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Volume 2013 (2013), Article ID 594636, 7 pages
A Study on Fuzzy Ideals of -Groups
1Department of Mathematics, Yazd University, Yazd, Iran
2Department of Mathematics, Manipur University, Imphal, Manipur 795003, India
Received 10 February 2013; Accepted 19 June 2013
Academic Editor: Antonio M. Cegarra
Copyright © 2013 B. Davvaz and O. Ratnabala Devi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Using the idea of the new sort of fuzzy subnear-ring of a near-ring, fuzzy subgroups, and their generalizations defined by various researchers, we try to introduce the notion of ()-fuzzy ideals of -groups. These fuzzy ideals are characterized by their level ideals, and some other related properties are investigated.
1. Introduction and Basic Definitions
The concept of a fuzzy set was introduced by Zadeh  in 1965, utilizing what Rosenfeld  defined as fuzzy subgroups. This was studied further in detail by different researchers in various algebraic systems. The concept of a fuzzy ideal of a ring was introduced by Liu . The notion of fuzzy subnear-ring and fuzzy ideals was introduced by Abou-Zaid . Then in many papers, fuzzy ideals of near-rings were discussed for example, see [5–11]. In , the idea of fuzzy point and its belongingness to and quasi coincidence with a fuzzy set were used to define -fuzzy subgroup, where , take one of the values from , . A fuzzy subgroup in the sense of Rosenfeld is in fact an -fuzzy subgroup. Thus, the concept of -fuzzy subgroup was introduced and discussed thoroughly in . Bhakat and Das  introduced the concept of -fuzzy subrings and ideals of a ring. Davvaz [14, 15], Narayanan and Manikantan , and Zhan and Davvaz  studied a new sort of fuzzy subnear-ring (ideal and prime ideal) called -fuzzy subnear-ring (ideal and prime ideal) and gave characterizations in terms of the level ideals. In [18, 19], the idea of fuzzy ideals of -groups was defined, and various properties such as fundamental theorem of fuzzy ideals and fuzzy congruence were studied, respectively. In the present paper, we extend the idea of -fuzzy ideals of near-rings to the case of -groups and introduce the idea of fuzzy cosets with some results.
We first recall some basic concepts for the sake of completeness.
By a near-ring we mean a nonempty set with two binary operations “+” and “” satisfying the following axioms:(i) is a group,(ii) is a semigroup, (iii) for all .It is in fact a right near-ring because it satisfies the right distributive law. We will use the word “near-ring” to mean “right near-ring.” is said to be zero symmetric if for all . We denote by .
Note that the missing left distributive law, , has to do with linearity if is considered as a function.
Example 1. Let be a group, and let be the set of all mappings from into . We define + and on by Then, is a near-ring.
Just in the same way as -modules or vector spaces are used in ring theory, -groups are used in near-ring theory.
By an -group we mean a nonempty set together with a map written as satisfying the following conditions:(i) is a group (not necessarily abelian), (ii), (iii) for all , .
Example 2. Let be a subnear-ring of . Then, is an -group via function application as operation.
Example 3. The additive group of a near-ring is an -group via the near-ring multiplication.
An ideal of -group is an additive normal subgroup of such that and for all , , . A mapping between two -groups and is called an -homomorphism if and for all , .
Throughout this study, we use to denote a zero-symmetric near-ring and to denote an -group.
For any fuzzy subset of , denotes the image of . For any subset of , denotes the characteristic function of .
Definition 4 (see ). A fuzzy subset of a group is called a fuzzy subgroup of if it satisfies the following conditions: (i), (ii), for all .
Definition 5. For a fuzzy subset of , , the subset is called a level subset of determined by and .
The set is called the support of and is denoted by . A fuzzy subset of of the form is said to be a fuzzy point denoted by . Here is called the support point, and is called its value. A fuzzy point is said to belong to (resp., quasi coincident with) a fuzzy set written as (resp., ) if (resp., ). If or , then we write . The symbols , mean that , do not hold, respectively.
Remark 8. For any -fuzzy subgroup of such that for some , then and if , then for all . So, is just the usual fuzzy subgroup in the sense of Rosenfeld.
Remark 9. It is noted that if is a fuzzy subgroup then it is an -fuzzy subgroup of . However the converse may not be true.
Here onwards we assume that is an -fuzzy subgroup in the nontrivial sense for which case we have .
Definition 10 (see ). An -fuzzy subgroup of a group is said to be -fuzzy normal subgroup if for any and ,
Remark 11 (see ). The condition of -fuzzy normal subgroup is given in the equivalent forms as(i),(ii),(iii), for all .
Here denotes the commutator of , in .
Definition 12 (see ). Let be a fuzzy subset of an -group . It is called a fuzzy -subgroup of if it satisfies the following conditions: (i), (ii), for all , .
Remark 13. If is a unitary -group, the above conditions are equivalent to conditions and for all , .
Definition 15 (see ). A fuzzy set of a near-ring is called an -fuzzy subnear-ring of if for all , and (i) (a) , (b) ,(ii). is called an -fuzzy ideal of if it is -fuzzy subnear-ring of and(iii), (iv), (v), for all .
2. Generalized Fuzzy Ideals
Definition 16. A fuzzy subset of an -group is said to be an -fuzzy subgroup of if , , ,(i), (ii), (iii), .
Lemma 17. Let be a fuzzy subset of and . Then,(i), (ii), for all , (iii), for all , .
Proof. (i) Let . Consider the case (a): .
Assume that . Choose such that which implies that , but [as and ]. Consider the case (b): . Assume that . Choose such that so that but .
Conversely, let . Then, . Thus if either or and if both and which means .
(ii) Let , . Suppose . Choose such that . Then, but which contradicts the hypothesis. So, for all .
Conversely, let . Then, . But we have or according as or .
(iii) Let and . Suppose . Choose such that . Then, that is, , but as and .
Conversely let , ; then . But or according as or or .
Theorem 18. Let be a fuzzy subset of . Then, is an -fuzzy subgroup of if and only if the following conditions are satisfied: (i), (ii), (iii), for all , .
Proof. It follows from the previous lemma.
Definition 19. A fuzzy subset of an -group is said to be -fuzzy ideal of if it is an -fuzzy subgroup and satisfies the following conditions:(i), (ii), for any , .
Lemma 20. Let be a fuzzy subset of and . Then,(i), (ii).
Proof. (i) Assume that . Choose such that . But or according as or . So, or or . But or , respectively, which contradicts the hypothesis.
Conversely, assume that , then . For any , we have or according as or or . So, .
(ii) Assume that or for some , . According as or . Choose such that . In either case, and . So, , a contradiction.
Conversely, assume that for all , . Let . Then, . So, or according as or . So, .
Theorem 21. Let be an fuzzy subgroup of . Then, is an -fuzzy ideal of if and only if (i), for all , (ii), for all , .
Proof. It is immediate from Lemma 20.
By definition, a fuzzy ideal of is an -fuzzy ideal of . But the converse is not true in general as shown by the following example.
Example 22. Consider (written additively) to be a -group. Define a fuzzy subset of as , , which is not fuzzy ideal as ; it contradicts the condition (iv) of Definition 14. As , , and or or , thus, the notion of -fuzzy ideal is a successful generalization of fuzzy ideals of as introduced in .
Theorem 23. Let be any family of -fuzzy ideals of . Then, is an -fuzzy ideal of .
Proof. It is straightforward.
Theorem 24. A nonempty subset of is an ideal of if and only if is an -fuzzy ideal of .
Proof. If is an ideal of , it is clear from [18, Proposition 2.11] that is fuzzy ideal of . Since every fuzzy ideal is -fuzzy ideal, is -fuzzy ideal of .
Conversely, let be an -fuzzy ideal of . Let , . So, . Let , , , , , , , . Then, is an ideal of .
Theorem 25. A fuzzy subset of is an -fuzzy (subgroup) ideal of if and only if the level subset is a (subgroup) ideal for .
Proof. We prove the result for -fuzzy ideal . Let be an -fuzzy ideal of . Let , , . (i), (ii), (iii), (iv), , . Hence, is an ideal of . Again, let be an ideal of for all . If possible, let there exist such that . Let be such that , and , a contradiction. So, , for all . For , let . If possible let be such that . This implies , but , a contradiction. Similarly, we can prove that , , , .
Remark 26. For , may be an -fuzzy ideal of , but may not be an ideal of . Let in Example 22. Then, . is not an ideal of as it is not a normal subgroup of .
We are looking for a corresponding result when is an ideal of for all .
Theorem 27. Let be a fuzzy subset of an -group . Then, is an ideal of for all if and only if satisfies the following conditions: (i), (ii), (iii), (iv), for all , .
Proof. Suppose that is an ideal of for all . In order to prove (i), suppose that for some , . Let . So, and . Since is an ideal, . So, , a contradiction. In order to prove (ii), suppose that , and (say). Then, , a contradiction. Similarly, we can prove (iii) and (iv).
Conversely, suppose that conditions (i) to (iv) hold. We show that is an ideal of for all . Let . Then, . So, . Let , . Then, so . For , , . Also, if , , , . Hence, . Then, is an ideal of .
A definition for the previous kind of fuzzy subset was given for the case of near-rings in . Now, we give the definition for -groups.
Definition 28. A fuzzy subset of is called an -fuzzy subgroup of if for all and for all , ,(i) (a) implies or , (b) implies ,(ii) implies . Moreover, is called an -fuzzy ideal of if is -fuzzy subgroup of and(iii) implies , (iv) implies .
Theorem 29. A fuzzy subset of is an -fuzzy ideal of if and only if (1), ,(2),(3),(4).
Proof. . Let be such that . Let ; then , . So we must have or . But and . Here or then and or and then , a contradiction.
Conversely, let . Then, . If , then . Hence, either or which implies or . Thus, or .
Again if , then by Suppose that or then or . It follows that either or , and thus or .
: Suppose that there exists such that . If then and so that . But then we must have either or . Also we have . So, which means that , a contradiction.
Conversely, suppose that then . If , then which gives . Again if by we have . Putting , then so that which means that . Similarly, we can prove the remaining parts.
Theorem 30. A fuzzy subset of is an -fuzzy ideal if and only if is an ideal of for all .
3. Fuzzy Cosets and Isomorphism Theorem
In this section, we first study the properties of -fuzzy ideals under a homomorphism. Then, we introduce the fuzzy cosets and prove the fundamental isomorphism theorem on -groups with respect to the structure induced by these fuzzy cosets.
Theorem 31. Let and be two -groups, and let be an -homomorphism. If is surjective and is an -fuzzy ideal of , then so is . If is a -fuzzy ideal of , then is a fuzzy ideal of .
Proof. We assume that is an -fuzzy ideal of . For any ; it follows that Also, Again, Therefore, is an -fuzzy ideal of . Similarly, we can show that is an -fuzzy ideal of .
Definition 32. Let be -fuzzy subgroup of . For any , let be defined by for all . This fuzzy subset is called the -fuzzy left coset of determined by and .
Remark 33. Let be an -fuzzy subgroup of . Then, is an -fuzzy normal if and only if for all . If is an -fuzzy ideal, we simply denote fuzzy coset by .
Lemma 34. Let be an -fuzzy ideal of . Then, if an only if .
Proof. Assume that and . which implies that . Similarly, we can verify that . Conversely we assume that . Then, .
Proposition 35. Every fuzzy coset is constant on every coset of .
Proof. Let . Now, we have . Also . Thus for all .
Theorem 36. For any -fuzzy ideal of , the set of all fuzzy cosets of in is an -group under the addition and scalar multiplication defined by , for all , . The function defined by for all is -fuzzy ideal of .
Proof. First, we show that the compositions are well defined. Let , such that and . Then, , . Now, we have which implies that . So, by Lemma 34, we have . Again . Similarly, we show that . Thus, for for all . Hence the compositions are well defined. It is now easy to verify that is an -group with null element and negative element . Next, we check that is -fuzzy ideal of . Let . We have(i), (ii), (iii), (iv).Hence, the proof is completed.
Lemma 37. Let , . Then, .
Proof. Let . Then, for all , . Also, for , . So, . As we have seen, if then . Conversely, if , for any , . Thus, , which means that is an ideal of .
Theorem 38. If is an -fuzzy ideal of , then the map given by is an -homomorphism with kernel and so is isomorphic to .
Proof. It is clear that is an onto -homomorphism from to with kernel .
Corollary 39. Let be an -fuzzy ideal of , and let be an -fuzzy ideal of . Then, defined by is an -fuzzy ideal of containing .
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