Cubic Z-Ideals in Z-Algebras

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Cubic Z-Ideals in Z-Algebras

S. Sowmiya

Assistant Professor, Department of Mathematics, Sri Ramakrishna Engineering College Vattamalaipalayam,Coimbatore-22,Tamilnadu,India

P. Jeyalakshmi

Professor and Head, Department of Mathematics, Avinashilingam Institute for Home Science and Higher Education for Women, Coimbatore-43,Tamilnadu,India.

AbstractIn this article, the notions of Cubic Z-Ideals in Z-algebras is introduced and some of their properties are investigated. The Z-homomorphic image and inverse image of

(Z3)

(Z4)

x x x

x y y x when x 0 and y 0

x, y X.

cubic Z-Ideals in Z- algebras is investigated. Also, the cartesian product of cubic Z-Ideals in Z-algebras are also discussed.

2010 Mathematics Subject Classification. 06F35, 03G25, 08A27

KeywordsZ-algebra, Z-ideal, Z-homomorphism, Cubic

Definition 2.2[1] Let (X,,0) and (Y,,0) be two

Z-algebras. A mapping h : (X,,0) (Y,,0) is said to be a Z-homomorphism of Z-algebras 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

Z-ideal .

  1. 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 interval-valued fuzzy set

    A A

    Chandramouleeswaran et al. [1] introduced the concept of

    Z-algebra 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 interval-valued fuzzy set whose membership

    For two cubic sets A (~A

    define

    , A

    ) and B (~B

    , B

    ) in X, we

    function is many-valued and form an interval in the

    membership scale. In our earlier paper [718] we have introduced the concept of cubic set to Z-Subalgebras in

    1. A B iff ~A ~B and

      A B

      Z-algebras and the concepts of fuzzy set, interval-valued fuzzy

    2. A B iff A B and B A.

      set, intuitionistic fuzzy set, intuitionistic L-fuzzy set, interval- valued intuitionistic fuzzy set to Z-Subalgebras and Z-ideals in

    3. Ac x,A

    (x),~A

    (x) | xX

    Z-algebras. In 2012, using a fuzzy set and an interval-valued 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 Z-Ideals of Z-algebras and investigated some of their properties.

  2. 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

    interval-valued upper [s1,s2 ] -level subset of A and

    Definition 2.1[1] A Z-algebra 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 t-level subset

    of A.

    Definition 2.5:[4] A cubic set A (~A , A ) in a nonempty set X is said to have the rsup-inf 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 pre-image) of B under h, denoted by

    (i) P-union and P-intersection 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 .

  3. CUBIC Z-IDEALS IN Z-ALGEBRAS

(ii) Union and intersection denoted by Ai

i

defined as follows.

and Ai are

i

In this section, the notion of Cubic Z-ideals in Z-algebras is defined and corresponding results are proved.

A x, ~ (x), (x) x X

Definition 3.1: Let (X,,0) be a Z-algebra. A cubic set

i

i

Ai

i

Ai

i

A (~A , A ) in X is called a cubic Z-ideal 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 Z-algebra 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 Z-ideal of a

and

Z-algebra X.

Theorem 3.3: The intersection of any set of cubic Z-ideals of a Z-algebra X is also a cubic Z-ideal of X.

Theorem 3.7: Cubic set A (~A , A ) of a Z-algebra X is a cubic Z-ideal 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 Z-ideals of X.

index set, be a set of cubic Z-ideals of a Z-algebra 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)

Z-algebra X. Then A is a cubic Z-ideal 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 Z-ideals of X.

Theorem 3.9: Let h be a Z-homomorphism from a

r min{r inf ~ A (x y), r inf ~ A (y)}

i i Z-algebra

(X,,0)

onto a Z-algebra

(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 Z-ideal of X with rsup-inf property. Then image of A denoted by h(A) is a cubic Z-ideal 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

Z-homomorphism of Z-algebras. If B is a cubic Z-ideal of Y,

i i

i i

max{A (x y), A (y)}

then p (B) is a cubic Z-ideal of X.

Hence Ai (~A , A )

is a cubic Z-ideal of a Z-

Theorem 3.11: Let h : (X,,0) (Y,,0) be an

i i

i

algebra X.

Z-epimorphism of Z-algebras. Let B be a cubic set of Y. If

i Ai

i Ai

Theorem 3.4: Let Ai (~ A , ) be a set of cubic Z-ideals

p (B)

is a cubic Z-ideal of X then B is a cubic Z-ideal of Y.

of a Z-algebra X, where i an index set. If

Theorem 3.12: If A and B be cubic Z-ideals of Z-algebra X

r sup{r min{~

(x y), ~

A A

A A

i i

(y)}}

then A B is a cubic Z-ideal 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 Z-algebra

i i X. If

A B

is a cubic Z-ideal 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 .

Z-ideal 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

Z-ideals of a Z-algebra 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 P-intersection of Ai

is again a

~AB (x, y) r min{~A (x), ~B (y)} r min{~B (0), ~A (0)}

cubic Z-ideal of X.

Theorem 3.6: Let

Ai (~ A , )

be a set of cubic

which is a contradiction.

~AB (0,0)

i Ai

i Ai

Z-ideals of a Z-algebra 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 P-union of Ai is again a cubic Z-Subalgebra 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 Z-algebra X such that A B is a cubic Z-ideal of X X .Then either A or B is a cubic Z-ideal of X.

IV CONCLUSION

In this article, we have introduced cubic Z-ideals in Z-algebras 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.

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