International Journal of Scientific & Engineering Research, Volume 3, Issue 1, January-2012 1

ISSN 2229-5518

Lower and Upper Approximation of Fuzzy Ideals in a Semiring

G. Senthil Kumar, V. Selvan

Abstract— In this paper, we introduce the rough fuzzy ideals of a semiring. We also introduce and study rough fuzzy prime ideals of a semiring.

Index Terms— Semiring, lower approximation, upper approximation, fuzzy ideal, fuzzy prime ideal, rough ideal.

1 INTRODUCTION

——————————  ——————————
he fuzzy set introduced by L.A.Zadeh [16] in 1965 and the rough set introduced by Pawlak [12] in 1982 are generali- zations of the classical set theory. Both these set theories
are new mathematical tool to deal the uncertain, vague, im- precise and inexact data. In Zadeh fuzzy set theory, the degree of membership of elements of a set plays the key role, whereas in Pawlak rough set theory, the equivalence classes of a set are used to define the lower and upper approximation of a set.
Rosenfeld [13] applied the notion of fuzzy sets to groups and

of addition and multiplication have been defined such that the following conditions are satisfied.

(i)  R,   is a commutative monoid with identity element 0; (ii)  R, is a monoid with identity element 1R ;

(iii) Multiplication distributives over addition from either side;

(iv) 0r  0  r0 , for all r  R .

Throughout this paper R denotes a semiring.

Definition 2.2. [6] Let  be an equivalence relation on R , then

introduced the notion of fuzzy subgroups. After this paper, many researchers applied the theory of fuzzy sets to several algebraic concepts such as rings, fields, vector spaces, etc.

 is called a congruence relation if a,b 

a  x,b  x,  x  a, x  b, ax,bx and  xa, xb

xR .

implies for all

The notion of rough subgroups was introduced by Biswas and

Theorem 2.3. [6] Let  be a congruence relation on R , then

Nanda [1]. The concept of rough ideal in a semigroup was

a,b  and c, d  

imply a  c,b  d  

and

introduced by Kuroki in [11]. B.Davvaz [3], [2], [4] studied the roughness in many algebraic system such as rings, modules, n-ary systems, HV -groups, etc. Osman Kazanci and B.Davvaz [10] introduced the rough prime and rough primary ideals in commutative rings and also discussed the roughness of fuzzy

ac,bd   for all a,b, c, d  R .

Lemma 2.4. [6] Let  be a congruence relation on a semiring R . If

a,b  R then

ideals in rings. The roughness of ideals in BCK algebras was
considered by Y.B. Jun in [8]. In [14] the present authors have
studied rough ideals in semirings.
(i)
(ii)

a  b / a a ,b b   a  b

ab / a a ,b b   ab

In this paper, we introduce the concept of rough fuzzy ideal of

Definition 2.5. A congruence relation  on R is called complete if

a semiring. Also we study the notion of rough fuzzy prime
(i) a  b

 a  b / a a ,b b  and

ideal in a semiring.

(ii) ab

  

 ab / a a ,b b  .

2 CONGRUENCE IN SEMIRINGS

Definition 2.1. A semiring is a nonempty set R on which operations

 G. Senthil Kumar

  

for all a,b  R .

Definition 2.6. A ideal I of a semiring R is a nonempty subset of

R satisfying the following condition:

Department of Mathematics,

(i) If a,b  I

then a  b  I .

Faculty of Engineering and Technology,

SRM University, Kattankulathur, Chennai - 603203, India

Email: gsenthilkumar77@gmail.com

 V. Selvan

Department of Mathematics,

(ii) If a  I and r  R then ar, ra  I .

A ideal I of a semiring R defines an equivalence relation  I on R , called the Bourne relation, given by r I r if and only if there exists elements a and a of I satisfying r  a  r  a .

R. K. M. Vivekananda College, Chennai - 600004, India

Email: venselvan@yahoo.co.in, venselvan@gmail.com
The relation  I
is an congruence relation on R [6], [7].

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 1, January-2012 2

ISSN 2229-5518

(4) If   

then Apr

  

 Apr

 

We denote the set of all equivalence classes of elements of R

I I

under this relation by

R /  I

and we will denote the equiva-

(5) If   

then Apr   

 Apr

 

lence class of an element r of R by r  .

(6)

Apr

Apr

   =Apr

  

Throughout this paper  I
denotes the Bourne congruence

(7)

Apr

 Apr

   = Apr

  

relation induced by an ideal I of a semiring R .

I I I

Definition 2.7. An ideal I of a semiring R is called a  -ideal if

(8)

Apr Apr    =Apr

I  I I

  

r  a  I

implies r  I

for each r  R and each a  I .

(9)

Apr

I

 Apr    = Apr

  

(10) Apr

I

    =Apr

I

    Apr

I

 

3 LOWER AND UPPER APPROXIMATION OF A FUZZY

(11)

Apr    

 Apr

    Apr  

I I I

IDEAL IN A SEMIRING

(12) Apr

I

   

 Apr

I

    Apr

I

 

A mapping  : R  0,1 is called a fuzzy subset of R .

(13)

Apr     = Apr

    Apr  

A fuzzy subset  of a semiring R is called a fuzzy ideal of R
if it has the following properties:

Proof. It is straightforward.

(i)
(ii)

  x  y     x    y 

  xy     x    y 

Theorem 3.4. Let  I be the Bourne congruence relation on R in- duced by the ideal I of R . If  is a fuzzy ideal of R , then

A fuzzy ideal  of R is said to be normal if  0  1 .

Apr   

is a fuzzy ideal of R .

Definition 3.1. A fuzzy ideal  of a semiring R is said to be prime if  is not a constant function and for any two fuzzy ideals 

Proof. We have,

and  of

R,    implies either    or    .

Apr    x  y  =



a x  y  I

  a 

Definition 3.2. Let  I

be the Bourne congruence relation on R

  b x I c y I

 b  c 

induced by I and  be a fuzzy subset of R . Then we define the

 

b x I

  b    c 

fuzzy sets Apr

I

   and Apr    as follows:

c y I

Apr

   x  = 

  a 

   

 I a x

  

 b     

 c  

Apr    x  =

 I

   a 

 b x I

  c y I 

a x I

 Apr    x  

Apr    y 

The fuzzy sets Apr

   and

Apr    are respectively called

I I

Hence,         

I I

Apr 

x  y 

Apr 

x  Apr  y

the  I -lower and  I -upper approximations of the fuzzy set  .
Also we have,
The pair

Apr   , Apr     Apr   

is called a rough

Apr    xy  =



a xy

  a 

fuzzy set with respect to  I
if Apr

I

    Apr    .

 I

   bc 

b x I

c y 

Theorem 3.3. For every approximation space  R, I 

fuzzy subsets , of R , we have:

and every

 I

  b x I c y

  b   c 

(1) Apr

I

  

  

Apr   

(2) Apr

  =

, where 

is the characteristic function of

  

  b



  c 

I      

   

empty set.

 b x I

  c y  I 

(3)

Apr

   = , where

 is the characteristic function of

 Apr

   x  Apr

   y 

I R R R

I I

R . Hence,

Apr    xy  

Apr    x  Apr   y  .

I I I

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 1, January-2012 3

ISSN 2229-5518

Therefore,

Apr    is a fuzzy ideal of R .

    x    x    x for all x  R .

Theorem 3.5. Let  I

be the Bourne congruence relation on R in-

If  and  are fuzzy ideals of a semiring of R , then   is

duced by the ideal I of R . If  is a fuzzy ideal of R , then

also a fuzzy ideal of R .

Apr

I

  

is a fuzzy ideal of R .

Corollary 3.6. If  is a fuzzy ideal of R then

Proof. We have,

Apr   , Apr    is a rough fuzzy ideal of R .

Apr

 I

   x  y  =



a x  y 

  a 

Corollary 3.7. If  and  are fuzzy ideals of R then

 I

   b  c 

b x I

c y

Apr

    , Apr      is a rough fuzzy ideal of R .
 I Let  be a fuzzy subset of R . Then the sets

    b    c 

t  x  R   x  t

and

 s   

    

where  

b x

t x R x t

t  0,1

 I

c y I

   
are called respectively, t-level subset and t-strong level subset
of  .

    b    

  c  

Theorem 3.8. [7] Let  be a fuzzy subset of R . Then  is a fuzzy

 b x

  c y  

  I
   I 

ideal of R iff

t and t

are, if they are non-empty, ideals of R

 Apr

 I

   x   Apr

 I

   y 

for every t [0,1] .

Hence, Apr

  x  y  

Apr

   x  Apr

   y  .

Theorem 3.9. Let  I

be a congruence relation on R . If  is a fuzzy

I

Also we have,

I I

subset of R and t 0,1 , then

Apr

   xy  =

   a 

(i)

Apr   Apr t 



a xy  I

I t I

   bc 

(ii)

 Apr I  

Apr   t 

  s

b x I

c y I

  b x I c y I

  b    c 

Proof.

(i) We have

x  Apr

    Apr

t I

   x  t

   

    b    

  c  

    a   t

 b x

  c y  

a x I

  I
   I 
   a   t,a   x
 Apr

 I

   x   Apr

 I

   y 

 a  t ,a   x

Hence, Apr

   xy  

Apr

   x  Apr

  y  .

  x

 t

I I I

 x  Apr

  

Therefore, Apr

I

   is a fuzzy ideal of R .

(ii) We have

 I t

Let  be a fuzzy subset of R and

Apr   , Apr    a

x   Apr   

 Apr    x   t

I t I

   a   t

rough fuzzy set. If Apr

I

   and

Apr    are fuzzy ideals of

a x I

R , then     , AprI    a rough fuzzy ideal of R .
Let  and  be two fuzzy subsets of R . The inclusion   
   a   t, for some a  x

is defined by by
  x    x for all x  R , and  
is defined

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 1, January-2012 4

ISSN 2229-5518

 a   s , for some a  x

4 ROUGH FUZZY PRIME IDEALS

  x

t  I

  s  
An ideal P of a semiring R is prime if and only if whenever
 x  Apr

 s 

HK  P , for ideals H and K of R , we must have either

H  P or K  P .

Lemma 3.10. If  is a normal fuzzy ideal of a semiring R and if

Apr

I

    

then I   .

Theorem 4.1. Let  be a fuzzy prime ideal of a semiring R .

(i) If  I
is a complete congruence relation on R and

Proof. Since Apr

I

     , we have      x     x 

Apr

I

     , then  is a lower rough fuzzy prime ideal
x  R
That is,

  a     x ,x  R .

ax I

of R .
(ii) If  I is a complete congruence relation on R then  is a

  a    x,a x

In particular,  a   0,a 0 .

As  is a normal fuzzy ideal of R ,  0  1 .
upper rough fuzzy prime ideal of R .

Proof.

(i) Since  is a fuzzy prime ideal, t t 0,1 is, if it is non- empty, a prime ideal of R .
Hence,  a   0  1,a 0

Thus, I   .
 I.
By Theorem 3.5. [14], we obtain that

Apr

 t 

if it is non-
The converse of the above leema is true if I is a  -ideal of R .
empty, is a prime ideal of R .
Hence by Theorem 3.9.,      is a prime ideal of R and

Lemma 3.11. If I is a  –ideal of R such that

I  

then

hence by Theorem 3.8.,

Apr

I t

   is a fuzzy prime ideal of R .

Apr

I

     .

I

(ii) It can be seen in a similar way.
Proof. Since I   , we have   x  1,x  I .
By Theorem 3.3, Apr

I

     .

Theorem 4.2. Let  I

be a complete congruence relation on a semir-

So, to complete the proof we need to show that   Apr

I

   .

ing R . Then  is a lower (an upper) rough fuzzy prime ideal iff

i.e., we need to show that   x  Apr

I

   x,x  R .

for t 0,1t , t

prime ideals of R .

are, if they are nonempty, lower [upper] rough

i.e., we need to show that   x   a,a x .

Case (i) Let x  I
and a x .

Proof. It is straightforward.

Then a  i1  x  i2 , for some i1,i2  I .

 a  i1     x  i2   min  x,  i2     x .

Theorem 4.3. [14] Let f be an onto homomorphism of a semiring

In particular,  a    x,a x .

R to a semiring R and let  2

and A be a subset of R . Then

be a congruence relation on R

This implies that   x       x  .

Case (ii) Let x  I , then   x  1 .
(i)

1  a,b  R  R /  f a , f b 2  is a congruence rela-

We have  x

 0 .

tion on R .

(ii) If  is complete and f is one – to – one, then  is com-

Since I is a  –ideal, 0  I .

2 1

plete.

We have,      x   

 a     a   1    x  .

(iii)

f  Apr  A  Apr  f  A

I a x I

aI

Hence in this case also, Apr

I

     .

(iv)

f Apr

  A  Apr

 f  A . If f is one-to-one, then

Combining Case (i) and (ii), we get   Apr

I

  

f Apr

1 2

 A  Apr

 f  A .

1 2

http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 3, Issue 1, January-2012 5

ISSN 2229-5518

Theorem 4.4. Let f be a surjective homomorphism of a semiring R to a semiring R . Let  2 be a complete congruence relation on R and  be a fuzzy subset of R .

If 1   x, y   R  R /  f  x, f  y  2  , then

[2] B.Davvaz, “Roughness in rings”, Inform. Sci., vol. 164, pp. 147-163,

2004.

[3] B.Davvaz, “A new view of approximations in Hv-groups”, Soft.

Comput., 10(11), pp. 1043-1046, 2006.

[4] B.Davvaz, “Roughness based on fuzzy ideals”, Inform. Sci., vol. 176, pp. 2417-2437, 2006.

(i)

Ap r    is a fuzzy ideal (fuzzy prime ideal) of R if and

[5] B.Davvaz, M.Mahdavipour, “Roughness in modules”, Inform. Sci.,

only if
of R .

Apr  f   

is a fuzzy ideal (fuzzy prime ideal)

vol. 176, pp. 3658-3674, 2006.

[6] U.Hebisch, H.J.Weinert, “Semi rings - Algebraic Theory and Applica-

(ii) If f is one – to – one, then Apr

1

  

is a fuzzy ideal

tions in Computer Science”, World Scientific Series in Algebra, 5

1998.

(fuzzy prime ideal) of R if and only if fuzzy ideal (fuzzy prime ideal) of R .

Proof.

Apr

1

 f   

is a

[7] Jonathan S. Golan, “Semirings and their applications”, Kluwer Aca- demic Publishers, 1999.

[8] Y.B.Jun, “Roughness of ideals in BCK-algebras”, Sci. Math. Japonica, vol. 57(1), pp. 165-169, 2003.

(i) By Theorem 3.5. [14] we obtain that

Apr   

is a fuzzy

[9] O.Kazanci, B.Davvaz, “On the structure of rough prime (primary)

ideal (fuzzy prime ideal) of R iff

Apr

  s , if it is non-

ideals and rough fuzzy prime (primary) ideals in commutative rings”, Information Sciences, vol. 178, pp. 1343-1354, 2008.

empty, an ideal (prime ideal) of R , for every t 0,1 .

[10] N.Kuroki, “Rough ideals in semigroups”, Inform. Sci., vol. 100, pp.

We have by Theorem 3.9., Apr

  s  Apr

 s  .

139-163, 1997.

Thus we obtain that

Apr

 s 

is an ideal (prime ideal) of R

[11] N.Kuroki, P.P.Wang, “The lower and upper approximations in a fuzzy group”, Inform. Sci., vol. 90, pp. 220-230, 1996.

iff

Apr  f

 s 

is an ideal (prime ideal) of R .

[12] Z.Pawlak, “Rough sets”, Int. J. Inf. Comp. Sci., vol. 11, pp. 341-356,

Since  s 

  s

  s 

  s 

1982.

f t

 f 

t , Apr2

f t

 Apr f 

2 t

  Apr2 f   t  .

[13] A.Rosenfeld, “Fuzzy groups”, J. Math. Anal. Appl., vol. 35, pp. 512-

517, 1971.

Therefore,

 Apr

f   

is an ideal (prime ideal) of R for

[14] V.Selvan, G.Senthil Kumar, “Rough ideals in semirings” - submitted for publication.

every t 0,1 .

[15] Q.M.Xiao, Z.L.Zhang, “Neighbourhood operator systems and aproxi

Thus,

Apr

f    is a fuzzy ideal (fuzzy prime ideal) of R .

mations”, Inform. Sci., vol. 144, pp. 201-217, 2002.

(ii) If f is one – to – one, then

Apr  A  Apr  f  A .

[16] L.A.Zadeh, “Fuzzy sets”, Inform. Control, vol. 8, pp. 338-353, 1965.

This proof is similar to that of (i).

5 CONCLUSION

The theory of semirings has wide applications in several areas such as optimization theory, discrete event dynamical systems, automata theory, formal language theory and parallel compu- ting. The theory of fuzzy sets and rough sets also has many applications in the above areas. In this paper, we developed the concept of a rough fuzzy ideal of a semiring. We certainly hope that our work will be very useful both in the theoretical and application aspect. We also propose to work further on this area to bring out many more interesting properties of rough fuzzy ideals in semirings.

REFERENCES

[1] R.Biswas, S.Nanda, “Rough groups and rough subgroups”, Bull.

Polish Acad. Sci. Math., vol. 42, pp. 251-254, 1994.

http://www.ijser.org