 Open Access
 Authors : S. Sowmiya , P. Jeyalakshmi
 Paper ID : IJERTV10IS020263
 Volume & Issue : Volume 10, Issue 02 (February 2021)
 Published (First Online): 03032021
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Cubic ZIdeals 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, the notions of Cubic ZIdeals in Zalgebras is introduced and some of their properties are investigated. The Zhomomorphic image and inverse image of
(Z3)
(Z4)
x x x
x y y x when x 0 and y 0
x, y X.
cubic ZIdeals in Z algebras is investigated. Also, the cartesian product of cubic ZIdeals in Zalgebras are also discussed.
2010 Mathematics Subject Classification. 06F35, 03G25, 08A27
KeywordsZalgebra, Zideal, Zhomomorphism, Cubic
Definition 2.2[1] Let (X,,0) and (Y,,0) be two
Zalgebras. A mapping h : (X,,0) (Y,,0) is said to be a Zhomomorphism of Zalgebras if h(x y) h(x) h(y) for all x, yX .
Definition 2.3:[6] Let X be a nonempty set . A cubic set A in
Zideal .

INTRODUCTION
X is a structure A {x, ~ A (x), A (x)
denoted by A (~A , A ) where
x X} briefly
Imai and Iseki [2, 3] introduced two new classes of algebras
that arise from the propositional logic. In 2017,
~A (x) :[L ,U ] : X D[0,1] is an intervalvalued fuzzy set
A A
Chandramouleeswaran et al. [1] introduced the concept of
Zalgebra as a new structure of algebra based on propositional
in X and A : X [0,1] is a fuzzy set in X .
logic. Zadeh [19] introduced the notion of fuzzy sets in 1965. In 1975, Zadeh [20] made an extension of the concept of fuzzy set by an intervalvalued fuzzy set whose membership
For two cubic sets A (~A
define
, A
) and B (~B
, B
) in X, we
function is manyvalued and form an interval in the
membership scale. In our earlier paper [718] we have introduced the concept of cubic set to ZSubalgebras in

A B iff ~A ~B and
A B
Zalgebras and the concepts of fuzzy set, intervalvalued fuzzy

A B iff A B and B A.
set, intuitionistic fuzzy set, intuitionistic Lfuzzy set, interval valued intuitionistic fuzzy set to ZSubalgebras and Zideals in

Ac x,A
(x),~A
(x)  xX
Zalgebras. In 2012, using a fuzzy set and an intervalvalued fuzzy set, Jun et al. [6] introduced a new notion called a cubic
4. A B { x, ~
AB
(x),
AB
(x)
x X}
set and investigated several properties. Meanwhile, in 2010, Jun et al. [5] introduced the notion of cubic subalgebras/cubic ideals in BCK/BCIalgebras and they investigated several
{x, r min(~ A (x), ~ B (x)), max( A
~
(x), B
(x))
x X}
properties.In 2011, Jun et al. [4] applied the notion called a cubic sets to a group and introduced the notion of cubic subgroup. In this paper, we have introduced the concept of cubic ZIdeals of Zalgebras and investigated some of their properties.


PRELIMINARIES
In this section, we recall some basic definitions that are required for our work
5. A B x, AB (x), AB x X}
{x, r max( ~ A (x), ~ B (x)), min(A (x), B (x))x X} Definition 2.4:[4] Let A (~A , A ) be a cubic set of X. For [s1, s2 ] D[0,1] and t [0,1] , the set
U(~A ;[s1, s2 ]) {x X  ~A (x) [s1, s2 ]} is called an
intervalvalued upper [s1,s2 ] level subset of A and
Definition 2.1[1] 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
(Z2) 0 x x
L(A ; t) {x X  A (x) t} is called lower tlevel subset
of A.
Definition 2.5:[4] A cubic set A (~A , A ) in a nonempty set X is said to have the rsupinf property if for any subset T
of X there exists t0 T such that
~ A (t 0 ) r sup ~ A (t) and
inf
A (z) if
p (y) {x  h(x) y}
tT
h(A)
(y) zp (y)
A (t 0 ) inf A (t) respectively.
tT
Definition 2.6:[6] Consider a collection of cubic sets
1
is a cubic set in Y.
otherwise
Ai { x, ~
(x),
A A
A A
i i
(x)
x X} where i ,
(ii) Let
B (~B , B )
be a cubic set in Y. Then the inverse
image (or preimage) of B under h, denoted by
(i) Punion and Pintersection denoted by PAi and
p (B) { x, ~
(x),
(x) x X} is a cubic set in X
i
p (B)
defined by
p (B)
h (B) B
h (B) B
~ 1 (x) ~ (h(x))
and
PAi are defined as follows.
i
p(B) (x) B (h(x)) for all x X .
~
Definition 2.8:[4] Let
A (~A , A )
and
B (~B , B ) be
PAi x, A (x), A (x) x X
i
i i
i i
any two cubic sets in X . Then, the Cartesian product of cubic
x, r sup~
(x), sup
A A
A A
i i
(x)
x X,
sets A and B is given by
A B (~AB , AB )
where
i
~
i
~AB
: X X D[0,1]
and
AB
: X X [0,1] are defined
PAi x, A (x), A (x)
x X
by ~
(x, y) r min{~
(x), ~
(y)}
and
i
i i
i i
AB A B
A
A
x, r inf ~
i i
(x), inf A (x) x X
i
i
i
AB (x, y) max{A (x), B (y)} for all (x, y) X X .

CUBIC ZIDEALS IN ZALGEBRAS
(ii) Union and intersection denoted by Ai
i
defined as follows.
and Ai are
i
In this section, the notion of Cubic Zideals in Zalgebras is defined and corresponding results are proved.
A x, ~ (x), (x) x X
Definition 3.1: Let (X,,0) be a Zalgebra. A cubic set
i
i
Ai
i
Ai
i
A (~A , A ) in X is called a cubic Zideal of X if it satisfies
x, r sup~
(x),inf
(x)
x X ,
the following conditions:
Ai i Ai ~ ~
i
(i) A (0) A (x)
and A(0) A (x)
Ai
i
x, ~
Ai
i
(x),
Ai
i
(x)
x X
(ii) ~A
(x) r min{~A
(x y), ~A
(y)}
~
(iii) A(x) max{A (x y), A (y)} , for all x, y X.
i Ai
i Ai
x, r inf A (x),sup (x) x X
Example 3.2: Consider a Zalgebra X= {0,1,2,3} with the
i
i
Definition 2.7:[4] Let h be a mapping from a set X into a set Y.
(i) Let A (~A , A ) be a cubic set in X. Then the image of
A under h, denoted by
following Cayley table :
h(A) { y, ~ h(A) (y), h(A) (y)
yY} , is defined by:
r sup ~A (z) if p (y) {x  h(x) y} ~
0 
1 
2 
3 

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

0 
0 
1 
2 
3 
1 
0 
1 
3 
1 
2 
0 
3 
2 
1 
3 
0 
1 
1 
3 
~ 1
Define a cubic set A in X by (x) [0.6,0.8]
and
h(A) (y) zh (y) A
[0,0]otherwise
A (x) 0.2
, for all
xX. Then, A is a cubic Zideal of a
and
Zalgebra X.
Theorem 3.3: The intersection of any set of cubic Zideals of a Zalgebra X is also a cubic Zideal of X.
Theorem 3.7: Cubic set A (~A , A ) of a Zalgebra X is a cubic Zideal of X where ~ A [L , U ] if and only if
Proof: Let
Ai { x, ~A (x), A (x)
x X} where
A A
i an L U c
i i A , A and (A )
are fuzzy Zideals of X.
index set, be a set of cubic Zideals of a Zalgebra X . Then for any x, yX ,
Analogously, the following theorems can be proved.
Theorem 3.8: Let A (~A , A ) be a cubic set in a
~A (0) r inf ~ A (0) r inf ~ A (x) ~A (x)
Zalgebra X. Then A is a cubic Zideal of X if and only if for
i
Ai
A
A
(0) sup
i
i
A
A
(0) sup
i
i
(x)
Ai
i
(x)
all [s1, s2 ] D[0,1]
and
t [0,1] , the sets
U(~A ;[s1 , s2 ])
~
~
Ai
(x) r inf ~
A
A
i
(x) r inf{r min{~
A
A
i
(x y), ~
A
A
i
(y)}}
and L(A ; t) of A are either empty or Zideals of X.
Theorem 3.9: Let h be a Zhomomorphism from a
r min{r inf ~ A (x y), r inf ~ A (y)}
i i Zalgebra
(X,,0)
onto a Zalgebra
(Y,,0)
and A be a
i i
i i
r min{~A (x y), ~A (y)}
and A (x) supA (x) sup{max{A (x y), A (y)}}
cubic Zideal of X with rsupinf property. Then image of A denoted by h(A) is a cubic Zideal of Y.
i i
max{sup
i
A A
A A
(x y), sup
i i
i
(y)}
Theorem 3.10: Let h : (X,,0) (Y,,0) be a
Zhomomorphism of Zalgebras. If B is a cubic Zideal of Y,
i i
i i
max{A (x y), A (y)}
then p (B) is a cubic Zideal of X.
Hence Ai (~A , A )
is a cubic Zideal of a Z
Theorem 3.11: Let h : (X,,0) (Y,,0) be an
i i
i
algebra X.
Zepimorphism of Zalgebras. Let B be a cubic set of Y. If
i Ai
i Ai
Theorem 3.4: Let Ai (~ A , ) be a set of cubic Zideals
p (B)
is a cubic Zideal of X then B is a cubic Zideal of Y.
of a Zalgebra X, where i an index set. If
Theorem 3.12: If A and B be cubic Zideals of Zalgebra X
r sup{r min{~
(x y), ~
A A
A A
i i
(y)}}
then A B is a cubic Zideal in
X X .
r min{r sup ~ A (x y), r sup ~ A (y)}
and
Theorem 3.13: Let A and B be two cubic sets of a Zalgebra
i i X. If
A B
is a cubic Zideal of
X X , the following are
inf{max{A (x y), A (y)}} max{inf A (x y), inf A (y)}
true.
i i i i
, for all
x, yX , then the union of Ai
is again a cubic
(i) ~A (0) ~B (y)
and ~B (0) ~A (x)
for all x, yX .
Zideal of X.
(ii) A (0) B (y)
and
B (0) A (x)
for all x, yX .
Theorem 3.5: Let
Ai (~ A , )
be a set of cubic
Proof: Assume that
~B (y) ~A (0)
and
~A (x) ~B (0)
i Ai
i Ai
Zideals of a Zalgebra X, where
i
an index set. If
for some x, yX .
A A A A
A A A A
inf{max{ (x y), (y)}} max{inf (x y), inf (y)} ,
i i i i
Then
for all
x, y X , then the Pintersection of Ai
is again a
~AB (x, y) r min{~A (x), ~B (y)} r min{~B (0), ~A (0)}
cubic Zideal of X.
Theorem 3.6: Let
Ai (~ A , )
be a set of cubic
which is a contradiction.
~AB (0,0)
i Ai
i Ai
Zideals of a Zalgebra X, where
i Ai
i Ai
r sup{r min{~ A (x y), ~ (y)}}
i
an index set. If
Similarly, assume that A (x) B (0)
for some x, yX .
and
B (y) A (0)
r min{r sup ~ A (x y), r sup ~ A (y)} , for all
x, yX , then
Then
i i
AB (x, y) max{A (x), B (y)} max{B (0), A (0)}
the Punion of Ai is again a cubic ZSubalgebra of X.
AB (0,0)
which is also a contradiction. Thus proving the result. Theorem 3.14: Let A and B be two cubic sets of a Zalgebra X such that A B is a cubic Zideal of X X .Then either A or B is a cubic Zideal of X.
IV CONCLUSION
In this article, we have introduced cubic Zideals in Zalgebras and discussed their properties. We extend this concept in our research work.
V 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

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

Y.Imai and K.Iseki:On axiom systems of propositional calculi XIV, Proceedings of the Japan Academy,42(1966),pp.1922.

K.Iseki:An algebra related with a propositional calculus, Proceedings of the Japan Academy, 42(1966), pp.2629.

Y.B.Jun,S.T.Jung and M.S.Kim: Cubic Subgroups,Annals of Fuzzy Mathematics and informatics,2(1)(2011),pp.915.

Y.B.Jun,C.S.Kim and M.S.Kang:Cubic Subalgebras and Ideals of BCK/BCIalgebras,Far East Journal of Mathematical Sciences,44(2)(2010),pp.239250.

Y.B.Jun,C.S.Kim and K.O.Yang:Cubic Sets,Annals of Fuzzy Mathematics and informatics,4(1)(2012), pp.8398.

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

S.Sowmiya and P.Jeyalakshmi: 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 o Zalgebras, AIP Conference Proceedings 2261 (2020), 0300981 – 0300985.

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, Intuitionistic LFuzzy Structures in ZAlgebras,International Journal of Engineering Research & Technology, 10(2)(2021), pp.497501.

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).

S.Sowmiya and P.Jeyalakshmi, IntervalValued Fuzzy Structures in ZAlgebras (Submitted).

S.Sowmiya and P.Jeyalakshmi, IntervalValued Intuitionistic Fuzzy Structures in ZAlgebras(Submitted).

S.Sowmiya and P.Jeyalakshmi, Cubic ZSubalgebras in ZAlgebras(Submitted).

L.A.Zadeh, Fuzzy Sets, Information and Control, 8 (1965), 338353.

L.A.Zadeh, The concept of a linguistic variable and its application to approximate reasoningI,Inform.Sci., 8 (1975), 199249.