have established some common properties related to them.

KEYWORDS AND PHRASES BCK/BCI-algebras, doubt fuzzy subalgebra, doubt fuzzy ideal, doubt intuitionistic fuzzy subalgebra, doubt intuitionistic fuzzy ideal.

1. INTRODUCTION Fuzzy set was introduced by Zadeh [1], and later on several researchers worked in this field. As a natural advancement of these research works we get, the idea of intuitionistic fuzzy sets propounded by Atanassov [2, 3], that is a generalisation of the notion of fuzzy set.

Imai and Iseki [4, 5, 6] introduced two classes of abstract algebras BCK-algebras and BCI-algebras.

It is known that the class of BCK-algebra is a proper subclass of the class of BCI-algebra. In 1991, Xi [7] introduced the concept of fuzzy set to BCK-algebras. Then in 1992, Huang [8] gave another concept of fuzzy set to BCI-algebras. Following the same rout in 1994, Jun [9] established the definition of doubt fuzzy subalgebra and doubt fuzzy ideal in BCK/BCI-algebras to avoid the confusion created in [8]Huang’s definition of fuzzy BCI-algebras and gave some results about it.

In 2000, Jun and Kim [10] explored the intuitionistic fuzzy subalgebra and intuitionistic fuzzy ideal in BCK-algebras. In the recent past in 2012 Solairaju and Begam [11] provided a noble relationship between an intuitionistic fuzzy ideal and an intuitionistic fuzzy P-ideal with some characterisation of intuitionistic fuzzy P-ideals. Also, Senapati et al. have presented several results on / -algebras, -algebra and -algebra [12, 13, 14, 15, 16].

In this ambitious paper, we are going to introduce the concept of doubt intuitionistic fuzzy subalgebra and doubt intuitionistic fuzzy ideal in BCK/BCI-algebras and to study some of their properties. We show that in BCI/BCK-algebras, an intuitionistic fuzzy subset is a doubt intuitionistic fuzzy ideal if and only if the complement of this intuitionistic fuzzy subset is an intuitionistic fuzzy ideal. Also we provide relations between a doubt intuitionistic fuzzy subalgebra DOI : 10.5121/ijfls.2015.5101

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

and a doubt intuitionistic fuzzy ideal in BCK/BCI-algebras.

2. PRELIMINARIES We first recall some basic concepts which are used to present the paper.

An algebra ( ;∗ ,0) of type (2,0) is called a -algebra if it satisfies the following axioms for all , , ∈ : (A1) (A2) (A3) (A4)

(( ∗ ) ∗ ( ∗ )) ∗ ( ∗ ) = 0 ( ∗ ( ∗ )) ∗ = 0 ∗ =0 ∗ = 0 and ∗ = 0 imply =

If a -algebra satisfies 0 ∗ = 0. Then is called a -algebra.

In a / -algebra, ∗ 0 = hold. A partial ordering "≤" on a / -algebra can be defined by ≤ if and only if ∗ = 0.

Any -algebra satisfies the following axioms for all , , ∈ : (i) ( ∗ ) ∗ = ( ∗ ) ∗ (ii) ∗ ≤ (iii) ( ∗ ) ∗ ≤ ( ∗ ) ∗ ( ∗ ) (iv) ≤ ⇒ ∗ ≤ ∗ , ∗ ≤ ∗ .

If a -algebra satisfies ∗ ( ∗ ) = ∗ ( ∗ ) for all , ∈ , then it is called commutative.

Throughout this paper, always means a / -algebra without any specification.

A non-empty subset of a / -algebra is called a subalgebra of if ∗ ∈ for any , ∈ .

A non-empty subset of a / -algebra is called an ideal of if (i) 0 ∈ (ii) ∗ ∈ and ∈ then ∈ , for all , ∈ .

The proposed work is done on intuitionistic fuzzy set. The formal definition of intuitionistic fuzzy set is given below: An intuitionistic fuzzy set in a non-empty set is an object having the form = { , ( ), ( )/ ∈ } , where the function : → [0,1] and : → [0,1] , denote the degree of membership and the degree of non-membership of each element ∈ to the set respectively and 0 ≤ ( ) + ( ) ≤ 1, for all ∈ .

For the sake of simplicity, we use the symbol form = ( , , ) or ( , ) for the intuitionistic fuzzy set = {〈 , ( ), ( )〉/ ∈ }.

The two operators used in this paper are defined as: If = ( , ), is an intuitionistic fuzzy set, then, 2

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

Π= {( , ( ), ̅ ( ))/ ∈ } ♦ = {( , ̅ ( ), ( ))/ ∈ }.

For the sake of simplicity, we also use ∨ for )* ( , ), and ∧ for ),-( , ).

Intuitionistic fuzzy subalgebra and intuitionistic fuzzy ideal are the extension of fuzzy subalgebra and fuzzy ideal which are defined below : An intuitionistic fuzzy set = ( , ) in , is called an intuitionistic fuzzy subalgebra [10] of if it satisfies the following two conditions: (i) ( ∗ ) ≥ ( ) ∧ ( ), (ii) ( ∗ ) ≤ ( ) ∨ ( ), for all , ∈ .

An intuitionistic fuzzy set = ( , ) in is called an intuitionistic fuzzy ideal [10] of , if it satisfies the following axioms: (i) (0) ≥ ( ), (0) ≤ ( ), (ii) ( ) ≥ ( ∗ ) ∧ ( ), (iii) ( ) ≤ ( ∗ ) ∨ ( ), for all , ∈ .

In order to resolve the contradiction popped up in Huang’s [8] definition of fuzzy -algebra, Jun [9] introduced the definition of doubt fuzzy subalgebra and doubt fuzzy ideals in / -algebras, which are as follows: A fuzzy set = {〈 , ( )〉: ∈ } in is called a doubt fuzzy subalgebra [9] of if ( ∗ ) ≤ ( ) ∨ ( ), for all , ∈ .

A fuzzy set = {〈 , ( )〉: ∈ } in is called a doubt fuzzy ideal [9] of if (i) (0) ≤ ( ) (ii) ( ) ≤ ( ∗ ) ∨ ( ), for all , ∈ .

3. DOUBT INTUITIONISTIC FUZZY IDEALS IN BCK/BCI-ALGEBRAS In this section, we define doubt intuitionistic fuzzy subalgebra and doubt intuitionistic fuzzy ideal in / -algebras and investigate their properties.

Definition 3.1 Let = ( , ) be an intuitionistic fuzzy subset of a / -algebra , then

is called a doubt intuitionistic fuzzy subalgebra of if (i) ( ∗ ) ≤ ( ) ∨ ( ), (ii) ( ∗ ) ≥ ( ) ∧ ( ), for all , ∈ .

Theorem 3.2 Let = ( , ) be a doubt intuitionistic fuzzy subalgebra of , then

(i) (0) ≤ ( ) and (ii) (0) ≥ ( ), for all ∈ .

Proof: By definition, (0) = ( ∗ ) ≤ ( ) ∨ ( ) ≤ ( ). Therefore, (0) ≤ ( ), for all ∈ .

Again, (0) = ( ∗ ) ≥ ( ) ∧ ( ) ≥ ( ). Therefore, (0) ≥ ( ), for all ∈ .

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

Theorem 3.3 Let = ( , ) be a doubt intuitionistic fuzzy subalgebra of a / -algebra . Then for any â&#x2C6;&#x2C6; , we have (i) ( 6 â&#x2C6;&#x2014; ) â&#x2030;¤ ( ), *-7 ( 6 â&#x2C6;&#x2014; ) â&#x2030;¥ ( ), ,8 - ,9 :77 (ii) ( 6 â&#x2C6;&#x2014; ) = ( ), *-7 ( 6 â&#x2C6;&#x2014; ) = ( ), ,8 - ,9 ;<;Proof: (i) Let â&#x2C6;&#x2C6; , then ( â&#x2C6;&#x2014; ) = (0) â&#x2030;¤ ( ).

If - is odd then let - = 2> â&#x2C6;&#x2019; 1 for some positive integer >.

Now assume that ( @ABC â&#x2C6;&#x2014; ) â&#x2030;¤ ( ) for some positive integer >.

Then, ( @(ADC)BC â&#x2C6;&#x2014; ) = ( @ADC â&#x2C6;&#x2014; ), = ( @ABC â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; ))) = ( @ABC â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; 0)) = ( @ABC â&#x2C6;&#x2014; ) â&#x2030;¤ ( ).

Hence, ( 6 â&#x2C6;&#x2014; ) â&#x2030;¤ ( ), if - is odd. This proves the first part. Similarly, we can prove the second part. (ii) Again, let - be even, and - = 2E. Now for E = 1, ( @ â&#x2C6;&#x2014; ) = ( â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; )) = ( â&#x2C6;&#x2014; 0) = ( ).

Also assume that, ( @F â&#x2C6;&#x2014; ) = ( ) for some positive integer E. Then, ( @(FDC) â&#x2C6;&#x2014; ) = ( @F â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; ))), = ( @F â&#x2C6;&#x2014; ) = ( ).

Hence, ( 6 â&#x2C6;&#x2014; ) = ( ), if - is even. This proves the first part. Similarly, we can prove the second part.

In a -algebra, 0 â&#x2C6;&#x2014; = 0. Hence we conclude the following proposition.

Proposition 3.4 Let = ( , ) be a doubt intuitionistic fuzzy-subalgebra of a -algebra . Then for any â&#x2C6;&#x2C6; , we have, ( â&#x2C6;&#x2014; 6 ) = (0), *-7 ( â&#x2C6;&#x2014; 6 ) = (0), for - = 1,2,3, â&#x20AC;¦ Example 3.5 Consider a -algebra * 0 1 2 3

= {0,1,2,3} with the following Cayley table: 0 1 2 3 0 0 0 0 1 0 0 1 2 1 0 2 3 3 3 0

Let = ( , ) be an intuitionistic fuzzy set of as defined by:

X

0 1 2 3 .5 .5 .6 .5 .5 .5 .3 .5

Then it can easily be verified that = ( , ) is a doubt intuitionistic fuzzy subalgebra of .

Definition 3.6 Let be a / -algebra. An intuitionistic fuzzy set = ( , ) in is called a doubt intuitionistic fuzzy ideal if 4

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

(F1) (0) â&#x2030;¤ ( ), (0) â&#x2030;¥ ( ) (F2) ( ) â&#x2030;¤ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ ( ) (F3) ( ) â&#x2030;¥ ( â&#x2C6;&#x2014; ) â&#x2C6;§ ( ), for all , â&#x2C6;&#x2C6; .

Theorem 3.7 Let an intuitionistic fuzzy set = ( , ) in be a doubt intuitionistic fuzzy ideal of . If the inequility â&#x2C6;&#x2014; > â&#x2030;¤ E holds in , then (i) ( ) â&#x2030;¤ )* { (>), (E)} (ii) ( ) â&#x2030;¥ ),-{ (>), (E)}.

Proof: Let , >, E â&#x2C6;&#x2C6; be such that â&#x2C6;&#x2014; > â&#x2030;¤ E then ( â&#x2C6;&#x2014; >) â&#x2C6;&#x2014; E = 0 and thus, ( ) â&#x2030;¤ )* { ( â&#x2C6;&#x2014; >), (>)}, â&#x2030;¤ )* {)* { (( â&#x2C6;&#x2014; >) â&#x2C6;&#x2014; E), (E)}, (>)} = )* {)* { (0), (E)}, (>)} = )* { (E), (>)} Therefore, ( ) â&#x2030;¤ )* { (>), (E)}.

I*,-, ( ) â&#x2030;¥ ),-{ ( â&#x2C6;&#x2014; >), (>)}, â&#x2030;¥ ),-{),-{ (( â&#x2C6;&#x2014; >) â&#x2C6;&#x2014; E), (E)}, (>)} = ),-{),-{ (0), (E)}, (>)} = ),-{ (E), (>)} = ),-{ (>), (E)} Therefore, ( ) â&#x2030;¥ ),-{ (>), (E)}. This completes the proof.

Corollary 3.8 If = ( , ) is a doubt intuitionistic fuzzy ideal of , then for any , >C , >@ , >J , â&#x20AC;¦ , >6 â&#x2C6;&#x2C6; and (â&#x2039;¯ ( â&#x2C6;&#x2014; >C ) â&#x2C6;&#x2014; >@ ) â&#x2C6;&#x2014; â&#x2039;¯ ) â&#x2C6;&#x2014; >6 = 0, (,) ( ) â&#x2030;¤ )* { (>C ), (>@ ), (>J ), â&#x20AC;¦ , (>6 )}, (,,) ( ) â&#x2030;¥ ),-{ (>C ), (>@ ), (>J ), â&#x20AC;¦ , (>6 )}.

Proposition 3.9 Let = ( , ) be a doubt intuitionistic fuzzy ideal of a / -algebra .Then the followings hold for all , , â&#x2C6;&#x2C6; , (i) if â&#x2030;¤ then ( ) â&#x2030;¤ ( ), ( ) â&#x2030;¥ ( ). (ii) ( â&#x2C6;&#x2014; ) â&#x2030;¤ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ ( â&#x2C6;&#x2014; ) *-7 ( â&#x2C6;&#x2014; ) â&#x2030;¥ ( â&#x2C6;&#x2014; ) â&#x2C6;§ ( â&#x2C6;&#x2014; ).

Proof: (i) If â&#x2030;¤ then â&#x2C6;&#x2014; = 0.

Hence ( ) â&#x2030;¤ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ ( ) = (0) â&#x2C6;¨ ( ) = ( ), and ( ) â&#x2030;¥ ( â&#x2C6;&#x2014; ) â&#x2C6;§ ( ) = (0) â&#x2C6;§ ( ) = ( ). (ii) Since ( â&#x2C6;&#x2014; ) â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; ) â&#x2030;¤ ( â&#x2C6;&#x2014; ).

It follows from (i) that, {( â&#x2C6;&#x2014; ) â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; )} â&#x2030;¤ ( â&#x2C6;&#x2014; ). Now ( â&#x2C6;&#x2014; ) â&#x2030;¤ {( â&#x2C6;&#x2014; ) â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; )} â&#x2C6;¨ ( â&#x2C6;&#x2014; ), [L;M*N9; = ( , ) is a doubt intuitionistic fuzzy ideal.].

Therefore ( â&#x2C6;&#x2014; ) â&#x2030;¤ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ ( â&#x2C6;&#x2014; ) .Again, ( â&#x2C6;&#x2014; ) â&#x2030;¥ {( â&#x2C6;&#x2014; ) â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; )} â&#x2C6;§ ( â&#x2C6;&#x2014; ) â&#x2030;¥ ( â&#x2C6;&#x2014; ) â&#x2C6;§ ( â&#x2C6;&#x2014; ).

Therefore ( â&#x2C6;&#x2014; ) â&#x2030;¥ ( â&#x2C6;&#x2014; ) â&#x2C6;§ ( â&#x2C6;&#x2014; ).

Theorem 3.10 Every doubt intuitionistic fuzzy ideal of a -algebra is a doubt intuitionistic fuzzy subalgebra of .

Proof: Let = ( , ) be a doubt intuitionistic fuzzy ideal of .

Since â&#x2C6;&#x2014; â&#x2030;¤ , for all , â&#x2C6;&#x2C6; . Then ( â&#x2C6;&#x2014; ) â&#x2030;¤ ( ) and ( â&#x2C6;&#x2014; ) â&#x2030;¥ ( ).

So, ( â&#x2C6;&#x2014; ) â&#x2030;¤ ( ) â&#x2030;¤ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ ( ) â&#x2030;¤ ( ) â&#x2C6;¨ ( ), for all , â&#x2C6;&#x2C6; , [because = ( , ) is a doubt intuitionistic fuzzy ideal.].

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

And ( â&#x2C6;&#x2014; ) â&#x2030;¥ ( ) â&#x2030;¥ ( â&#x2C6;&#x2014; ) â&#x2C6;§ ( ) â&#x2030;¥ ( ) â&#x2C6;§ ( ), for all , â&#x2C6;&#x2C6; .

This shows that, = ( , ) is a doubt intuitionistic fuzzy subalgebra of .

Example 3.11 Let = {0,1,2,3,4} be a -algebra with the following Cayley table: 0 0 1 2 3 4

* 0 1 2 3 4

1 0 0 2 3 2

2 0 0 0 0 1

3 0 0 2 0 2

4 0 0 0 0 0

Let = ( , ) be an intuitionistic fuzzy set of as defined by:

X

0 0 1

1 0.4 0.5

2 0.6 0.4

3 0.6 0.4

4 0.6 0.4

Then = ( , ) is a doubt intuitionistic fuzzy ideal of .

Thus is a doubt intuitionistic fuzzy ideal of as well as doubt intuitionistic fuzzy subalgebra of .

The converse of Theorem 3.10 may not be true, to justify it let us consider the Example 3.5.

As (2) = 0.3, (2) = 0.3 â&#x2030;± 0 .5 = (2 â&#x2C6;&#x2014; 1) â&#x2C6;§ (1), Therefore, = ( , ) is not a doubt intuitionistic fuzzy ideal.

We now give a condition for the intuitionistic fuzzy set = ( , ) which is a doubt intuitionistic fuzzy subalgebra of to be a doubt intuitionistic fuzzy ideal of .

Theorem 3.12 Let an intuitionistic fuzzy set = ( , ) be a doubt intuitionistic fuzzy subalgebra of . If the inequiality â&#x2C6;&#x2014; â&#x2030;¤ holds in , then would be a doubt intuitionistic fuzzy ideal of .

Proof: Let = ( , ) be a doubt intuitionistic fuzzy subalgebra of X. Then from Theorem 3.2, (0) â&#x2030;¤ ( ) and (0) â&#x2030;¥ ( ), where â&#x2C6;&#x2C6; .

As â&#x2C6;&#x2014; â&#x2030;¤ holds in , then from Theorem 3.7, we get, ( ) â&#x2030;¤ )* { ( ), ( )} and ( ) â&#x2030;¥ ),-{ ( ), ( )}, for all , , â&#x2C6;&#x2C6; .

Again since, â&#x2C6;&#x2014; ( â&#x2C6;&#x2014; ) â&#x2030;¤ , then, ( ) â&#x2030;¤ )* { ( â&#x2C6;&#x2014; ), ( )} and ( ) â&#x2030;¥ ),-{ ( â&#x2C6;&#x2014; ), ( )}.

Hence, = ( , ) is a doubt intuitionistic fuzzy ideal of .

Proposition 3.13 Let an intuitionistic fuzzy set = ( , ) be a doubt intuitionistic fuzzy ideal of a / -algebra . Then (0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )) â&#x2030;¤ ( ) and (0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )) â&#x2030;¥ ( ), for all â&#x2C6;&#x2C6; .

Proof: (0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )) â&#x2030;¤ {(0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )) â&#x2C6;&#x2014; } â&#x2C6;¨ ( ) â&#x2030;¤ (0) â&#x2C6;¨ ( ) = ( ), 8:X *YY â&#x2C6;&#x2C6; Therefore, (0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )) â&#x2030;¤ ( ), for all â&#x2C6;&#x2C6; .

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International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

Again, (0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )) â&#x2030;¥ {(0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )) â&#x2C6;&#x2014; } â&#x2C6;§ ( ) â&#x2030;¥ (0) â&#x2C6;§ ( ) = ( ), 8:X *YY â&#x2C6;&#x2C6; Therefore, Z0 â&#x2C6;&#x2014; (0 â&#x2C6;&#x2014; )[ â&#x2030;¥ ( ), for all â&#x2C6;&#x2C6; .

Theorem 3.14 Let = ( , ) be a doubt intuitionistic fuzzy ideal in . Then so is ] = {â&#x152;© , ( ), Ì&#x2026; ( )â&#x152;ª/ â&#x2C6;&#x2C6; }.

Proof: Since = ( , ) is a doubt intuitionistic fuzzy ideal of , then (0) â&#x2030;¤ ( ) and ( ) â&#x2030;¤ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ ( ).

Now, (0) â&#x2030;¤ ( ), :X 1 â&#x2C6;&#x2019; Ì&#x2026; (0) â&#x2030;¤ 1 â&#x2C6;&#x2019; Ì&#x2026; ( ), :X Ì&#x2026; (0) â&#x2030;¥ Ì&#x2026; ( ), for any â&#x2C6;&#x2C6; .

Consider, for any , â&#x2C6;&#x2C6; , ( ) â&#x2030;¤ )* { ( â&#x2C6;&#x2014; ), ( )} ^â&#x201E;&#x17D;,9 I,<;9, 1 â&#x2C6;&#x2019; Ì&#x2026; ( ) â&#x2030;¤ )* {1 â&#x2C6;&#x2019; Ì&#x2026; ( â&#x2C6;&#x2014; ),1 â&#x2C6;&#x2019; Ì&#x2026; ( )} :X, Ì&#x2026; ( ) â&#x2030;¥ 1 â&#x2C6;&#x2019; )* {1 â&#x2C6;&#x2019; Ì&#x2026; ( â&#x2C6;&#x2014; ),1 â&#x2C6;&#x2019; Ì&#x2026; ( )} `,-*YY , Ì&#x2026; ( ) â&#x2030;¥ ),-{ Ì&#x2026; ( â&#x2C6;&#x2014; ), Ì&#x2026; ( )}.

Hence, Î = {( , ( ), Ì&#x2026; ( ))/ â&#x2C6;&#x2C6; } is a doubt intuitionistic fuzzy ideal of .

Theorem 3.15 Let = ( , ) be a doubt intuitionistic fuzzy ideal in . Then so is â&#x2122;¦ = {â&#x152;© , Ì&#x2026; ( ), ( )â&#x152;ª/ â&#x2C6;&#x2C6; }.

Proof: Since = ( , ) is a doubt intuitionistic fuzzy ideal of , then (0) â&#x2030;¥ ( ). Also ( ) â&#x2030;¥ ( â&#x2C6;&#x2014; ) â&#x2C6;§ ( ).

Again we have, (0) â&#x2030;¥ ( ), :X 1 â&#x2C6;&#x2019; Ì&#x2026; (0) â&#x2030;¥ 1 â&#x2C6;&#x2019; Ì&#x2026; ( ), :X Ì&#x2026; (0) â&#x2030;¤ Ì&#x2026; ( ) , for any â&#x2C6;&#x2C6; .

Consider, for any , â&#x2C6;&#x2C6; , ( ) â&#x2030;¥ ),-{ ( â&#x2C6;&#x2014; ), ( )} This implies, 1 â&#x2C6;&#x2019; Ì&#x2026; ( ) â&#x2030;¥ ),-{1 â&#x2C6;&#x2019; Ì&#x2026; ( â&#x2C6;&#x2014; ),1 â&#x2C6;&#x2019; Ì&#x2026; ( )} That is, Ì&#x2026; ( ) â&#x2030;¤ 1 â&#x2C6;&#x2019; ),-{1 â&#x2C6;&#x2019; Ì&#x2026; ( â&#x2C6;&#x2014; ),1 â&#x2C6;&#x2019; Ì&#x2026; ( )} or, Ì&#x2026; ( ) â&#x2030;¤ )* { Ì&#x2026; ( â&#x2C6;&#x2014; ), Ì&#x2026; ( )}.

Hence, â&#x2122;¦ = {â&#x152;© , Ì&#x2026; ( ), ( )â&#x152;ª/ â&#x2C6;&#x2C6; } is a doubt intuitionistic fuzzy ideal of .

Theorem 3.16 Let = ( , ) be an intuitionistic fuzzy set in X. Then = ( , ) is a doubt

intuitionistic fuzzy ideal of if and only if ] = {â&#x152;© , ( ), Ì&#x2026; ( )â&#x152;ª/ â&#x2C6;&#x2C6; } and â&#x2122;¦ = {â&#x152;© , Ì&#x2026; ( ), ( )â&#x152;ª/ â&#x2C6;&#x2C6; } are doubt intuitionistic fuzzy ideals of .

Proof : The proof is same as Theorem 3.14 and Theorem 3.15 Let us illustrate the Theorem 3.16 using the following example.

Example 3.17 Let = {0,1,2,3,4} be a -algebra with the following Cayley table: * 0 1 2

0 0 1 2

1 0 0 2

2 0 1 0

3 0 0 2

4 0 1 0 7

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

3 4

3 4

1 4

3 2

0 4

3 0

Let = ( , ) be a doubt intuitionistic fuzzy ideal of as defined by:

X

0 1 0.4 0.5 0.5 0.4

2 0.6 0.3

3 0.5 0.4

4 0.6 0.3

Then Î = {â&#x152;© , ( ), Ì&#x2026; ( )â&#x152;ª/ â&#x2C6;&#x2C6; }, where ( ) and Ì&#x2026; ( ) are defined as follows:

X

Ì&#x2026;

0 1 0.4 0.5 0.6 0.5

2 0.6 0.4

3 0.5 0.5

4 0.6 0.4

Also â&#x2122;¦ = {â&#x152;© , Ì&#x2026; ( ), ( )â&#x152;ª/ â&#x2C6;&#x2C6; },whose ( ) and Ì&#x2026; ( ) are defined by:

X

Ì&#x2026;

0 1 0.5 0.6 0.5 0.4

2 0.7 0.3

3 0.6 0.4

4 0.7 0.3

So, it can be verified that Î = {â&#x152;© , ( ), Ì&#x2026; ( )â&#x152;ª/ â&#x2C6;&#x2C6; } and â&#x2122;¦ = {â&#x152;© , Ì&#x2026; ( ), ( )â&#x152;ª/ â&#x2C6;&#x2C6; } are doubt intuitionistic fuzzy ideals.

Theorem 3.18 An intuitionistic fuzzy set = ( , ) is a doubt intuitionistic fuzzy ideal of a / -algebra if and only if the fuzzy sets and Ì&#x2026; are doubt fuzzy ideals of .

Proof: Let = ( , ) be a doubt intuitionistic fuzzy ideal of . Then it is obvious that is a doubt fuzzy ideal of . And from Theorem 3.15, we can prove that Ì&#x2026; is a doubt fuzzy ideal of .

Conversely, let be a doubt fuzzy ideal of . Therefore (0) â&#x2030;¤ ( ) and ( ) â&#x2030;¤ )* { ( â&#x2C6;&#x2014; ), ( )}, for all , â&#x2C6;&#x2C6; .

Again since Ì&#x2026; is a doubt intuitionistic fuzzy ideal of , so, Ì&#x2026; (0) â&#x2030;¤ Ì&#x2026; ( ), I,<;9 1 â&#x2C6;&#x2019; (0) â&#x2030;¤ 1 â&#x2C6;&#x2019; ( ), ,)>Y,;9 (0) â&#x2030;¥ ( ).

Also

Ì&#x2026; ( ) â&#x2030;¤ )* { Ì&#x2026; ( â&#x2C6;&#x2014; ), Ì&#x2026; ( )} :X, 1 â&#x2C6;&#x2019; ( ) â&#x2030;¤ )* {1 â&#x2C6;&#x2019; ( â&#x2C6;&#x2014; ),1 â&#x2C6;&#x2019; ( )} :X, ( ) â&#x2030;¥ 1 â&#x2C6;&#x2019; )* {1 â&#x2C6;&#x2019; ( â&#x2C6;&#x2014; ),1 â&#x2C6;&#x2019; ( )} `,-*YY , ( ) â&#x2030;¥ ),-{ ( â&#x2C6;&#x2014; ), ( )}, for all , â&#x2C6;&#x2C6; .

Hence, = ( , ) is a doubt intuitionistic fuzzy ideal of

Example 3.19 Let = {0,1,2,3} be a -algebra with the following Cayley table: * 0 1

0 0 1

1 0 0

2 0 1

.

3 0 1 8

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2 3

2 3

1 1

0 3

0 0

Let = ( , ) be an intuitionistic fuzzy set of as defined by:

X

0 0.1 0.8

1 0.5 0.4

2 0.7 0.2

3 0.6 0.4

Since (2) = 0.7 â&#x2030;° 0 .6 = )* { (2 â&#x2C6;&#x2014; 3), (3)}, and (2) = 0.2 â&#x2030;± 0 .4 = ),-{ (2 â&#x2C6;&#x2014; 3), (3)}. Therefore, = ( , ) is not a doubt intuitionistic fuzzy ideal of .

Corollary 3.20 Let = ( , ) be a doubt intuitionistic fuzzy ideal of a / -algebra .

Then the sets, cde = { â&#x2C6;&#x2C6; / ( ) = (0)}, *-7 cfe = { â&#x2C6;&#x2C6; / ( ) = (0)} are ideals of .

Proof: Let = ( , ) be a doubt intuitionistic fuzzy ideal of . Obviously, 0 â&#x2C6;&#x2C6; cde *-7 cfe .

Now, let , â&#x2C6;&#x2C6; , such that â&#x2C6;&#x2014; , â&#x2C6;&#x2C6; cde . Then ( â&#x2C6;&#x2014; ) = (0) = ( ).Now, ( ) â&#x2030;¤ )* { ( â&#x2C6;&#x2014; ), ( )} = (0). Again, since is a doubt fuzzy ideal of X, (0) â&#x2030;¤ ( ).

Therefore, (0) = ( ). Hence, â&#x2C6;&#x2C6; cde .

Therefore, cde is an ideal of .

Following the same way we can prove that cfe is also an ideal of .

Theorem 3.21 If and are two doubt intuitionistic fuzzy ideals of , if one is contained another then â&#x2C6;© is also a doubt intuitionistic fuzzy ideal of .

Proof: Let = ( , ) and = ( h , h ) be two doubt intuitionistic fuzzy ideals of .

Again let, = â&#x2C6;© = ( i , i ), where i = â&#x2C6;§ h and i = â&#x2C6;¨ h .

Let , â&#x2C6;&#x2C6; , then i (0) = (0) â&#x2C6;§ h (0) â&#x2030;¤ ( ) â&#x2C6;§ h ( ) = i ( ) and i (0) = (0) â&#x2C6;¨ h (0) â&#x2030;¥ ( ) â&#x2C6;¨ h ( ) = i ( ) Y9:, i ( ) = ( ) â&#x2C6;§ h ( ) â&#x2030;¤ )* [ ( â&#x2C6;&#x2014; ), ( )] â&#x2C6;§ )* [ h ( â&#x2C6;&#x2014; ), h ( )] = )* {[ ( â&#x2C6;&#x2014; ) â&#x2C6;§ h ( â&#x2C6;&#x2014; )], [ ( ) â&#x2C6;§ h ( )]} [ L;M*N9; :-; ,9 M:-j*,-;7 *-:jâ&#x201E;&#x17D;;X ] = )* [ i ( â&#x2C6;&#x2014; ), i ( )].

Similarly, we can prove that, i ( ) â&#x2030;¥ ),-[ i ( â&#x2C6;&#x2014; ), i ( )]. This completes proof.

the

Theorem 3.22 Union of any two doubt intuitionistic fuzzy ideals of X, is also a doubt intuitionistic fuzzy ideal of X. 9

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

Proof: Let = ( , ) and = ( h , h ) be two doubt intuitionistic fuzzy ideals of . Again let, = â&#x2C6;ª = ( i , i ), where i = â&#x2C6;¨ h and l = â&#x2C6;§ h .

Let , â&#x2C6;&#x2C6; , then, i (0) = (0) â&#x2C6;¨ h (0) â&#x2030;¤ ( ) â&#x2C6;¨ h ( ) = i ( ) *-7 i (0) = (0) â&#x2C6;§ h (0) â&#x2030;¥ ( ) â&#x2C6;§ h ( ) = i ( ) Y9:, i ( ) = ( ) â&#x2C6;¨ h ( ) â&#x2030;¤ )* [ ( â&#x2C6;&#x2014; ), ( )] â&#x2C6;¨ )* [ h ( â&#x2C6;&#x2014; ), h ( )] = )* {[ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ h ( â&#x2C6;&#x2014; )], [ ( ) â&#x2C6;¨ h ( )]} = )* [ i ( â&#x2C6;&#x2014; ), i ( )].

Similarly, we can prove that, i ( ) â&#x2030;¥ ),-[ i ( â&#x2C6;&#x2014; ), i ( )].

This completes the proof.

Example 3.23 Let = {0,1,2,3,4} be a -algebra [by Example 3.11] with the following Cayley table: 0 0 1 2 3 4

* 0 1 2 3 4

1 0 0 2 3 2

2 0 0 0 0 1

3 0 0 2 0 2

4 0 0 0 0 0

Let = ( , ) be an intuitionistic fuzzy set of as defined by:

X

0 0 1

1 2 3 4 0.62 0.65 0.62 0.65 0.34 0.32 0.34 0.32

Then = ( , ) is a doubt intuitionistic fuzzy ideal of .

Again, let = ( h , h ) be an intuitionistic fuzzy set of as defined by:

X h h

0 1 2 3 4 0. 22 0.56 0.58 0.56 0.58 0.72 0.44 0.42 0.44 0.42

Then = ( h , h ) is a doubt intuitionistic fuzzy ideal of .

Now let, m = â&#x2C6;© = ( n , n ) where n = â&#x2C6;§ h and n = â&#x2C6;¨ h .

Then m is an intuitionistic fuzzy set of which can be defined as:

X n

0 0

1 2 3 4 0.56 0.58 0.56 0.58 10

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n

1

0.44 0.42 0.44 0.42

Then it is clear that m = ( n , n ) is a doubt intuitionistic fuzzy ideal of .

We also assume that o = â&#x2C6;ª = ( l , l ) where l = â&#x2C6;¨ h and l = â&#x2C6;§ h and o is defined as:

X l l

0 1 2 3 4 0. 22 0.62 0.65 0.62 0.65 0.72 0.34 0.32 0.34 0.32

Then o = ( l , l ) is a doubt intuitionistic fuzzy ideal of .

Definition 3.24 Let = ( , ) be an intuitionistic fuzzy set of , and j, 9 â&#x2C6;&#x2C6; [0,1], then level j-cut and level 9-cut of , is as followes:

q ,p = { â&#x2C6;&#x2C6; / ( ) â&#x2030;¤ j} s and ,r = { â&#x2C6;&#x2C6; / ( ) â&#x2030;¥ 9}. q s Theorem 3.25 If = ( , ) be a doubt intuitionistic fuzzy ideal of , then ,p and ,r are

ideals of for j, 9 â&#x2C6;&#x2C6; [0,1].

Proof: Let = ( , ) be a doubt intuitionistic fuzzy ideal of , and let j â&#x2C6;&#x2C6; [0,1] with q (0) â&#x2030;¤ j. Then we have, (0) â&#x2030;¤ ( ), 8:X *YY â&#x2C6;&#x2C6; , but ( ) â&#x2030;¤ j, 8:X *YY â&#x2C6;&#x2C6; ,p . q So, 0 â&#x2C6;&#x2C6; ,p . q q q q Let , â&#x2C6;&#x2C6; be such that â&#x2C6;&#x2014; â&#x2C6;&#x2C6; ,p and â&#x2C6;&#x2C6; ,p , then, ( â&#x2C6;&#x2014; ) â&#x2C6;&#x2C6; ,p and ( ) â&#x2C6;&#x2C6; ,p .

Therefore, ( â&#x2C6;&#x2014; ) â&#x2030;¤ j and ( ) â&#x2030;¤ j .

Since is a doubt fuzzy ideal of , it follows that, ( ) â&#x2030;¤ ( â&#x2C6;&#x2014; ) â&#x2C6;¨ ( ) â&#x2030;¤ j and hence q â&#x2C6;&#x2C6; ,p . q Therefore, ,p is ideal of for j â&#x2C6;&#x2C6; [0,1]. s Similarly, we can prove that ,r is an ideal of for 9 â&#x2C6;&#x2C6; [0,1] .

q s Theorem 3.26 If ,p and ,r are either empty or ideals of for j, 9 â&#x2C6;&#x2C6; [0,1], then = [ , ] is a doubt intuitionistic fuzzy ideal of . q s Proof: Let ,p and ,r be either empty or ideals of for j, 9 â&#x2C6;&#x2C6; [0,1]. For any â&#x2C6;&#x2C6; , let q s q s q s ( ) = j *-7 ( ) = 9. Then â&#x2C6;&#x2C6; ,p â&#x2C6;§ ,r , so ,p â&#x2030; u â&#x2030; ,r . Since ,p and ,r are q s ideals of , therefore 0 â&#x2C6;&#x2C6; ,p â&#x2C6;§ ,r .

Hence, (0) â&#x2030;¤ j = ( ) and (0) â&#x2030;¥ 9 = ( ), where â&#x2C6;&#x2C6; .

If there exist v , v â&#x2C6;&#x2C6; such that ( v ) > )* { ( v â&#x2C6;&#x2014; yv ), ( v )}, then by taking, jx = C ( ( v ) + )* { ( v â&#x2C6;&#x2014; v ), ( v )}). @ q q We have, ( v ) > jx > )* { ( v â&#x2C6;&#x2014; v ), ( v )} . Hence v â&#x2C6;&#x2030; ,p , ( v â&#x2C6;&#x2014; v ) â&#x2C6;&#x2C6; ,p and z z q q v â&#x2C6;&#x2C6; ,pz , that is ,pz is not an ideal of , which is a contradiction.

Finally, assume that there exist >, E â&#x2C6;&#x2C6; such that (>) < ),-{ (> â&#x2C6;&#x2014; E), (E)}. Taking C 9x = @ ( (>) + ),-{ (> â&#x2C6;&#x2014; E), (E)}), then ),-{ (> â&#x2C6;&#x2014; E), (E)} > 9x > (>). s s s Therefore, > â&#x2C6;&#x2014; E â&#x2C6;&#x2C6; ,r and E â&#x2C6;&#x2C6; ,r but > â&#x2C6;&#x2030; λ ,r . Again a contradiction. This completes the 11

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015

proof.

But, if an intuitionistic fuzzy set = ( , ), is not a doubt intuitionistic fuzzy ideal of , then q s and ,r are not ideals of for j, 9 â&#x2C6;&#x2C6; [0,1], which is illustrated in the following example. ,p

Example 3.27 Let = {0,1,2,3} be a -algebra in Example 3.19 with the following Cayley table: * 0 1 2 3

0 0 1 2 3

1 0 0 1 1

2 0 1 0 3

3 0 1 0 0

Let = ( , ) be an intuitionistic fuzzy set of as defined by:

X

0 0.1 0.8

1 0.5 0.4

2 0.7 0.2

3 0.6 0.4

which is not a doubt intuitionistic fuzzy ideal of . q s For j = 0.67 and 9 = 0.25, we get ,~ = ,r = {0,1,3}, which are not ideals of , as 2 â&#x2C6;&#x2014; 1 = 1 â&#x2C6;&#x2C6; {0,1,3}, and 1 â&#x2C6;&#x2C6; {0,1,3}, but 2 â&#x2C6;&#x2030; {0,1,3}.

4. CONCLUSIONS We see that to develope the theory of / -algebras the ideal theory plays an important role.

In this paper, we have defined doubt intuitionistic fuzzy subalgebras and doubt intuitionistic fuzzy ideals in / -algebras and have developed several characterizations of doubt intuitionistic fuzzy subalgebras and doubt intuitionistic fuzzy ideals in / -algebras. In terms of the above notion we can come to this conclusion that the research along this direction can be continued, and in fact some results in this paper have already constituted a foundation for further investigation concerning the further developments of intuitionistic fuzzy / -algebras and their applications in other branches of algebra. In the future study of intuitionistic fuzzy / -algebras, perhaps the following topic are worth to be considered: 1. To characterize other classes of / -algebras by using this notion.

2. To apply this notion to some other algebraic structures.

3. To consider these results to some possible application in computer sciences and information systems in the future.

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K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.

K. T. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems, 61 (1994), 137-142.

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Y. Imai and K. Iseki, On axiom systems of propositional calculi, Proc. Japan Academy, 42 (1966), 19-22.

K. Iseki, An algebra related with a propositional calculus, Proc. Of Japan Academy, 42 (1966), 26-29.

K. Iseki, On BCI-algebras, Math. Seminar Notes, 8 (1980), 125-130.

O. G. Xi, Fuzzy BCK-algebras, Math. Japon. 24(36) (1991), 935-942.

F. Y. Huang, Another definition of fuzzy BCI-algebras, Selected Papers on BCK/BCI-Algebras, China, 1 (1992), 91-92.

Y. B. Jun, Doubt fuzzy BCK/BCI-algebras, Soochow J. Math., 20(3) (1994), 351-358.

Y. B. Jun and K. H. Kim, Intuitionistic fuzzy ideals of BCK-algebras, Internat. J. Math and Math. Sci., 24 (2000), 839-849.

A. Solairaju and R. Begam, Intuitionistic fuzzifications of P-ideals in BCI-algebras, Internat. J. Math Research, 4(2) (2000), 133-144.

T. Senapati, M. Bhowmik and M. Pal, Atanassov’s intuitionistic fuzzy translations of intuitionistic fuzzy H-ideals in BCK/BCI-algebras, Notes on Intuitionistic Fuzzy Sets, 19 (1) (2013), 32-47.

T. Senapati, M. Bhowmik and M. Pal, Interval-valued intuitionistic fuzzy BG-subalgebras, The Journal of Fuzzy Mathematics, 20 (3) (2012), 707-720.

T. Senapati, M. Bhowmik and M. Pal, Interval-valued intuitionistic fuzzy closed ideals of BG-algebra and their products, Int. J. Fuzzy Logic Systems, 2 (2) (2012), 27-44.

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Authors Mrs. Tripti Bej received her Bachelor of Science degree with Honours in Mathematics in 2004 from Garbeta College, Paschim Medinipur, West Bengal, India and Master of Science in Mathematics from Vidyasagar University, West Bengal, India in 2006. Her research interest includes fuzzy sets, intuitionist fuzzy sets and fuzzy algebras.

Dr. Madhumangal Pal is currently a Professor of Applied Mathematics, Vidyasagar University, India. He has received Gold and Silver medals from Vidyasagar University for rank first and second in M.Sc. and B.Sc. examinations respectively. Also he received jointly with Prof. G.P.Bhattacherjee, “Computer Division Medal” from Institute of Engineers (India) in 1996 for best research work. Prof. Pal has successfully guided 19 research scholars for Ph.D. degrees and has published more than 175 articles in international and national journals. His specializations include Algorithmic Graph Theory, Fuzzy Correlation & Regression, Fuzzy Game Theory, Fuzzy Matrices, Genetic Algorithms and Parallel Algorithms.Prof. Pal is the Editor-in-Chief of “Journal of Physical Sciences” and “Annals of Pure and Applied Mathematics”, and member of the editorial Boards of several journals. Prof. Pal is the author of eight books published from Indian and abroad.He organized several national seminars/ conferences/ workshops. Also, visited China, London, Malaysia, Thailand, Hong Kong and Bangladesh to participated, delivered invited talks and chaired in national and international seminars/ conferences/ winter school/ refresher course.He is the member of several administrative and academic bodies in Vidyasagar University and other institutes. He is also member of the Calcutta Mathematical Society, Advanced Discrete Mathematics and Application, etc.

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