DOUBT INTUITIONISTIC FUZZY IDEALS IN BCK/BCI-ALGEBRAS

International Journal of Fuzzy Logic Systems (IJFLS) Vol.5, No.1, January 2015
DOI : 10.5121/ijfls.2015.5101 1
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 ‫/ܭܥܤ‬
‫-ܫܥܤ‬algebras, ‫-ܩܤ‬algebra and ‫-ܤ‬algebra [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

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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 ‫-ܫܥܤ‬algebra if it satisfies the following axioms for all
‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ ∈ ܺ :
(A1) ((‫ݔ‬ ∗ ‫)ݕ‬ ∗ (‫ݔ‬ ∗ ‫))ݖ‬ ∗ (‫ݖ‬ ∗ ‫)ݕ‬ = 0
(A2) (‫ݔ‬ ∗ (‫ݔ‬ ∗ ‫))ݕ‬ ∗ ‫ݕ‬ = 0
(A3) ‫ݔ‬ ∗ ‫ݔ‬ = 0
(A4) ‫ݔ‬ ∗ ‫ݕ‬ = 0 and ‫ݕ‬ ∗ ‫ݔ‬ = 0 imply ‫ݔ‬ = ‫ݕ‬
If a ‫-ܫܥܤ‬algebra ܺ satisfies 0 ∗ ‫ݔ‬ = 0. Then ܺ is called a ‫-ܭܥܤ‬algebra.
In a ‫-ܫܥܤ/ܭܥܤ‬algebra, ‫ݔ‬ ∗ 0 = ‫ݔ‬ hold. A partial ordering "≤" on a ‫-ܫܥܤ/ܭܥܤ‬algebra ܺ can be
defined by ‫ݔ‬ ≤ ‫ݕ‬ if and only if ‫ݔ‬ ∗ ‫ݕ‬ = 0.
Any ‫-ܭܥܤ‬algebra ܺ satisfies the following axioms for all ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ ∈ ܺ :
(i) (‫ݔ‬ ∗ ‫)ݕ‬ ∗ ‫ݖ‬ = (‫ݔ‬ ∗ ‫)ݖ‬ ∗ ‫ݕ‬
(ii) ‫ݔ‬ ∗ ‫ݕ‬ ≤ ‫ݔ‬
(iii) (‫ݔ‬ ∗ ‫)ݕ‬ ∗ ‫ݖ‬ ≤ (‫ݔ‬ ∗ ‫)ݖ‬ ∗ (‫ݕ‬ ∗ ‫)ݖ‬
(iv) ‫ݔ‬ ≤ ‫ݕ‬ ⇒ ‫ݔ‬ ∗ ‫ݖ‬ ≤ ‫ݕ‬ ∗ ‫,ݖ‬ ‫ݖ‬ ∗ ‫ݕ‬ ≤ ‫ݖ‬ ∗ ‫.ݔ‬
If a ‫ܭܥܤ‬ -algebra satisfies ‫ݔ‬ ∗ (‫ݔ‬ ∗ ‫)ݕ‬ = ‫ݕ‬ ∗ (‫ݕ‬ ∗ ‫)ݔ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ, then it is called
commutative.
Throughout this paper, ܺ always means a ‫-ܫܥܤ/ܭܥܤ‬algebra without any specification.
A non-empty subset ܵ of a ‫-ܫܥܤ/ܭܥܤ‬algebra ܺ is called a subalgebra of ܺ if ‫ݔ‬ ∗ ‫ݕ‬ ∈ ܵ for any
‫,ݔ‬ ‫ݕ‬ ∈ ܵ.
A non-empty subset ‫ܫ‬ of a ‫-ܫܥܤ/ܭܥܤ‬algebra ܺ is called an ideal of ܺ if
(i) 0 ∈ ‫ܫ‬
(ii) ‫ݔ‬ ∗ ‫ݕ‬ ∈ ‫ܫ‬ and ‫ݕ‬ ∈ ‫ܫ‬ then ‫ݔ‬ ∈ ‫,ܫ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
The proposed work is done on intuitionistic fuzzy set. The formal definition of intuitionistic fuzzy
set is given below:
An intuitionistic fuzzy set ‫ܣ‬ in a non-empty set ܺ is an object having the form ‫ܣ‬ =
{‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߣ஺(‫ݔ/)ݔ‬ ∈ ܺ} , where the function ߤ஺: ܺ → [0,1] and ߣ஺: ܺ → [0,1] , denote the
degree of membership and the degree of non-membership of each element ‫ݔ‬ ∈ ܺ to the set ‫ܣ‬
respectively and 0 ≤ ߤ஺(‫)ݔ‬ + ߣ஺(‫)ݔ‬ ≤ 1, for all ‫ݔ‬ ∈ ܺ.
For the sake of simplicity, we use the symbol form ‫ܣ‬ = (ܺ, ߤ஺, ߣ஺) or (ߤ஺, ߣ஺) for the
intuitionistic fuzzy set ‫ܣ‬ = {〈‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߣ஺(‫ݔ/〉)ݔ‬ ∈ ܺ}.
The two operators used in this paper are defined as:
If ‫ܣ‬ = (ߤ஺, ߣ஺), is an intuitionistic fuzzy set, then,

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Π‫ܣ‬ = {(‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߤ̅஺(‫ݔ/))ݔ‬ ∈ ܺ}
♦‫ܣ‬ = {(‫,ݔ‬ ߣ̅஺(‫,)ݔ‬ ߣ஺(‫ݔ/))ݔ‬ ∈ ܺ}.
For the sake of simplicity, we also use ‫ݔ‬ ∨ ‫ݕ‬ for ݉ܽ‫,ݔ(ݔ‬ ‫,)ݕ‬ and ‫ݔ‬ ∧ ‫ݕ‬ for ݉݅݊(‫,ݔ‬ ‫.)ݕ‬
Intuitionistic fuzzy subalgebra and intuitionistic fuzzy ideal are the extension of fuzzy subalgebra
and fuzzy ideal which are defined below :
An intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) in ܺ, is called an intuitionistic fuzzy subalgebra [10] of ܺ
if it satisfies the following two conditions:
(i) ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≥ ߤ஺(‫)ݔ‬ ∧ ߤ஺(‫,)ݕ‬ (ii) ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߣ஺(‫)ݔ‬ ∨ ߣ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
An intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) in ܺ is called an intuitionistic fuzzy ideal [10] of ܺ, if it
satisfies the following axioms:
(i) ߤ஺(0) ≥ ߤ஺(‫,)ݔ‬ ߣ஺(0) ≤ ߣ஺(‫,)ݔ‬
(ii) ߤ஺(‫)ݔ‬ ≥ ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∧ ߤ஺(‫,)ݕ‬
(iii) ߣ஺(‫)ݔ‬ ≤ ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߣ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
In order to resolve the contradiction popped up in Huang’s [8] definition of fuzzy ‫-ܫܥܤ‬algebra, Jun
[9] introduced the definition of doubt fuzzy subalgebra and doubt fuzzy ideals in ‫/ܭܥܤ‬
‫-ܫܥܤ‬algebras, which are as follows:
A fuzzy set ‫ܣ‬ = {〈‫,ݔ‬ ߤ஺(‫:〉)ݔ‬ ‫ݔ‬ ∈ ܺ} in ܺ is called a doubt fuzzy subalgebra [9] of ܺ if
ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߤ஺(‫)ݔ‬ ∨ ߤ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
A fuzzy set ‫ܣ‬ = {〈‫,ݔ‬ ߤ஺(‫:〉)ݔ‬ ‫ݔ‬ ∈ ܺ} in ܺ is called a doubt fuzzy ideal [9] of ܺ if
(i) ߤ஺(0) ≤ ߤ஺(‫)ݔ‬
(ii) ߤ஺(‫)ݔ‬ ≤ ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߤ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
3. DOUBT INTUITIONISTIC FUZZY IDEALS IN BCK/BCI-ALGEBRAS
In this section, we define doubt intuitionistic fuzzy subalgebra and doubt intuitionistic fuzzy ideal
in ‫-ܫܥܤ/ܭܥܤ‬algebras and investigate their properties.
Definition 3.1 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be an intuitionistic fuzzy subset of a ‫-ܫܥܤ/ܭܥܤ‬algebra ܺ, then
‫ܣ‬ is called a doubt intuitionistic fuzzy subalgebra of ܺ if
(i) ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߤ஺(‫)ݔ‬ ∨ ߤ஺(‫,)ݕ‬
(ii) ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≥ ߣ஺(‫)ݔ‬ ∧ ߣ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
Theorem 3.2 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy subalgebra of ܺ, then
(i) ߤ஺(0) ≤ ߤ஺(‫)ݔ‬ and
(ii) ߣ஺(0) ≥ ߣ஺(‫,)ݔ‬ for all ‫ݔ‬ ∈ ܺ.
Proof: By definition, ߤ஺(0) = ߤ஺(‫ݔ‬ ∗ ‫)ݔ‬ ≤ ߤ஺(‫)ݔ‬ ∨ ߤ஺(‫)ݔ‬ ≤ ߤ஺(‫.)ݔ‬ Therefore, ߤ஺(0) ≤
ߤ஺(‫,)ݔ‬ for all ‫ݔ‬ ∈ ܺ.
Again, ߣ஺(0) = ߣ஺(‫ݔ‬ ∗ ‫)ݔ‬ ≥ ߣ஺(‫)ݔ‬ ∧ ߣ஺(‫)ݔ‬ ≥ ߣ஺(‫.)ݔ‬ Therefore, ߣ஺(0) ≥ ߣ஺(‫,)ݔ‬ for all ‫ݔ‬ ∈ ܺ.

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Theorem 3.3 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy subalgebra of a ‫/ܭܥܤ‬
‫-ܫܥܤ‬algebra ܺ. Then for any ‫ݔ‬ ∈ ܺ, we have
(i) ߤ஺(‫ݔ‬௡
∗ ‫)ݔ‬ ≤ ߤ஺(‫ߣ ݀݊ܽ ,)ݔ‬஺(‫ݔ‬௡
∗ ‫)ݔ‬ ≥ ߣ஺(‫,)ݔ‬ ݂݅ ݊ ݅‫݀݀݋ ݏ‬
(ii) ߤ஺(‫ݔ‬௡
∗ ‫)ݔ‬ = ߤ஺(‫ߣ ݀݊ܽ ,)ݔ‬஺(‫ݔ‬௡
∗ ‫)ݔ‬ = ߣ஺(‫,)ݔ‬ ݂݅ ݊ ݅‫݊݁ݒ݁ ݏ‬
Proof: (i) Let ‫ݔ‬ ∈ ܺ, then ߤ஺(‫ݔ‬ ∗ ‫)ݔ‬ = ߤ஺(0) ≤ ߤ஺(‫.)ݔ‬
If ݊ is odd then let ݊ = 2‫݌‬ − 1 for some positive integer ‫.݌‬
Now assume that ߤ஺(‫ݔ‬ଶ௣ିଵ
∗ ‫)ݔ‬ ≤ ߤ஺(‫)ݔ‬ for some positive integer ‫.݌‬
Then,
ߤ஺(‫ݔ‬ଶ(௣ାଵ)ିଵ
∗ ‫)ݔ‬ = ߤ஺(‫ݔ‬ଶ௣ାଵ
∗ ‫,)ݔ‬
= ߤ஺(‫ݔ‬ଶ௣ିଵ
∗ (‫ݔ‬ ∗ (‫ݔ‬ ∗ ‫)))ݔ‬
∗ (‫ݔ‬ ∗ 0))
∗ ‫)ݔ‬
≤ ߤ஺(‫.)ݔ‬
Hence, ߤ஺(‫ݔ‬௡
∗ ‫)ݔ‬ ≤ ߤ஺(‫,)ݔ‬ if ݊ is odd. This proves the first part. Similarly, we can prove the
second part.
(ii) Again, let ݊ be even, and ݊ = 2‫.ݍ‬
Now for ‫ݍ‬ = 1, ߤ஺(‫ݔ‬ଶ
∗ ‫)ݔ‬ = ߤ஺(‫ݔ‬ ∗ (‫ݔ‬ ∗ ‫))ݔ‬ = ߤ஺(‫ݔ‬ ∗ 0) = ߤ஺(‫.)ݔ‬
Also assume that, ߤ஺(‫ݔ‬ଶ௤
∗ ‫)ݔ‬ = ߤ஺(‫)ݔ‬ for some positive integer ‫.ݍ‬
Then,
ߤ஺(‫ݔ‬ଶ(௤ାଵ)
∗ ‫)ݔ‬ = ߤ஺(‫ݔ‬ଶ௤
∗ (‫ݔ‬ ∗ (‫ݔ‬ ∗ ‫,)))ݔ‬
= ߤ஺(‫ݔ‬ଶ௤
∗ ‫)ݔ‬
= ߤ஺(‫.)ݔ‬
Hence, ߤ஺(‫ݔ‬௡
∗ ‫)ݔ‬ = ߤ஺(‫,)ݔ‬ if ݊ is even. This proves the first part. Similarly, we can prove the
second part.

In a ‫-ܭܥܤ‬algebra, 0 ∗ ‫ݔ‬ = 0. Hence we conclude the following proposition.
Proposition 3.4 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy-subalgebra of a ‫-ܭܥܤ‬algebra
ܺ. Then for any ‫ݔ‬ ∈ ܺ, we have, ߤ஺(‫ݔ‬ ∗ ‫ݔ‬௡
) = ߤ஺(0), ܽ݊݀ ߣ஺(‫ݔ‬ ∗ ‫ݔ‬௡
) = ߣ஺(0), for ݊ =
1,2,3, …
Example 3.5 Consider a ‫-ܭܥܤ‬algebra ܺ = {0,1,2,3} with the following Cayley table:
Let ‫ܣ‬ = (ߤ஺, ߣ஺) be an intuitionistic fuzzy set of ܺ as defined by:
X 0 1 2 3
ߤ஺ .5 .5 .6 .5
ߣ஺ .5 .5 .3 .5
Then it can easily be verified that ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy subalgebra of ܺ.
Definition 3.6 Let ܺ be a ‫-ܫܥܤ/ܭܥܤ‬algebra. An intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) in ܺ is
called a doubt intuitionistic fuzzy ideal if
* 0 1 2 3
0 0 0 0 0
1 1 0 0 1
2 2 1 0 2
3 3 3 3 0

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(F1) ߤ஺(0) ≤ ߤ஺(‫,)ݔ‬ ߣ஺(0) ≥ ߣ஺(‫)ݔ‬
(F2) ߤ஺(‫)ݔ‬ ≤ ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߤ஺(‫)ݕ‬
(F3) ߣ஺(‫)ݔ‬ ≥ ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∧ ߣ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
Theorem 3.7 Let an intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) in ܺ be a doubt intuitionistic fuzzy
ideal of ܺ. If the inequility ‫ݔ‬ ∗ ‫݌‬ ≤ ‫ݍ‬ holds in ܺ, then
(i) ߤ஺(‫)ݔ‬ ≤ ݉ܽ‫ߤ{ݔ‬஺(‫,)݌‬ ߤ஺(‫})ݍ‬
(ii) ߣ஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫,)݌‬ ߣ஺(‫.})ݍ‬
Proof: Let ‫,ݔ‬ ‫,݌‬ ‫ݍ‬ ∈ ܺ be such that ‫ݔ‬ ∗ ‫݌‬ ≤ ‫ݍ‬ then (‫ݔ‬ ∗ ‫)݌‬ ∗ ‫ݍ‬ = 0 and thus,
ߤ஺(‫)ݔ‬ ≤ ݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ ∗ ‫,)݌‬ ߤ஺(‫,})݌‬
≤ ݉ܽ‫ߤ{ݔܽ݉{ݔ‬஺((‫ݔ‬ ∗ ‫)݌‬ ∗ ‫,)ݍ‬ ߤ஺(‫,})ݍ‬ ߤ஺(‫})݌‬
= ݉ܽ‫ߤ{ݔܽ݉{ݔ‬஺(0), ߤ஺(‫,})ݍ‬ ߤ஺(‫})݌‬
= ݉ܽ‫ߤ{ݔ‬஺(‫,)ݍ‬ ߤ஺(‫})݌‬
Therefore, ߤ஺(‫)ݔ‬ ≤ ݉ܽ‫ߤ{ݔ‬஺(‫,)݌‬ ߤ஺(‫.})ݍ‬
‫ߣ ,݊݅ܽ݃ܣ‬஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫ݔ‬ ∗ ‫,)݌‬ ߣ஺(‫,})݌‬
≥ ݉݅݊{݉݅݊{ߣ஺((‫ݔ‬ ∗ ‫)݌‬ ∗ ‫,)ݍ‬ ߣ஺(‫,})ݍ‬ ߣ஺(‫})݌‬
= ݉݅݊{݉݅݊{ߣ஺(0), ߣ஺(‫,})ݍ‬ ߣ஺(‫})݌‬
= ݉݅݊{ߣ஺(‫,)ݍ‬ ߣ஺(‫})݌‬
= ݉݅݊{ߣ஺(‫,)݌‬ ߣ஺(‫})ݍ‬
Therefore, ߣ஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫,)݌‬ ߣ஺(‫.})ݍ‬ This completes the proof.
Corollary 3.8 If ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of ܺ, then for any
‫,ݔ‬ ‫݌‬ଵ, ‫݌‬ଶ, ‫݌‬ଷ, … , ‫݌‬௡ ∈ ܺ and (⋯ (‫ݔ‬ ∗ ‫݌‬ଵ) ∗ ‫݌‬ଶ) ∗ ⋯ ) ∗ ‫݌‬௡ = 0,
(݅)ߤ஺(‫)ݔ‬ ≤ ݉ܽ‫ߤ{ݔ‬஺(‫݌‬ଵ), ߤ஺(‫݌‬ଶ), ߤ஺(‫݌‬ଷ), … , ߤ஺(‫݌‬௡)},
(݅݅)ߣ஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫݌‬ଵ), ߣ஺(‫݌‬ଶ), ߣ஺(‫݌‬ଷ), … , ߣ஺(‫݌‬௡)}.
Proposition 3.9 Let ‫ܣ‬ = (ߤ, ߣ) be a doubt intuitionistic fuzzy ideal of a ‫-ܫܥܤ/ܭܥܤ‬algebra
ܺ.Then the followings hold for all ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ ∈ ܺ,
(i) if ‫ݔ‬ ≤ ‫ݕ‬ then ߤ(‫)ݔ‬ ≤ ߤ(‫,)ݕ‬ ߣ(‫)ݔ‬ ≥ ߣ(‫.)ݕ‬
(ii) ߤ(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߤ(‫ݔ‬ ∗ ‫)ݖ‬ ∨ ߤ(‫ݖ‬ ∗ ‫ݔ(ߣ ݀݊ܽ )ݕ‬ ∗ ‫)ݕ‬ ≥ ߣ(‫ݔ‬ ∗ ‫)ݖ‬ ∧ ߣ(‫ݖ‬ ∗ ‫.)ݕ‬
Proof: (i) If ‫ݔ‬ ≤ ‫ݕ‬ then ‫ݔ‬ ∗ ‫ݕ‬ = 0.
Hence ߤ(‫)ݔ‬ ≤ ߤ(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߤ(‫)ݕ‬ = ߤ(0) ∨ ߤ(‫)ݕ‬ = ߤ(‫,)ݕ‬
and ߣ(‫)ݔ‬ ≥ ߣ(‫ݔ‬ ∗ ‫)ݕ‬ ∧ ߣ(‫)ݕ‬ = ߣ(0) ∧ ߣ(‫)ݕ‬ = ߣ(‫.)ݕ‬
(ii) Since (‫ݔ‬ ∗ ‫)ݕ‬ ∗ (‫ݔ‬ ∗ ‫)ݖ‬ ≤ (‫ݖ‬ ∗ ‫.)ݕ‬
It follows from (i) that, ߤ{(‫ݔ‬ ∗ ‫)ݕ‬ ∗ (‫ݔ‬ ∗ ‫})ݖ‬ ≤ ߤ(‫ݖ‬ ∗ ‫.)ݕ‬ Now ߤ(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߤ{(‫ݔ‬ ∗ ‫)ݕ‬ ∗ (‫ݔ‬ ∗ ‫})ݖ‬ ∨
ߤ(‫ݔ‬ ∗ ‫,)ݖ‬ [ܾ݁ܿܽ‫ܣ ݁ݏݑ‬ = (ߤ, ߣ) is a doubt intuitionistic fuzzy ideal.].
Therefore ߤ(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߤ(‫ݖ‬ ∗ ‫)ݕ‬ ∨ ߤ(‫ݔ‬ ∗ ‫ )ݖ‬ .Again, ߣ(‫ݔ‬ ∗ ‫)ݕ‬ ≥ ߣ{(‫ݔ‬ ∗ ‫)ݕ‬ ∗ (‫ݔ‬ ∗ ‫})ݖ‬ ∧ ߣ(‫ݔ‬ ∗ ‫)ݖ‬ ≥
ߣ(‫ݖ‬ ∗ ‫)ݕ‬ ∧ ߣ(‫ݔ‬ ∗ ‫.)ݖ‬
Therefore ߣ(‫ݔ‬ ∗ ‫)ݕ‬ ≥ ߣ(‫ݔ‬ ∗ ‫)ݖ‬ ∧ ߣ(‫ݖ‬ ∗ ‫.)ݕ‬
Theorem 3.10 Every doubt intuitionistic fuzzy ideal of a ‫-ܭܥܤ‬algebra ܺ is a doubt intuitionistic
fuzzy subalgebra of ܺ.
Proof: Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal of ܺ.
Since ‫ݔ‬ ∗ ‫ݕ‬ ≤ ‫,ݔ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ. Then ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߤ஺(‫)ݔ‬ and ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≥ ߣ஺(‫.)ݔ‬
So, ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ߤ஺(‫)ݔ‬ ≤ ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߤ஺(‫)ݕ‬ ≤ ߤ஺(‫)ݔ‬ ∨ ߤ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ, [because ‫ܣ‬ =
(ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal.].

6
And ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≥ ߣ஺(‫)ݔ‬ ≥ ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∧ ߣ஺(‫)ݕ‬ ≥ ߣ஺(‫)ݔ‬ ∧ ߣ஺(‫,)ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
This shows that, ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy subalgebra of ܺ.
Example 3.11 Let ܺ = {0,1,2,3,4} be a ‫-ܭܥܤ‬algebra with the following Cayley table:
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 0 0 0
2 2 2 0 2 0
3 3 3 0 0 0
4 4 2 1 2 0
X 0 1 2 3 4
ߤ஺ 0 0.4 0.6 0.6 0.6
ߣ஺ 1 0.5 0.4 0.4 0.4
Then ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of ܺ.
Thus ‫ܣ‬ 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 ߣ஺(2) = 0.3, ߣ஺(2) = 0.3 ≱ 0.5 = ߣ஺(2 ∗ 1) ∧ ߣ஺(1),
Therefore, ‫ܣ‬ = (ߤ஺, ߣ஺) is not a doubt intuitionistic fuzzy ideal.
We now give a condition for the intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) which is a doubt
intuitionistic fuzzy subalgebra of ܺ to be a doubt intuitionistic fuzzy ideal of ܺ.
Theorem 3.12 Let an intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy
subalgebra of ܺ. If the inequiality ‫ݔ‬ ∗ ‫ݕ‬ ≤ ‫ݖ‬ holds in ܺ, then ‫ܣ‬ would be a doubt intuitionistic
fuzzy ideal of ܺ.
Proof: Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy subalgebra of X. Then from Theorem 3.2,
ߤ஺(0) ≤ ߤ஺(‫)ݔ‬ and ߣ஺(0) ≥ ߣ஺(‫,)ݔ‬ where ‫ݔ‬ ∈ ܺ.
As ‫ݔ‬ ∗ ‫ݕ‬ ≤ ‫ݖ‬ holds in ܺ, then from Theorem 3.7, we get, ߤ஺(‫)ݔ‬ ≤ ݉ܽ‫ߤ{ݔ‬஺(‫,)ݕ‬ ߤ஺(‫})ݖ‬ and
ߣ஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫,)ݕ‬ ߣ஺(‫,})ݖ‬ for all ‫,ݔ‬ ‫,ݕ‬ ‫ݖ‬ ∈ ܺ.
Again since, ‫ݔ‬ ∗ (‫ݔ‬ ∗ ‫)ݕ‬ ≤ ‫,ݕ‬ then, ߤ஺(‫)ݔ‬ ≤ ݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஺(‫})ݕ‬ andߣ஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫ݔ‬ ∗
‫,)ݕ‬ ߣ஺(‫.})ݕ‬
Hence, ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of ܺ .

Proposition 3.13 Let an intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal
of a ‫-ܫܥܤ/ܭܥܤ‬algebra ܺ. Then
ߤ஺(0 ∗ (0 ∗ ‫))ݔ‬ ≤ ߤ஺(‫)ݔ‬ and ߣ஺(0 ∗ (0 ∗ ‫))ݔ‬ ≥ ߣ஺(‫,)ݔ‬ for all ‫ݔ‬ ∈ ܺ.
Proof: ߤ஺(0 ∗ (0 ∗ ‫))ݔ‬ ≤ ߤ஺{(0 ∗ (0 ∗ ‫))ݔ‬ ∗ ‫}ݔ‬ ∨ ߤ஺(‫)ݔ‬
≤ ߤ஺(0) ∨ ߤ஺(‫)ݔ‬
= ߤ஺(‫ݔ ݈݈ܽ ݎ݋݂ ,)ݔ‬ ∈ ܺ
Therefore, ߤ஺(0 ∗ (0 ∗ ‫))ݔ‬ ≤ ߤ஺(‫,)ݔ‬ for all ‫ݔ‬ ∈ ܺ.

7
Again, ߣ஺(0 ∗ (0 ∗ ‫))ݔ‬ ≥ ߣ஺{(0 ∗ (0 ∗ ‫))ݔ‬ ∗ ‫}ݔ‬ ∧ ߣ஺(‫)ݔ‬
≥ ߣ஺(0) ∧ ߣ஺(‫)ݔ‬
= ߣ஺(‫ݔ ݈݈ܽ ݎ݋݂ ,)ݔ‬ ∈ ܺ
Therefore, ߣ஺൫0 ∗ (0 ∗ ‫)ݔ‬൯ ≥ ߣ஺(‫,)ݔ‬ for all ‫ݔ‬ ∈ ܺ.

Theorem 3.14 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal in ܺ. Then so is ߎ‫ܣ‬ =
{〈‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߤ̅஺(‫ݔ/〉)ݔ‬ ∈ ܺ}.
Proof: Since ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of ܺ, then ߤ஺(0) ≤ ߤ஺(‫)ݔ‬ and
ߤ஺(‫)ݔ‬ ≤ ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߤ஺(‫.)ݕ‬
Now, ߤ஺(0) ≤ ߤ஺(‫1 ݎ݋ ,)ݔ‬ − ߤ̅஺(0) ≤ 1 − ߤ̅஺(‫̅ߤ ݎ݋ ,)ݔ‬஺(0) ≥ ߤ̅஺(‫,)ݔ‬ for any ‫ݔ‬ ∈ ܺ.
Consider, for any ‫,ݔ‬ ‫ݕ‬ ∈ ܺ,
ߤ஺(‫)ݔ‬ ≤ ݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஺(‫})ݕ‬
ܶℎ݅‫1 ,ݏ݁ݒ݅݃ ݏ‬ − ߤ̅஺(‫)ݔ‬ ≤ ݉ܽ‫1{ݔ‬ − ߤ̅஺(‫ݔ‬ ∗ ‫1,)ݕ‬ − ߤ̅஺(‫})ݕ‬
‫̅ߤ ,ݎ݋‬஺(‫)ݔ‬ ≥ 1 − ݉ܽ‫1{ݔ‬ − ߤ̅஺(‫ݔ‬ ∗ ‫1,)ݕ‬ − ߤ̅஺(‫})ݕ‬
‫̅ߤ ,ݕ݈݈ܽ݊݅ܨ‬஺(‫)ݔ‬ ≥ ݉݅݊{ߤ̅஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ̅஺(‫.})ݕ‬
Hence, Π‫ܣ‬ = {(‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߤ̅஺(‫ݔ/))ݔ‬ ∈ ܺ} is a doubt intuitionistic fuzzy ideal of ܺ.
Theorem 3.15 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal in ܺ. Then so is ♦‫ܣ‬ =
{〈‫,ݔ‬ ߣ̅஺(‫,)ݔ‬ ߣ஺(‫ݔ/〉)ݔ‬ ∈ ܺ}.
Proof: Since ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of ܺ, then ߣ஺(0) ≥ ߣ஺(‫.)ݔ‬ Also
ߣ஺(‫)ݔ‬ ≥ ߣ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∧ ߣ஺(‫.)ݕ‬
Again we have, ߣ஺(0) ≥ ߣ஺(‫1 ݎ݋ ,)ݔ‬ − ߣ̅஺(0) ≥ 1 − ߣ̅஺(‫ߣ ݎ݋ ,)ݔ‬̅஺(0) ≤ ߣ̅஺(‫)ݔ‬ , for any
‫ݔ‬ ∈ ܺ.
Consider, for any ‫,ݔ‬ ‫ݕ‬ ∈ ܺ,
ߣ஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߣ஺(‫})ݕ‬
This implies, 1 − ߣ̅஺(‫)ݔ‬ ≥ ݉݅݊{1 − ߣ̅஺(‫ݔ‬ ∗ ‫1,)ݕ‬ − ߣ̅஺(‫})ݕ‬
That is, ߣ̅஺(‫)ݔ‬ ≤ 1 − ݉݅݊{1 − ߣ̅஺(‫ݔ‬ ∗ ‫1,)ݕ‬ − ߣ̅஺(‫})ݕ‬
or, ߣ̅஺(‫)ݔ‬ ≤ ݉ܽ‫ߣ{ݔ‬̅஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߣ̅஺(‫.})ݕ‬
Hence, ♦‫ܣ‬ = {〈‫,ݔ‬ ߣ̅஺(‫,)ݔ‬ ߣ஺(‫ݔ/〉)ݔ‬ ∈ ܺ} is a doubt intuitionistic fuzzy ideal of ܺ.
Theorem 3.16 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be an intuitionistic fuzzy set in X. Then ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt
intuitionistic fuzzy ideal of ܺ if and only if ߎ‫ܣ‬ = {〈‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߤ̅஺(‫ݔ/〉)ݔ‬ ∈ ܺ} and ♦‫ܣ‬ =
{〈‫,ݔ‬ ߣ̅஺(‫,)ݔ‬ ߣ஺(‫ݔ/〉)ݔ‬ ∈ ܺ} 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 ‫-ܭܥܤ‬algebra with the following Cayley table:
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 1 0 1
2 2 2 0 2 0

8
3 3 1 3 0 3
4 4 4 2 4 0
Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal of ܺ as defined by:
X 0 1 2 3 4
ߤ஺ 0.4 0.5 0.6 0.5 0.6
ߣ஺ 0.5 0.4 0.3 0.4 0.3
Then Π‫ܣ‬ = {〈‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߤ̅஺(‫ݔ/〉)ݔ‬ ∈ ܺ}, where ߤ஺(‫)ݔ‬ and ߤ̅஺(‫)ݔ‬ are defined as follows:
X 0 1 2 3 4
ߤ஺ 0.4 0.5 0.6 0.5 0.6
ߤ̅஺ 0.6 0.5 0.4 0.5 0.4
Also ♦‫ܣ‬ = {〈‫,ݔ‬ ߣ̅஺(‫,)ݔ‬ ߣ஺(‫ݔ/〉)ݔ‬ ∈ ܺ},whose ߣ஺(‫)ݔ‬ and ߣ̅஺(‫)ݔ‬ are defined by:
X 0 1 2 3 4
ߣ̅஺ 0.5 0.6 0.7 0.6 0.7
ߣ஺ 0.5 0.4 0.3 0.4 0.3
So, it can be verified that Π‫ܣ‬ = {〈‫,ݔ‬ ߤ஺(‫,)ݔ‬ ߤ̅஺(‫ݔ/〉)ݔ‬ ∈ ܺ} and ♦‫ܣ‬ = {〈‫,ݔ‬ ߣ̅஺(‫,)ݔ‬ ߣ஺(‫ݔ/〉)ݔ‬ ∈ ܺ}
are doubt intuitionistic fuzzy ideals.
Theorem 3.18 An intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of a
‫-ܫܥܤ/ܭܥܤ‬algebra ܺ if and only if the fuzzy sets ߤ஺and ߣ̅஺ are doubt fuzzy ideals of ܺ.
Proof: Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal of ܺ. Then it is obvious that ߤ஺ is
a doubt fuzzy ideal of ܺ. And from Theorem 3.15, we can prove that ߣ̅஺ is a doubt fuzzy ideal of
ܺ.
Conversely, let ߤ஺ be a doubt fuzzy ideal of ܺ . Therefore ߤ஺(0) ≤ ߤ஺(‫)ݔ‬ and ߤ஺(‫)ݔ‬ ≤
݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஺(‫,})ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
Again since ߣ̅஺ is a doubt intuitionistic fuzzy ideal of ܺ , so, ߣ̅஺(0) ≤ ߣ̅஺(‫1 ݏ݁ݒ݅݃ ,)ݔ‬ −
ߣ஺(0) ≤ 1 − ߣ஺(‫ߣ ݏ݈݁݅݌݉݅ ,)ݔ‬஺(0) ≥ ߣ஺(‫.)ݔ‬
Also
ߣ̅஺(‫)ݔ‬ ≤ ݉ܽ‫ߣ{ݔ‬̅஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߣ̅஺(‫})ݕ‬
‫1 ,ݎ݋‬ − ߣ஺(‫)ݔ‬ ≤ ݉ܽ‫1{ݔ‬ − ߣ஺(‫ݔ‬ ∗ ‫1,)ݕ‬ − ߣ஺(‫})ݕ‬
‫ߣ ,ݎ݋‬஺(‫)ݔ‬ ≥ 1 − ݉ܽ‫1{ݔ‬ − ߣ஺(‫ݔ‬ ∗ ‫1,)ݕ‬ − ߣ஺(‫})ݕ‬
‫ߣ ,ݕ݈݈ܽ݊݅ܨ‬஺(‫)ݔ‬ ≥ ݉݅݊{ߣ஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߣ஺(‫,})ݕ‬ for all ‫,ݔ‬ ‫ݕ‬ ∈ ܺ.
Hence, ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of ܺ .

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

9
2 2 1 0 0
3 3 1 3 0
X 0 1 2 3
ߤ஺ 0.1 0.5 0.7 0.6
ߣ஺ 0.8 0.4 0.2 0.4
Since ߤ஺(2) = 0.7 ≰ 0.6 = ݉ܽ‫ߤ{ݔ‬஺(2 ∗ 3), ߤ஺(3)}, and ߣ஺(2) = 0.2 ≱ 0.4 = ݉݅݊{ߣ஺(2 ∗
3), ߣ஺(3)}. Therefore, ‫ܣ‬ = (ߤ஺, ߣ஺) is not a doubt intuitionistic fuzzy ideal of ܺ.
Corollary 3.20 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal of a ‫-ܫܥܤ/ܭܥܤ‬algebra ܺ.
Then the sets, ‫ܦ‬ఓಲ
= {‫ݔ‬ ∈ ܺ/ߤ஺(‫)ݔ‬ = ߤ஺(0)}, ܽ݊݀ ‫ܦ‬ఒಲ
= {‫ݔ‬ ∈ ܺ/ߣ஺(‫)ݔ‬ = ߣ஺(0)} are ideals
of ܺ.
Proof: Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal of ܺ . Obviously,
0 ∈ ‫ܦ‬ఓಲ
ܽ݊݀ ‫ܦ‬ఒಲ
.
Now, let ‫,ݔ‬ ‫ݕ‬ ∈ ܺ, such that ‫ݔ‬ ∗ ‫,ݕ‬ ‫ݕ‬ ∈ ‫ܦ‬ఓಲ
. Then ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ = ߤ஺(0) = ߤ஺(‫.)ݕ‬Now, ߤ஺(‫)ݔ‬ ≤
݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஺(‫})ݕ‬ = ߤ஺(0). Again, since ߤ஺ is a doubt fuzzy ideal of X, ߤ஺(0) ≤ ߤ஺(‫.)ݔ‬
Therefore, ߤ஺(0) = ߤ஺(‫.)ݔ‬ Hence, ‫ݔ‬ ∈ ‫ܦ‬ఓಲ
.
Therefore, ‫ܦ‬ఓಲ
is an ideal of ܺ.
Following the same way we can prove that ‫ܦ‬ఒಲ
is also an ideal of ܺ.
Theorem 3.21 If ‫ܣ‬ and ‫ܤ‬ are two doubt intuitionistic fuzzy ideals of ܺ, if one is contained
another then ‫ܣ‬ ∩ ‫ܤ‬is also a doubt intuitionistic fuzzy ideal of ܺ.
Proof: Let ‫ܣ‬ = (ߤ஺, ߣ஺) and ‫ܤ‬ = (ߤ஻, ߣ஻) be two doubt intuitionistic fuzzy ideals of ܺ.
Again let, ‫ܥ‬ = ‫ܣ‬ ∩ ‫ܤ‬ = (ߤ஼, ߣ஼), where ߤ஼ = ߤ஺ ∧ ߤ஻ and ߣ஼ = ߣ஺ ∨ ߣ஻.
Let ‫,ݔ‬ ‫ݕ‬ ∈ ܺ, then
ߤ஼(0) = ߤ஺(0) ∧ ߤ஻(0)
≤ ߤ஺(‫)ݔ‬ ∧ ߤ஻(‫)ݔ‬
= ߤ஼(‫)ݔ‬
and ߣ஼(0) = ߣ஺(0) ∨ ߣ஻(0)
≥ ߣ஺(‫)ݔ‬ ∨ ߣ஻(‫)ݔ‬
= ߣ஼(‫)ݔ‬
‫ߤ ,݋ݏ݈ܣ‬஼(‫)ݔ‬ = ߤ஺(‫)ݔ‬ ∧ ߤ஻(‫)ݔ‬
≤ ݉ܽ‫ߤ[ݔ‬஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஺(‫])ݕ‬ ∧ ݉ܽ‫ߤ[ݔ‬஻(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஻(‫])ݕ‬
= ݉ܽ‫ߤ[{ݔ‬஺(‫ݔ‬ ∗ ‫)ݕ‬ ∧ ߤ஻(‫ݔ‬ ∗ ‫,])ݕ‬ [ߤ஺(‫)ݕ‬ ∧ ߤ஻(‫}])ݕ‬
[ ܾ݁ܿܽ‫ݐ݋݊ܽ ݀݁݊݅ܽݐ݊݋ܿ ݏ݅ ݁݊݋ ݁ݏݑ‬ℎ݁‫] ݎ‬
= ݉ܽ‫ߤ[ݔ‬஼(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஼(‫.])ݕ‬
Similarly, we can prove that, ߣ஼(‫)ݔ‬ ≥ ݉݅݊[ߣ஼(‫ݔ‬ ∗ ‫,)ݕ‬ ߣ஼(‫.])ݕ‬ This completes the
proof.
Theorem 3.22 Union of any two doubt intuitionistic fuzzy ideals of X, is also a doubt intuitionistic
fuzzy ideal of X.

10
Proof: Let ‫ܣ‬ = (ߤ஺, ߣ஺) and ‫ܤ‬ = (ߤ஻, ߣ஻) be two doubt intuitionistic fuzzy ideals of ܺ. Again
let, ‫ܥ‬ = ‫ܣ‬ ∪ ‫ܤ‬ = (ߤ஼, ߣ஼), where ߤ஼ = ߤ஺ ∨ ߤ஻ and ߣொ = ߣ஺ ∧ ߣ஻.
Let ‫,ݔ‬ ‫ݕ‬ ∈ ܺ, then,
ߤ஼(0) = ߤ஺(0) ∨ ߤ஻(0)
≤ ߤ஺(‫)ݔ‬ ∨ ߤ஻(‫)ݔ‬
= ߤ஼(‫)ݔ‬
ܽ݊݀ ߣ஼(0) = ߣ஺(0) ∧ ߣ஻(0)
≥ ߣ஺(‫)ݔ‬ ∧ ߣ஻(‫)ݔ‬
= ߣ஼(‫)ݔ‬
‫ߤ ,݋ݏ݈ܣ‬஼(‫)ݔ‬ = ߤ஺(‫)ݔ‬ ∨ ߤ஻(‫)ݔ‬
≤ ݉ܽ‫ߤ[ݔ‬஺(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஺(‫])ݕ‬ ∨ ݉ܽ‫ߤ[ݔ‬஻(‫ݔ‬ ∗
‫,)ݕ‬ ߤ஻(‫])ݕ‬
= ݉ܽ‫ߤ[{ݔ‬஺(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߤ஻(‫ݔ‬ ∗ ‫,])ݕ‬ [ߤ஺(‫)ݕ‬ ∨
ߤ஻(‫}])ݕ‬
= ݉ܽ‫ߤ[ݔ‬஼(‫ݔ‬ ∗ ‫,)ݕ‬ ߤ஼(‫.])ݕ‬
Similarly, we can prove that, ߣ஼(‫)ݔ‬ ≥ ݉݅݊[ߣ஼(‫ݔ‬ ∗ ‫,)ݕ‬ ߣ஼(‫.])ݕ‬
This completes the proof.
Example 3.23 Let ܺ = {0,1,2,3,4} be a ‫-ܭܥܤ‬algebra [by Example 3.11] with the following
Cayley table:
* 0 1 2 3 4
0 0 0 0 0 0
1 1 0 0 0 0
2 2 2 0 2 0
3 3 3 0 0 0
4 4 2 1 2 0
X 0 1 2 3 4
ߤ஺ 0 0.62 0.65 0.62 0.65
ߣ஺ 1 0.34 0.32 0.34 0.32
Then ‫ܣ‬ = (ߤ஺, ߣ஺) is a doubt intuitionistic fuzzy ideal of ܺ.
Again, let ‫ܤ‬ = (ߤ஻, ߣ஻) be an intuitionistic fuzzy set of ܺ as defined by:
X 0 1 2 3 4
ߤ஻ 0. 22 0.56 0.58 0.56 0.58
ߣ஻ 0.72 0.44 0.42 0.44 0.42
Then ‫ܤ‬ = (ߤ஻, ߣ஻) is a doubt intuitionistic fuzzy ideal of ܺ.
Now let, ܲ = ‫ܣ‬ ∩ ‫ܤ‬ = (ߤ௉, ߣ௉) where ߤ௉ = ߤ஺ ∧ ߤ஻ and ߣ௉ = ߣ஺ ∨ ߣ஻.
Then ܲ is an intuitionistic fuzzy set of ܺ which can be defined as:
X 0 1 2 3 4
ߤ௉ 0 0.56 0.58 0.56 0.58

11
ߣ௉ 1 0.44 0.42 0.44 0.42
Then it is clear that ܲ = (ߤ௉, ߣ௉) is a doubt intuitionistic fuzzy ideal of ܺ.
We also assume that ܳ = ‫ܣ‬ ∪ ‫ܤ‬ = (ߤொ, ߣொ) where ߤொ = ߤ஺ ∨ ߤ஻ and ߣொ = ߣ஺ ∧ ߣ஻ and ܳ is
defined as:
X 0 1 2 3 4
ߤொ 0. 22 0.62 0.65 0.62 0.65
ߣொ 0.72 0.34 0.32 0.34 0.32
Then ܳ = (ߤொ, ߣொ) is a doubt intuitionistic fuzzy ideal of ܺ.
Definition 3.24 Let ‫ܣ‬ = (ߤ஺, ߣ஺) be an intuitionistic fuzzy set of ܺ, and ‫,ݐ‬ ‫ݏ‬ ∈ [0,1], then ߤ
level ‫-ݐ‬cut and ߣ level ‫-ݏ‬cut of ‫,ܣ‬ is as followes:
ߤ஺,௧
ஸ
= {‫ݔ‬ ∈ ܺ/ߤ஺(‫)ݔ‬ ≤ ‫}ݐ‬
and ߣ஺,௦
ஹ
= {‫ݔ‬ ∈ ܺ/ߣ஺(‫)ݔ‬ ≥ ‫.}ݏ‬
Theorem 3.25 If ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal of ܺ, then ߤ஺,௧
ஸ
and ߣ஺,௦
ஹ
are
ideals of ܺ for ‫,ݐ‬ ‫ݏ‬ ∈ [0,1].
Proof: Let ‫ܣ‬ = (ߤ஺, ߣ஺) be a doubt intuitionistic fuzzy ideal of ܺ, and let ‫ݐ‬ ∈ [0,1] with
ߤ஺(0) ≤ ‫.ݐ‬ Then we have, ߤ஺(0) ≤ ߤ஺(‫ݔ ݈݈ܽ ݎ݋݂ ,)ݔ‬ ∈ ܺ, but ߤ஺(‫)ݔ‬ ≤ ‫ݔ ݈݈ܽ ݎ݋݂ ,ݐ‬ ∈ ߤ஺,௧
ஸ
.
So, 0 ∈ ߤ஺,௧
ஸ
.
Let ‫,ݔ‬ ‫ݕ‬ ∈ ܺ be such that ‫ݔ‬ ∗ ‫ݕ‬ ∈ ߤ஺,௧
ஸ
and ‫ݕ‬ ∈ ߤ஺,௧
ஸ
, then, ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∈ ߤ஺,௧
ஸ
and ߤ஺(‫)ݕ‬ ∈ ߤ஺,௧
ஸ
.
Therefore, ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ≤ ‫ݐ‬ and ߤ஺(‫)ݕ‬ ≤ ‫ݐ‬ .
Since ߤ஺ is a doubt fuzzy ideal of ܺ, it follows that, ߤ஺(‫)ݔ‬ ≤ ߤ஺(‫ݔ‬ ∗ ‫)ݕ‬ ∨ ߤ஺(‫)ݕ‬ ≤ ‫ݐ‬ and hence
‫ݔ‬ ∈ ߤ஺,௧
ஸ
.
Therefore, ߤ஺,௧
ஸ
is ideal of ܺ for ‫ݐ‬ ∈ [0,1].
Similarly, we can prove that ߣ஺,௦
ஹ
is an ideal of ܺ for ‫ݏ‬ ∈ [0,1] .

Theorem 3.26 If ߤ஺,௧
ஸ
and ߣ஺,௦
ஹ
are either empty or ideals of ܺ for ‫,ݐ‬ ‫ݏ‬ ∈ [0,1], then ‫ܣ‬ =
[ߤ஺, ߣ஺] is a doubt intuitionistic fuzzy ideal of ܺ.
Proof: Let ߤ஺,௧
ஸ
and ߣ஺,௦
ஹ
be either empty or ideals of ܺ for ‫,ݐ‬ ‫ݏ‬ ∈ [0,1]. For any ‫ݔ‬ ∈ ܺ, let
ߤ஺(‫)ݔ‬ = ‫ߣ ݀݊ܽ ݐ‬஺(‫)ݔ‬ = ‫.ݏ‬ Then ‫ݔ‬ ∈ ߤ஺,௧
ஸ
∧ ߣ஺,௦
ஹ
, so ߤ஺,௧
ஸ
≠ ߶ ≠ ߣ஺,௦
ஹ
. Since ߤ஺,௧
ஸ
and ߣ஺,௦
ஹ
are
ideals of ܺ, therefore 0 ∈ ߤ஺,௧
ஸ
∧ ߣ஺,௦
ஹ
.
Hence, ߤ஺(0) ≤ ‫ݐ‬ = ߤ஺(‫)ݔ‬ and ߣ஺(0) ≥ ‫ݏ‬ = ߣ஺(‫,)ݔ‬ where ‫ݔ‬ ∈ ܺ.
If there exist ‫ݔ‬ᇱ
, ‫ݕ‬ᇱ
∈ ܺ such that ߤ஺(‫ݔ‬ᇱ
) > ݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ᇱ
∗ yᇱ
), ߤ஺(‫ݕ‬ᇱ
)}, then by taking, ‫ݐ‬଴ =
ଵ
ଶ
(ߤ஺(‫ݔ‬ᇱ) + ݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ᇱ
∗ ‫ݕ‬ᇱ), ߤ஺(‫ݕ‬ᇱ)}).
We have, ߤ஺(‫ݔ‬ᇱ
) > ‫ݐ‬଴ > ݉ܽ‫ߤ{ݔ‬஺(‫ݔ‬ᇱ
∗ ‫ݕ‬ᇱ
), ߤ஺(‫ݕ‬ᇱ
)} . Hence ‫ݔ‬ᇱ
∉ ߤ஺,௧బ
ஸ
, (‫ݔ‬ᇱ
∗ ‫ݕ‬ᇱ
) ∈ ߤ஺,௧బ
ஸ
and
‫ݕ‬ᇱ
∈ ߤ஺,௧బ
ஸ
, that is ߤ஺,௧బ
ஸ
is not an ideal of ܺ, which is a contradiction.
Finally, assume that there exist ‫,݌‬ ‫ݍ‬ ∈ ܺ such that ߣ஺(‫)݌‬ < ݉݅݊{ߣ஺(‫݌‬ ∗ ‫,)ݍ‬ ߣ஺(‫.})ݍ‬ Taking
‫ݏ‬଴ =
ଵ
ଶ
(ߣ஺(‫)݌‬ + ݉݅݊{ߣ஺(‫݌‬ ∗ ‫,)ݍ‬ ߣ஺(‫,)})ݍ‬ then ݉݅݊{ߣ஺(‫݌‬ ∗ ‫,)ݍ‬ ߣ஺(‫})ݍ‬ > ‫ݏ‬଴ > ߣ஺(‫.)݌‬
Therefore, ‫݌‬ ∗ ‫ݍ‬ ∈ ߣ஺,௦
ஹ
and ‫ݍ‬ ∈ ߣ஺,௦
ஹ
but ‫݌‬ ∉ λ஺,௦
ஹ
. Again a contradiction. This completes the

12
proof.
But, if an intuitionistic fuzzy set ‫ܣ‬ = (ߤ஺, ߣ஺), is not a doubt intuitionistic fuzzy ideal of ܺ, then
ߤ஺,௧
ஸ
and ߣ஺,௦
ஹ
are not ideals of ܺ for ‫,ݐ‬ ‫ݏ‬ ∈ [0,1], which is illustrated in the following example.
Example 3.27 Let ܺ = {0,1,2,3} be a ‫-ܭܥܤ‬algebra in Example 3.19 with the following Cayley
table:
* 0 1 2 3
0 0 0 0 0
1 1 0 1 1
2 2 1 0 0
3 3 1 3 0
X 0 1 2 3
ߤ஺ 0.1 0.5 0.7 0.6
ߣ஺ 0.8 0.4 0.2 0.4
which is not a doubt intuitionistic fuzzy ideal of ܺ.
For ‫ݐ‬ = 0.67 and ‫ݏ‬ = 0.25, we get ߤ஺,୲
ஸ
= ߣ஺,௦
ஹ
= {0,1,3}, which are not ideals of ܺ, as 2 ∗ 1 =
1 ∈ {0,1,3}, and 1 ∈ {0,1,3}, but 2 ∉ {0,1,3}.
4. CONCLUSIONS
We see that to develope the theory of ‫-ܫܥܤ/ܭܥܤ‬algebras the ideal theory plays an important role.
In this paper, we have defined doubt intuitionistic fuzzy subalgebras and doubt intuitionistic fuzzy
ideals in ‫-ܫܥܤ/ܭܥܤ‬algebras and have developed several characterizations of doubt intuitionistic
fuzzy subalgebras and doubt intuitionistic fuzzy ideals in ‫-ܫܥܤ/ܭܥܤ‬algebras. 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 ‫ܫܥܤ/ܭܥܤ‬ -algebras and their
applications in other branches of algebra. In the future study of intuitionistic fuzzy ‫/ܭܥܤ‬
‫-ܫܥܤ‬algebras, perhaps the following topic are worth to be considered:
1. To characterize other classes of ‫-ܫܥܤ/ܭܥܤ‬algebras 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.
REFERENCES
[1] L. A. Zadeh, Fuzzy sets, Inform. and Control, 8 (1965), 338-353.
[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
[3] K. T. Atanassov, New operations defined over the intuitionistic fuzzy sets, Fuzzy Sets and Systems,
61 (1994), 137-142.

13
[4] Y. Imai and K. Iseki, On axiom systems of propositional calculi, Proc. Japan Academy, 42 (1966),
19-22.
[5] K. Iseki, An algebra related with a propositional calculus, Proc. Of Japan Academy, 42 (1966), 26-29.
[6] K. Iseki, On BCI-algebras, Math. Seminar Notes, 8 (1980), 125-130.
[7] O. G. Xi, Fuzzy BCK-algebras, Math. Japon. 24(36) (1991), 935-942.
[8] F. Y. Huang, Another definition of fuzzy BCI-algebras, Selected Papers on BCK/BCI-Algebras,
China, 1 (1992), 91-92.
[9] Y. B. Jun, Doubt fuzzy BCK/BCI-algebras, Soochow J. Math., 20(3) (1994), 351-358.
[10] Y. B. Jun and K. H. Kim, Intuitionistic fuzzy ideals of BCK-algebras, Internat. J. Math and Math. Sci.,
24 (2000), 839-849.
[11] A. Solairaju and R. Begam, Intuitionistic fuzzifications of P-ideals in BCI-algebras, Internat. J. Math
Research, 4(2) (2000), 133-144.
[12] 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.
[13] T. Senapati, M. Bhowmik and M. Pal, Interval-valued intuitionistic fuzzy BG-subalgebras, The
Journal of Fuzzy Mathematics, 20 (3) (2012), 707-720.
[14] 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.
[15] T. Senapati, M. Bhowmik and M. Pal, Intuitionistic fuzzifications of ideals in BG-algebras,
Mathematica Aeterna, 2 (9) (2012), 761-778.
[16] T. Senapati, M. Bhowmik and M. Pal, Fuzzy closed ideals of B-algebras, Int. J. Computer Science
Engineering and Technology, 1 (10) (2011), 669-673.
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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