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Lower and Upper Approximation of Fuzzy Ideals in a Semiring
« on: February 13, 2012, 04:24:02 am »
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Author : G. Senthil Kumar, V. Selvan
International Journal of Scientific & Engineering Research Volume 3, Issue 1, January-2012
ISSN 2229-5518
Download Full Paper : PDF

Abstractó 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 Termsó Semiring, lower approximation, upper approximation, fuzzy ideal, fuzzy prime ideal, rough ideal.

1   INTRODUCTION                                                                      
The 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 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.

The notion of rough subgroups was introduced by Biswas and Nanda [1]. The concept of rough ideal in a semigroup was introduced by Kuroki in [11]. B.Davvaz [3], [2], [4] studied the roughness in many algebraic system such as rings, modules, n-ary systems,  -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 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.

In this paper, we introduce the concept of rough fuzzy ideal of a semiring. Also we study the notion of rough fuzzy prime ideal in a semiring.

2    CONGRUENCE IN SEMIRINGS
Definition 2.1. A semiring is a nonempty set R on which operations of addition and multiplication have been defined such that the following conditions are satisfied.
(i)   is a commutative monoid with identity element 0;
(ii)   is a monoid with identity element   ;
(iii) Multiplication distributives over addition from either side;
(iv)  ,  for all  .

Throughout this paper   denotes a semiring.

 Definition 2.2.  [6] Let   be an equivalence relation on  , then   is called a congruence relation if   implies   for all  .

Theorem 2.3. [6] Let   be a congruence relation on  , then  and   imply    and   for all  .

Lemma 2.4. [6] Let   be a congruence relation on a semiring  . If   then
(i)    
(ii)   

Definition 2.5. A congruence relation   on   is called complete if
(i)     and
(ii)    .
for all  .

Definition 2.6. A ideal  of a semiring   is a nonempty subset of   satisfying the following condition:
(i)   If   then  .
(ii)   If   and   then  .

A ideal   of a semiring   defines an equivalence relation   on  , called the Bourne relation, given by   if and only if there exists elements   and   of   satisfying  . The relation   is an congruence relation on   [6], [7].



We denote the set of all equivalence classes of elements of   under this relation by   and we will denote the equiva-lence class of an element   of   by  .   

 Throughout this paper   denotes the Bourne congruence relation induced by an ideal   of a semiring  .
   
Definition 2.7. An ideal   of a semiring   is called a  -ideal if   implies   for each   and each  .             


3   LOWER AND UPPER APPROXIMATION OF A FUZZY IDEAL IN A SEMIRING

A mapping   is called a fuzzy subset of  .
A fuzzy subset   of a semiring   is called a fuzzy ideal of   if it has the following properties:   
(i)    
(ii)   
 A fuzzy ideal   of   is said to be normal if  .

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

Definition 3.2. Let   be the Bourne congruence relation on   induced by   and   be a fuzzy subset of . Then we define the fuzzy sets  as follows:
         

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