Author : G. Senthil Kumar, V. Selvan
International Journal of Scientific & Engineering Research Volume 3, Issue 1, January-2012
ISSN 2229-5518
Download Full Paper : PDFAbstract— 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 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 SEMIRINGSDefinition 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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