Intuitionistic L-Fuzzy Structures in Z-Algebras

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Intuitionistic L-Fuzzy Structures 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, we initiate and explore the idea of intuitionistic L-Fuzzy Z-Subalgebras and intuitionistic L-Fuzzy Z-ideals in Z-algebras. 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 L-Fuzzy Z-Subalgebras and

  1. algebras. A mapping

    h : (X,,0) (Y,,0)

    is said to

    intuitionistic L-Fuzzy Z-ideals under Z-homomorphism and cartesian product in Z-algebras.

    2010 Mathematics Subject Classification. 03B20, 03B52

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

    ,intuitionistic L-fuzzy Z-Subalgebra, intuitionistic L-fuzzy Z-ideal .

    1. INTRODUCTION

      Atanassov and Stoeva [1] introduced the notion of intuitionistic L-fuzzy sets in 1984 as an extension of Goguens

      [3] notion of L-fuzzy set. Here they have defined both membership and non-membership 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 L-fuzzy structures. Currently, in the year 2017, Chandramouleeswaran et al.[2] introduced a new class of algebra called Z-algebra that arise from the notion of propositional calculi. In our previous articles [4, 5, 6, 7, 8, 9, 10, 11] we have introduced fuzzy Z-Subalgebras, fuzzy Z- ideals, fuzzy H-ideals, fuzzy p-ideals, fuzzy implicative ideals, intuitionistic fuzzy Z-Subalgebras and intuitionistic fuzzy Z- ideals in Z-algebras. In this article, we have initiated intuitionistic L-fuzzy Z-Subalgebras and intuitionistic L-fuzzy Z-Ideals in Z-algebras.

    2. PRELIMINARIES

      be a Z-homomorphism of Z-algebras 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 µA 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 non-membership 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 L-fuzzy 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

      1. 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 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

        is called sup-inf property of A.

      2. 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

      1. Let A x,A (x),A (x) | x X be an intuitionistic

        0.3 if x 2,3

        L-fuzzy 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

        L-fuzzy 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 L-fuzzy Z-Subalgebra of X.

        By applying the definition of intuitionistic L-fuzzy 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 L-fuzzy

        inf

        A (z) if

        p (y) {x | h(x) y}

        Z- Subalgebras of a Z-algebra X. Then

        A1 A2

        is an

        h(A)

        (y) zp ( y)

        intuitionistic L-fuzzy Z-Subalgebra of X.

        1 otherwise

        is

        We can generalize the above theorem as follows.

        Corollary 3.4: Let Ai | i be a family of intuitionistic

      2. Let B y,A (y),A (y) | yY be an intuitionistic

      L-fuzzy Z-Subalgebras of a Z-algebra X. Then Ai is an

      fuzzy set in Y. The pre-image of B under h, symbolized by

      i

      p (B) { x,

      (x),

      (x)

      xX}

      defined by:

      intuitionistic L-fuzzy Z-Subalgebra 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 L-fuzzy set of X.

      Theorem 3.5: An intuitionistic L-fuzzy set A A ,A is

      an intuitionistic L-fuzzy Z-Subalgebra of a Z-algebra X if

    3. – –

and only if the L-fuzzy sets A

and

(A )c

are L-fuzzy

INTUITIONISTIC L FUZZY Z SUBALGEBRAS IN

Z-ALGEBRAS

Z-Subalgebras of X.

Theorem 3.6: A A ,A

is an intuitionistic L-fuzzy

In this section we introduce the notion of Intuitionistic L-Fuzzy Z-Subalgebra of a Z-algebra. Also we prove some interesting results.

Definition 3.1 : An Intuitionistic L-fuzzy Set

Z-Subalgebra of a Z-algebra X if and only if

(i) A A , (A )c and (ii) A (A )c , A , both are

intuitionistic L-fuzzy Z-Subalgebras of X.

Proof: Let A A ,A be an intuitionistic L-fuzzy

A A ,A in a Z-algebra X,,0 is called an

Z-Subalgebra of a Z-algebra X.

Intuitionistic L-fuzzy Z-Subalgebra 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 Z-algebra 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 L-fuzzy set A A

,A

in X defined by

From (i) and (iii), we get A is an intuitionistic L-fuzzy Z-Subalgebra of X.

And, from (ii) and (iv), we get A is an intuitionistic

L-fuzzy Z-Subalgebra of X.

Conversely, assume that A A , (A )c and

A (A )c , A are intuitionistic L-fuzzy Z-Subalgebras

of a Z-algebra 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 L-fuzzy Z-Subalgebra of X.

Analogously, we can prove the following result.

Theorem 3.7: An intuitionistic L-fuzzy set A A ,A in

a Z-algebra X is an intuitionistic L-fuzzy Z-Subalgebra of X if and only if UA;s and LA ; t are Z-Subalgebras of

with sup-inf property. Then the image

hA y,hAy,hAy yYof A under h is an

intuitionistic L-fuzzy Z-Subalgebra of Y.

Theorem 3.15 : Let h : (X,,0) (Y,,0) be a

h

h

B

B

h

h

B

B

Z-homomorphism of Z-algebras and B be an intuitionistic L-fuzzy Z-Subalgebra 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 Z-Subalgebra of a Z-algebra X can be

L-fuzzy Z-Subalgebra of X. Converse is true if h is an Z-epimorphism.

realized as both the upper s-level and lower t-level

Z-Subalgebras of some intuitionistic L-fuzzy Z-Subalgebras

Proof: If h is an Z-epimorphism and pB

is an

of X.

Analogously, the following theorems can be proved. Theorem 3.9: Let X be a Z-algebra. Then any given chain of Z-Subalgebras Q0 Q1 … Qr X , there exists an

intuitionistic L-fuzzy Z-Subalgebra of a Z-algebra 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 L-fuzzy Z-Subalgebra A of X whose upper s-level and lower t-level Z-Subalgebras are exactly the Z-Subalgebras of this chain.

Theorem 3.10: Let A be an intuitionistic L-fuzzy Z-Subalgebra of a Z-algebra X. Then

  1. two upper s-level 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 )

  2. two lower t-level 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 Z-algebra and A be an intuitionistic L-fuzzy Z-Subalgebra of X.

  1. If Im(A ) s1,…,sn , then the family of

    Z-Subalgebras UA;si , i 1,2,,n constitutes all the upper s-level Z-Subalgebras of A.

    Thus B is an intuitionistic L-fuzzy Z-Subalgebra of a Z-algebra Y.

    IV INTUITIONISTIC L-FUZZY Z-IDEALS IN Z-ALGEBRAS

    In this section we introduce the notion of Intuitionistic

    L-fuzzy Z-ideal of a Z-algebra and some interesting results are obtained.

    Definition 4.1: An intuitionistic L-fuzzy set A A ,A

  2. If Im(A ) t1,…, tr, then the family of

Z-Subalgebras LA ; ti , i 1,2,, n constitutes all the lower t-level Z-Subalgebras of A.

in a Z-algebra X,,0 is called an intuitionistic L-fuzzy 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 L-fuzzy

(ii) A (x)

A (x y) A (y)

Z-Subalgebra of a Z-algebra 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

L-fuzzy Z-Subalgebras of a Z-algebra X. Then A B is an intuitionistic L-fuzzy Z-Subalgebra of X X .

Theorem 3.14: Let h be a Z-homomorphism from a Z-algebra X,,0 onto a Z-algebra Y,,0 and

A A ,A be an intuitionistic L-fuzzy Z-Subalgebra of X

Then X,,0 is a Z-algebra. Define an intuitionistic L-fuzzy

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 L-fuzzy Z-ideal of a

(ii) B(y) B(h(x)) pBx pB0= B (h(0))

Z-algebra X.

By applying the definition of an intuitionistic L-fuzzy 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 L-fuzzy Z-ideals of a Z-algebra X is again an inuitionistic L-fuzzy

pB

a b

pB

b

Z-ideal 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)

L-fuzzy Z-ideals of a Z-algebra X. Then Ai is an

(iv)

B (x) B (h(a)) 1 a

i h B

intuitionistic L-fuzzy Z-Ideal of X.

pBa b pBb

Lemma 4.5: An intuitionistic L-fuzzy set A A ,A is an intuitionistic L-fuzzy Z-ideal of a Z-algebra X if and only

B (h(a b)) B (h(b))

(h(a) h(b)) (h(b))

if the L-fuzzy sets and ( )c are L-fuzzy Z-ideals of B B

A A B (x y) B (y)

X. Hence B is an intuitionistic L-fuzzy Z-ideal of Y.

Theorem 4.6: Let A A ,A be an intuitionistic

L-fuzzy set in a Z-algebra X. Then A A ,A is an intuitionistic L-fuzzy Z-ideal 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 L-fuzzy

Z-ideals in a Z-algebra X. Then A B is an intuitionistic L-fuzzy Z-ideal of X X .

L-fuzzy Z-ideals of X.

Theorem 4.7: An intuitionistic L-fuzzy set A A ,A is

an intuitionistic L-fuzzy Z-ideal of a Z-algebra X if and only if for all s, t L , the sets UA;s and LA ; t are either

empty or Z-ideals 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 Z-algebra

X,,0 onto a Z-algebra Y,,0 and A be an intuitionistic

L-fuzzy Z-ideal of X with sup-inf property. Then image of A,

AB

x1, x2

hA y,hAy,hAy yY is an intuitionistic L-fuzzy Z-ideal of Y.

Theorem 4.9: Let h : (X,,0) (Y,,0) be a

Z-homomorphism of Z-algebras and B be an intuitionistic L-fuzzy Z-ideal 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 )

L-fuzzy Z-ideal of X.

Theorem 4.10: Let h : (X,,0) (Y,,0) be an

AB

((x1, x2

) (y1, y2

))

AB

(y1, y2 )

Z-epimorphism of Z-algebras. Let B be an intuitionistic

AB (x1, x2 ) A (x1 ) B (x2 )

L-fuzzy set of Y. If p B is an intuitionistic L-fuzzy

Z-ideal of X then B is an intuitionistic L-fuzzy Z-ideal of Y.

Proof: Assume that If p B is an intuitionistic L-fuzzy Z-ideal 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 L-fuzzy Z-ideal of X X . Theorem 4.12: Let A and B be two intuitionistic L-fuzzy sets in a Z-algebra X. If A B is an intuitionistic L-fuzzy Z-ideal of X X , the following are true.

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

  2. A 0 B y and B 0 A x for all x, yX . Theorem 4.13: Let A and B be two intuitionistic L-fuzzy sets in a Z-algebra X such that A B is an intuitionistic L-fuzzy Z-ideal of X X . Then either A or B is an intuitionistic

L-fuzzy Z-Ideal of X.

V CONCLUSION

In this article, we have introduced intuitionistic L-fuzzy Z-Subalgebras and intuitionistic L-fuzzy Z-ideals in Z-algebras 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.

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