 Open Access
 Authors : S. Sowmiya , P. Jeyalakshmi
 Paper ID : IJERTV10IS020227
 Volume & Issue : Volume 10, Issue 02 (February 2021)
 Published (First Online): 02032021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Intuitionistic LFuzzy Structures in ZAlgebras
S. Sowmiya
Assistant Professor, Department of Mathematics,
Sri Ramakrishna Engineering College Vattamalaipalayam,Coimbatore22, Tamilnadu,India
P. Jeyalakshmi
Professor and Head, Department of Mathematics,
Avinashilingam Institute for Home Science and Higher Education for Women, Coimbatore43, Tamilnadu,India.
AbstractIn this article, we initiate and explore the idea of intuitionistic LFuzzy ZSubalgebras and intuitionistic LFuzzy Zideals in Zalgebras. We further explore some of their
(Z4) x y yx when x 0 and y 0 x, y X.
Definition 2.2[2] Let (X,,0) and (Y,,0) be two
properties of intuitionistic LFuzzy ZSubalgebras and

algebras. A mapping
h : (X,,0) (Y,,0)
is said to
intuitionistic LFuzzy Zideals under Zhomomorphism and cartesian product in Zalgebras.
2010 Mathematics Subject Classification. 03B20, 03B52
KeywordsZalgebra, Zideal, Zhomomorphism,
,intuitionistic Lfuzzy ZSubalgebra, intuitionistic Lfuzzy Zideal .

INTRODUCTION
Atanassov and Stoeva [1] introduced the notion of intuitionistic Lfuzzy sets in 1984 as an extension of Goguens
[3] notion of Lfuzzy set. Here they have defined both membership and nonmembership functions from the Universe of discourse X to the set L, where (L, , , ) is a complete lattice. Motivated by this, many mathematicians started to review various concepts and theorems in intuitionistic Lfuzzy structures. Currently, in the year 2017, Chandramouleeswaran et al.[2] introduced a new class of algebra called Zalgebra that arise from the notion of propositional calculi. In our previous articles [4, 5, 6, 7, 8, 9, 10, 11] we have introduced fuzzy ZSubalgebras, fuzzy Z ideals, fuzzy Hideals, fuzzy pideals, fuzzy implicative ideals, intuitionistic fuzzy ZSubalgebras and intuitionistic fuzzy Z ideals in Zalgebras. In this article, we have initiated intuitionistic Lfuzzy ZSubalgebras and intuitionistic Lfuzzy ZIdeals in Zalgebras. 
PRELIMINARIES
be a Zhomomorphism of Zalgebras if
h(x y) h(x) h(y) for all x, yX .
Definition 2.3:[12] Let X be a nonempty set. A fuzzy set A in X is characterized by a membership function ÂµA which associates with each point x in X, a real number in the interval [0,1] with the value A (x) representing the grade of membership of x in A. That is, a fuzzy set A in X is characterized by a membership function A (x) : X [0, 1]. Definition 2.4:[1] Let (L, , , ) be a complete lattice with least element 0 and greatest element 1 and an involutive
order reversing operation N : L L . Then Intuitionistic L Fuzzy Set A x,A (x),A (x)  x X in a nonempty set
X is an object having the form where A : X L is the degree of membership function and A : X L is the degree of nonmembership function of the element x X satisfying
A (x) N(A (x)) .
Definition 2.5:[1] Let A A ,A and B B,B be
any two intuitionistic Lfuzzy set of a set X. Then we have 1. Ac x, Ax,A x xX
2. A B x,A x B x, A x B x x X
3. A A , A c x, A x,1 A x x X
4. A A c ,A x,1 A x,A x xX

For any subset T of X x0 T such that
In this section, we recall some basic definitions that are
required for our work
A x0
supA tT
t
and
A x0
inf A tT
t.
Definition 2.1[2] A Zalgebra X,,0 is a nonempty set X
with a constant 0 and a binary operation satisfying the following conditions:
(Z1) x 0 0
is called supinf property of A.

The Cartesian product A B AB, AB whose
membership function AB : X X L and non
membership function AB : XX L are defined by
(Z2)
0 x x
AB
x, y A
x B
y and
(Z3)
x x x
AB
x, y A
x B
yfor all x, yX .
Definition 2.6:[1] Let h be a mapping from a set X into a set Y.
0.6 if x 0
A
A
(x) 0.4 if x 1
and

Let A x,A (x),A (x)  x X be an intuitionistic
0.3 if x 2,3
Lfuzzy set in X. Then the image of A under h, denoted by
0.4 if x 0
h(A) { y,h(A) (y),h(A) (y) yY} is an intuitionistic
(x) 0.5 if x 1
A
A
Lfuzzy set in Y, defined by:
0.6 if x 2 , 3
sup
1
A (z) if
p(y) {x  h(x) y}
is an intuitionistic Lfuzzy ZSubalgebra of X.
By applying the definition of intuitionistic Lfuzzy set, we
h(A) (y) zh
0
d
(y)
otherwise
an
can prove easily the following result.
Theorem 3.3: Let A1 and A2 be two intuitionistic Lfuzzy
inf
A (z) if
p (y) {x  h(x) y}
Z Subalgebras of a Zalgebra X. Then
A1 A2
is an
h(A)
(y) zp ( y)
intuitionistic Lfuzzy ZSubalgebra of X.
1 otherwise
is
We can generalize the above theorem as follows.
Corollary 3.4: Let Ai  i be a family of intuitionistic

Let B y,A (y),A (y)  yY be an intuitionistic
Lfuzzy ZSubalgebras of a Zalgebra X. Then Ai is an
fuzzy set in Y. The preimage of B under h, symbolized by
i
p (B) { x,
(x),
(x)
xX}
defined by:
intuitionistic Lfuzzy ZSubalgebra of X.
p (B)
p (B)
By using the definition of Ac , we can prove the following
p(B) (x) B (h(x)) and p(B) (x) B (h(x))
for all
result.
x X is an intuitionistic Lfuzzy set of X.
Theorem 3.5: An intuitionistic Lfuzzy set A A ,A is
an intuitionistic Lfuzzy ZSubalgebra of a Zalgebra X if


– –

and only if the Lfuzzy sets A
and
(A )c
are Lfuzzy
INTUITIONISTIC L FUZZY Z SUBALGEBRAS IN
ZALGEBRAS
ZSubalgebras of X.
Theorem 3.6: A A ,A
is an intuitionistic Lfuzzy
In this section we introduce the notion of Intuitionistic LFuzzy ZSubalgebra of a Zalgebra. Also we prove some interesting results.
Definition 3.1 : An Intuitionistic Lfuzzy Set
ZSubalgebra of a Zalgebra X if and only if
(i) A A , (A )c and (ii) A (A )c , A , both are
intuitionistic Lfuzzy ZSubalgebras of X.
Proof: Let A A ,A be an intuitionistic Lfuzzy
A A ,A in a Zalgebra X,,0 is called an
ZSubalgebra of a Zalgebra X.
Intuitionistic Lfuzzy ZSubalgebra of X if it
Let
x, yX. Then,
satisfies the following conditions: (i) A (x y) A (x) A (y)
(i)
(ii)
A (x y) A (x) A (y)
A (x y) A (x) A (y)
(ii) A (x y) A (x) A (y)
for all x, yX .
(iii)
(A )c (x y) 1 A (x y) (A )c (x) (A )c (y)
Example 3.2: Consider a Zalgebra X= {0,1,2,3} with the following Cayley table as in [8]:
(iv)
(A )c (x y) 1 A (x y)
0 
1 
2 
3 

0 
0 
1 
2 
3 
1 
0 
1 
3 
2 
2 
0 
3 
2 
1 
3 
0 
2 
1 
3 
0 
1 
2 
3 

0 
0 
1 
2 
3 
1 
0 
1 
3 
2 
2 
0 
3 
2 
1 
3 
0 
2 
1 
3 
(A
)c (x) (A
)c (y)
An intuitionistic Lfuzzy set A A
,A
in X defined by
From (i) and (iii), we get A is an intuitionistic Lfuzzy ZSubalgebra of X.
And, from (ii) and (iv), we get A is an intuitionistic
Lfuzzy ZSubalgebra of X.
Conversely, assume that A A , (A )c and
A (A )c , A are intuitionistic Lfuzzy ZSubalgebras
of a Zalgebra X. For any x, yA ,
A (x y) A (x) A (y) and A (x y) A (x) A (y)
Hence A A ,A is an intuitionistic Lfuzzy ZSubalgebra of X.
Analogously, we can prove the following result.
Theorem 3.7: An intuitionistic Lfuzzy set A A ,A in
a Zalgebra X is an intuitionistic Lfuzzy ZSubalgebra of X if and only if UA;s and LA ; t are ZSubalgebras of
with supinf property. Then the image
hA y,hAy,hAy yYof A under h is an
intuitionistic Lfuzzy ZSubalgebra of Y.
Theorem 3.15 : Let h : (X,,0) (Y,,0) be a
h
h
B
B
h
h
B
B
Zhomomorphism of Zalgebras and B be an intuitionistic Lfuzzy ZSubalgebra of Y. Then the inverse image of B,
X for all s, t [0,1] .
pB x, 1 x, 1 x xXis an intuitionistic
As a consequence, we have the following corollary.
Corollary 3.8: Any ZSubalgebra of a Zalgebra X can be
Lfuzzy ZSubalgebra of X. Converse is true if h is an Zepimorphism.
realized as both the upper slevel and lower tlevel
ZSubalgebras of some intuitionistic Lfuzzy ZSubalgebras
Proof: If h is an Zepimorphism and pB
is an
of X.
Analogously, the following theorems can be proved. Theorem 3.9: Let X be a Zalgebra. Then any given chain of ZSubalgebras Q0 Q1 … Qr X , there exists an
intuitionistic Lfuzzy ZSubalgebra of a Zalgebra X and
for y1, y2 Y there exists x1, x2 X such that h(x1) = y1 and h(x2) = y2.
This implies x1=h 1(y1) and x2= h 1(y2).
Now, (y y ) (h(x ) h(x )) (h(x x ))
intuitionistic Lfuzzy ZSubalgebra A of X whose upper slevel and lower tlevel ZSubalgebras are exactly the ZSubalgebras of this chain.
Theorem 3.10: Let A be an intuitionistic Lfuzzy ZSubalgebra of a Zalgebra X. Then

two upper slevel Z Subalgebras UA;s1 and
UA;s2 (with s1 s2 ) of A are equal if and only if there is no x X such that s1 A x s2 .
B 1 2 B 1
Analogously, we can prove that
B(y1 y2 ) B (y1) B (y2 )
2 B 1 2
pBx1 x2
pBx1 pBx2
B (h(x1 )) B (h(x 2 ))
B (y1 ) B (y2 )

two lower tlevel Z Subalgebras LA ; t1 and
LA; t 2 ( with t1 t2 ) of A are equal if and only if there is no x X such that t1 A x t 2
Theorem 3.11: Let X be a finite Zalgebra and A be an intuitionistic Lfuzzy ZSubalgebra of X.

If Im(A ) s1,…,sn , then the family of
ZSubalgebras UA;si , i 1,2,,n constitutes all the upper slevel ZSubalgebras of A.
Thus B is an intuitionistic Lfuzzy ZSubalgebra of a Zalgebra Y.
IV INTUITIONISTIC LFUZZY ZIDEALS IN ZALGEBRAS
In this section we introduce the notion of Intuitionistic
Lfuzzy Zideal of a Zalgebra and some interesting results are obtained.
Definition 4.1: An intuitionistic Lfuzzy set A A ,A

If Im(A ) t1,…, tr, then the family of
ZSubalgebras LA ; ti , i 1,2,, n constitutes all the lower tlevel ZSubalgebras of A.
in a Zalgebra X,,0 is called an intuitionistic Lfuzzy Z
ideal of X if it satisfies the following conditions: (i) A (0) A (x) and A (0) A (x)
Theorem 3.12: Let A be an intuitionistic Lfuzzy
(ii) A (x)
A (x y) A (y)
ZSubalgebra of a Zalgebra X. Then
(i) If Im(A ) is finite, say s1,…, sn, then for any
si , s j Im(A ), UA ; si UA ; s j implies si s j .
(iii) A (x) A (x y) A (y) , for all x, yX .
Example 4.2: Let X={0,1,2,3} be a set with the following Cayley table as in [8]:
0 
1 
2 
3 

0 
0 
1 
2 
3 
1 
0 
1 
1 
3 
2 
0 
1 
2 
2 
3 
0 
3 
2 
3 
0 
1 
2 
3 

0 
0 
1 
2 
3 
1 
0 
1 
1 
3 
2 
0 
1 
2 
2 
3 
0 
3 
2 
3 
(ii) If Im(A ) is finite, say t1,…, tn, then for any
t i , t j Im(A ), LA ; t i LA ; t j implies ti t j .
Theorems 3.13: Let A and B be any two intuitionistic
Lfuzzy ZSubalgebras of a Zalgebra X. Then A B is an intuitionistic Lfuzzy ZSubalgebra of X X .
Theorem 3.14: Let h be a Zhomomorphism from a Zalgebra X,,0 onto a Zalgebra Y,,0 and
A A ,A be an intuitionistic Lfuzzy ZSubalgebra of X
Then X,,0 is a Zalgebra. Define an intuitionistic Lfuzzy
Let y Y, there exists x X such that h (x) = y. Then (i)
set A A ,A in X as follows: A
(x)=0.8 for all
B(y) B(h(x)) pBx pB0 = B (h(0))
x = 0,1,2,3 and A (x)=0.1 for all x = 0,1,2,3 .
= B (0)
Then, A A ,A
is an intuitionistic Lfuzzy Zideal of a
(ii) B(y) B(h(x)) pBx pB0= B (h(0))
Zalgebra X.
By applying the definition of an intuitionistic Lfuzzy set, we can easily prove the following result.
= B (0)
Let x, y Y. Then there exists a, b X such that h(a) = x and h (b) = y. It follows that
(iii) B(x) B (h(a)) pBa
Theorem 4.3: Intersection of any two intuitionistic Lfuzzy Zideals of a Zalgebra X is again an inuitionistic Lfuzzy
pB
a b
pB
b
Zideal of X.
We generalize the above theorem as follows.
B (h(a b)) B (h(b))
B (h(a) h(b)) B (h(b))
Theorem 4.4: Let Ai  i be a family of intuitionistic
B (x y) B (y)
Lfuzzy Zideals of a Zalgebra X. Then Ai is an
(iv)
B (x) B (h(a)) 1 a
i h B
intuitionistic Lfuzzy ZIdeal of X.
pBa b pBb
Lemma 4.5: An intuitionistic Lfuzzy set A A ,A is an intuitionistic Lfuzzy Zideal of a Zalgebra X if and only
B (h(a b)) B (h(b))
(h(a) h(b)) (h(b))
if the Lfuzzy sets and ( )c are Lfuzzy Zideals of B B
A A B (x y) B (y)
X. Hence B is an intuitionistic Lfuzzy Zideal of Y.
Theorem 4.6: Let A A ,A be an intuitionistic
Lfuzzy set in a Zalgebra X. Then A A ,A is an intuitionistic Lfuzzy Zideal of X if and only if
A A , (A )c and A (A )c , A are intuitionistic
Theorem 4.11: Let A and B be two intuitionistic Lfuzzy
Zideals in a Zalgebra X. Then A B is an intuitionistic Lfuzzy Zideal of X X .
Lfuzzy Zideals of X.
Theorem 4.7: An intuitionistic Lfuzzy set A A ,A is
an intuitionistic Lfuzzy Zideal of a Zalgebra X if and only if for all s, t L , the sets UA;s and LA ; t are either
empty or Zideals of X.
Proof: Take x1, x2 X X .
Then AB0,0 A 0 B0 A x1 B x2
AB x1, x 2
and AB 0,0 A 0 B 0 A x1 B x 2
Theorem 4.8: Let h be a homomorphism from a Zalgebra
X,,0 onto a Zalgebra Y,,0 and A be an intuitionistic
Lfuzzy Zideal of X with supinf property. Then image of A,
AB
x1, x2
hA y,hAy,hAy yY is an intuitionistic Lfuzzy Zideal of Y.
Theorem 4.9: Let h : (X,,0) (Y,,0) be a
Zhomomorphism of Zalgebras and B be an intuitionistic Lfuzzy Zideal of Y. Then the inverse image of B,
pB x, 1 x, 1 x xX is an intuitionistic
Now take x1, x2 , y1, y2 X X . Then
AB (x1, x 2 ) A (x1 ) B (x 2 )
A (x1 y1 ) A (y1) B (x2 y2 ) B (y2 )
(A (x1 y1 ) B (x2 y2 )) (A (y1 ) B (y2 ))
h B h B
AB
((x1
y1 ), (x2

y2
))
AB
(y1, y2 )
Lfuzzy Zideal of X.
Theorem 4.10: Let h : (X,,0) (Y,,0) be an
AB
((x1, x2
) (y1, y2
))
AB
(y1, y2 )
Zepimorphism of Zalgebras. Let B be an intuitionistic
AB (x1, x2 ) A (x1 ) B (x2 )
Lfuzzy set of Y. If p B is an intuitionistic Lfuzzy
Zideal of X then B is an intuitionistic Lfuzzy Zideal of Y.
Proof: Assume that If p B is an intuitionistic Lfuzzy Zideal of X.
(A (x1 y1) A (y1 )) (B (x2 y2 ) B (y2 ))
(A (x1 y1 ) B (x2 y2 )) (A (y1 ) B (y2 ))
AB ((x1 y1 ),(x2 y2 )) AB (y1, y2 )
AB ((x1, x 2 ) (y1, y2 )) AB (y1, y2 )
Hence A B is an intuitionistic Lfuzzy Zideal of X X . Theorem 4.12: Let A and B be two intuitionistic Lfuzzy sets in a Zalgebra X. If A B is an intuitionistic Lfuzzy Zideal of X X , the following are true.

A 0 By and B 0 A x for all x, yX .

A 0 B y and B 0 A x for all x, yX . Theorem 4.13: Let A and B be two intuitionistic Lfuzzy sets in a Zalgebra X such that A B is an intuitionistic Lfuzzy Zideal of X X . Then either A or B is an intuitionistic
Lfuzzy ZIdeal of X.
V CONCLUSION
In this article, we have introduced intuitionistic Lfuzzy ZSubalgebras and intuitionistic Lfuzzy Zideals in Zalgebras and discussed their properties. We extend this concept in our research work.
VI ACKNOWLEDGMENT
Authors wish to thank Dr.M.Chandramouleeswaran, Professor and Head, PG Department of Mathematics, Sri Ramanas College of Arts and Science for Women, Aruppukottai, for his valuable suggestions to improve this paper a successful one.
REFERENCES

K.T. Atanassov and S.Stoeva, Intuitionistic LFuzzy Sets, Cybernetics and Systems Research,2(1984) , pp.539540.

M.Chandramouleeswaran,P.Muralikrishna,K.Sujatha and S.Sabarinathan, A note on Zalgebra, Italian Journal of Pure and Applied MathematicsN. 38 (2017) , pp.707714.

J.A.Goguen, LFuzzy Sets,J.Math.Anal.Appl. 18 (1967), pp. 145 174.

S.Sowmiya and P.Jeyalakshmi, Fuzzy Algebraic Structure in Z Algebras, World Journal of Engineering Research and Technology, 5(4) (2019) ,pp.7488.

S.Sowmiya and P.Jeyalakshmi.P., On Fuzzy Zideals in Zalgebras, Global Journal of Pure and Applied Mathematics, 15(4),(2019), pp. 505516.

S.Sowmiya and P.Jeyalakshmi, Fuzzy Translations and Fuzzy Multiplications of Zalgebras, Advances in Mathematics : Scientific Journal, 9(3) (2020) ,pp.12871292.

S.Sowmiya and P.Jeyalakshmi, ZHomomorphism and Cartesian product on Fuzzy Translations and Fuzzy Multiplications of Zalgebras,AIP Conference Proceedings 2261 (2020) , pp.03009810300985.

S.Sowmiya and P.Jeyalakshmi, Intuitionistic Fuzzy Sets in ZAlgebras, Journal of Advanced Mathematical Studies, 13(3) (2020), pp.302310.

S.Sowmiya and P.Jeyalakshmi, On Fuzzy Hideals in ZAlgebras(Submitted).

S.Sowmiya and P.Jeyalakshmi, On Fuzzy pideals in ZAlgebras(Submitted).

S.Sowmiya and P.Jeyalakshmi, On Fuzzy implicative ideals in ZAlgebras(Submitted).

L.A.Zadeh, Fuzzy sets, Information and Control,8 (1965), pp.338 353