3. Fuzzy Multi-Ideal of Near Rings
This section combines the notions of fuzzy multiset [
4,
25] and fuzzy ideals of near rings to define fuzzy multi-ideals of near rings. It presents several results related to the new defined concept. Since every fuzzy set can be considered as a fuzzy multiset then some of the results in this section can be considered as a generalization for those in [
26] that are related to fuzzy ideals of near rings.
Definition 4. [
10]
Let be a near ring. A fuzzy multiset A (with fuzzy count function ) over R is a fuzzy multi-subnear ring of R if for all , the following conditions hold.- 1.
;
- 2.
.
Definition 5. [
10]
Let be a near ring. A fuzzy multiset A (with fuzzy count function ) over R is a fuzzy multi-ideal of R if for all , the following conditions hold.- 1.
;
- 2.
;
- 3.
;
- 4.
;
- 5.
for all.
Proposition 1. [
10]
Let be a near ring with zero element and A be a fuzzy multi-ideal of R. Then- 1.
;
- 2.
;
- 3.
;
- 4.
for all.,
- 5.
for all.
Remark 1. Letbe a near ring with zero elementand A be a fuzzy multiset of R withfor all. Then A is a fuzzy multi-ideal of R and it is called the constant fuzzy multi-ideal.
Next, we deal with some operations on fuzzy multi-subnear rings (multi-ideals) of near rings such as intersection, union, and product.
Proposition 2. [
10]
Let be a near ring and be fuzzy multi-subnear rings of R. Then is a fuzzy multi-subnear ring of R. Corollary 1. [
10]
Let be a near ring and be a fuzzy multi-subnear ring of R for . Then is a fuzzy multi-subnear ring of R. Proposition 3. [
10]
Let be a near ring and be fuzzy multi-ideals of R. Then is a fuzzy multi-ideal of R. Corollary 2. [
10]
Let be a near ring and be a fuzzy multi-ideal of R for . Then is a fuzzy multi-ideal of R. Proposition 4. [
10]
Let be near rings with fuzzy multisets , respectively. If are fuzzy multi-subnear rings (multi-ideals) of then is fuzzy multi-subnear rings (multi-ideal) of , where for all . Proposition 5. [
10]
Let be near rings with fuzzy multiset for . If is a fuzzy multi-subnear ring (multi-ideal) of then is a fuzzy multi-ideal of , where for all . Notation 1. Letbe a near ring, A be a fuzzy multiset of R, and. Then
if,
if,
ifwhere
Definition 6. [
10]
Let be a near ring and A be a fuzzy multiset of H. Then and . Proposition 6. [
10]
Let be a near ring and A be a fuzzy multi-subnear ring (multi-ideal) of R. Then is either the empty set or a subnear ring (ideal) of R. Proposition 7. [
10]
Let be a near ring with a fuzzy multi-ideal A. Then the fuzzy multiset of defined as is a fuzzy multi-ideal of . Notation 2. Letbe a near ring, A be a fuzzy multiset of R and. We say thatifandfor all. Ifandthen we say thatandare not comparable.
Notation 3. Letbe a near ring, A a fuzzy multiset of R with fuzzy count function, andwhereforand. Then.
Theorem 1. [
10]
Let be a near ring, A a fuzzy multiset of R with fuzzy count function and where for and . Then A is a fuzzy multi-subnear ring of R if and only if is either the empty set or a subnear ring of R. Theorem 2. [
10]
Let be a near ring, A a fuzzy multiset of R with fuzzy count function and where for and . Then A is a fuzzy multi-ideal of R if and only if is either the empty set or an ideal of R. Corollary 3. [
10]
Let be a near ring. If every subnear ring of R is an ideal of R then every fuzzy multi-subnear ring of R is a fuzzy multi-ideal of R. Proposition 8. [
10]
Let be a near ring and I be an ideal of R. Then for some fuzzy multi-ideal of R, where for and . Corollary 4. [
10]
Let be a near ring. Then R has at least one fuzzy multi-ideal beside the constant fuzzy multi-ideal. Definition 7. Letbe a near ring, A and B be fuzzy multi-subnear rings (ideals) of R with fuzzy count functionsand, respectively. If for all:
- 1.
if and only if,
- 2.
if and only if, and
- 3.
are not comparable if and only ifare not comparable then A and B are equivalent fuzzy multi-subnear rings (ideals) of R.
4. Anti-Fuzzy Multi-Ideals of Near Rings
This section defines the complement of a fuzzy multiset and discusses some of its properties. Moreover, we introduce anti-fuzzy multi-subnear rings (multi-ideals) of near rings and prove some results related to the new defined concepts.
Definition 8. Letbe a near ring. A fuzzy multiset A (with count function) over R is an anti-fuzzy multi-subnear ring of R if for all, the following conditions hold.
- 1.
;
- 2.
.
Definition 9. Letbe a near ring. A fuzzy multiset A (with count function) over R is an anti-fuzzy multi-ideal of R if for all, the following conditions hold.
- 1.
;
- 2.
;
- 3.
;
- 4.
;
- 5.
for all.
Proposition 9. Letbe a near ring with zero elementand A be an anti-fuzzy multi-ideal of R. Then
- 1.
;
- 2.
;
- 3.
;
- 4.
for all.
Proof. The proof is straightforward. □
Definition 10. Let R be a non-empty set. Let A be a fuzzy multiset of R. The complement of A will be denotedand it is a fuzzy multiset defined as: For all, Example 1. Letbe a set. Let A be a fuzzy multiset and the fuzzy count functionbe defined as:. Then.
Proposition 10. Let R be a non-empty set, A be a fuzzy multiset of R, and. Then Proof. Let and . It is clear that and . Then . Since , it follows that for all . The latter implies that for all . Thus, . Similarly, we can prove that if then . □
Proposition 11. Let R be a non-empty set,fuzzy multisets of R,, and. Then
- 1.
ifandthen;
- 2.
ifthen.
Proof. (1) We have and . Since , it follows that for all . The latter implies that for all . Thus, .
(2) Since , it follows that , , and . By (1), we get that and . Thus, . □
Remark 2. Let R be a non-empty set,fuzzy multisets of R,, and. Ifandthenmay not hold.
Remark 2 can be illustrated by the following example.
Example 2. Let. Moreover, let,be fuzzy multisets of R. It is clear thatand. Havingandimplies that.
Proposition 12. Let R be a non-empty set,fuzzy multisets of R,, and. Then
- 1.
;
- 2.
;
- 3.
.
Proof. (1) Having for every implies that .
(2) Without loss of generality, let
. Having
implies that
. However,
. The latter implies that
.
(3) Without loss of generality, let
. Having
implies that
. However,
. The latter implies that
. □
Remark 3. Let R be a non-empty set,fuzzy multisets of R. Thenandmay not hold.
We illustrate Remark 3 by the following example.
Example 3. Letandbe fuzzy multisets of R withOne can easily see that:and that Remark 4. Letbe a near ring and A be the constant fuzzy multiset of R. Then A is an anti-fuzzy multi-ideal of R.
Next, we deal with some operations on anti-fuzzy multi-subnear rings (anti-fuzzy multi-ideals) of near rings.
Proposition 13. Letbe a near ring andbe anti-fuzzy multi-subnear rings (anti-fuzzy multi-ideals) of R. Thenis an anti fuzzy multi-subnear ring (anti-fuzzy multi-ideal) of R.
Proof. The proof is similar to that of Proposition 2 and Proposition 3. □
Corollary 5. Letbe a near ring andbe an anti-fuzzy multi-subnear ring (multi-ideal) of R for. Thenis an anti-fuzzy multi-subnear ring (multi-ideal) of R.
Proof. The proof follows by using induction and Proposition 13. □
Remark 5. Letbe anti-fuzzy multi-ideals of a near ring R. Thenmay not be an anti-fuzzy multi-ideal of R.
We illustrate Remark 5 by the following example.
Example 4. Letbe the near ring of integers under standard addition ”+” and with ”·” defined byfor all. Letbe the anti-fuzzy multi-ideals ofgiven by: Having,, andimplies thatis not an anti-fuzzy multi-ideal of.
Proposition 14. Let X be a non-empty set and A a fuzzy multiset of X. Then for all,
- 1.
;
- 2.
.
Proof. The proof is straightforward. □
Theorem 3. Letbe a near ring and A be a fuzzy multiset of R. Then A is a fuzzy multi-subnear ring (multi-ideal) of R if and only ifis an anti-fuzzy multi-subnear ring (multi-ideal) of R.
Proof. Let A be a fuzzy multi-ideal of R and . We need to prove that conditions of Definition 9 are satisfied for . (1) . Having A a fuzzy multi-ideal of R implies that . The latter implies that . (2) . Having A a fuzzy multi-ideal of R implies that . The latter implies that . (3) . Having A a fuzzy multi-ideal of R implies that . The latter implies that . (4) . Having A a fuzzy multi-ideal of R implies that . The latter implies that . (5) Let . . Having A a fuzzy multi-ideal of R implies that . The latter implies that .
Conversely, let be an anti-fuzzy multi-ideal of R and . We need to prove that conditions of Definition 5 are satisfied for A. (1) . Having an anti-fuzzy multi-ideal of R implies that . The latter implies that . (2) . Having an anti-fuzzy multi-ideal of R implies that . The latter implies that (3) . Having an anti-fuzzy multi-ideal of R implies that . The latter implies that . (4) . Having an anti-fuzzy multi-ideal of R implies that . The latter implies that . (5) Let . . Having an anti-fuzzy multi-ideal of R implies that . The latter implies that . □
Example 5. Letbe the near ring defined by the following tables: ![Mathematics 09 00494 i001]()
In [10], Al-Tahan et al. showed thatis a fuzzy multi-ideal of R. Theorem 3 asserts thatis an anti-fuzzy multi-ideal of R. Corollary 6. Letbe a near ring and A be a fuzzy multiset of R. Then A andare fuzzy multi-ideals of R if and only if A is the constant fuzzy multi-ideal of R.
Proof. Let A be the constant fuzzy multi-ideal of R. Then A and are fuzzy multi-ideals of R.
Let A and be fuzzy multi-ideals of R. Then, for all , we have and (by Propositions 1 and 9). The latter and Proposition 10 imply that and . Thus, . □
Corollary 7. Letbe a near ring and A be a fuzzy multiset of R. Then A andare anti-fuzzy multi-ideals of R if and only if A is the constant fuzzy multi-ideal of R.
Proof. The proof is similar to that of Corollary 6. □
Notation 4. Letbe a near ring, A a fuzzy multiset of R with fuzzy count function, andwhereforand. Then.
Theorem 4. Letbe a near ring, A a fuzzy multiset of R with fuzzy count functionandwhereforand. Then A is an anti-fuzzy multi-subnear ring (multi-ideal) of R if and only ifis either the empty set or a subnear ring (ideal) of R.
Proof. The proof is similar to that of Theorems 1 and 2. □
Proposition 15. Letbe a near ring and I be an ideal of R. Thenfor some anti-fuzzy multi-idealof R,whereforand.
Proof. Let
and
where
for
and define the fuzzy multiset
A of
R as follows:
It is clear that
and
. We still need to prove that
is an anti-fuzzy multi-ideal of
R. Using Theorem 4, it suffices to show that
is an ideal of
R for all
with
,
, and
for
. One can easily see that
Thus, is either the empty set or an ideal of R. □
Remark 6. Letbe a near ring and A and B be anti-fuzzy multi-subnear rings (ideals) of R with fuzzy count functionsand, respectively. If for all:
- 1.
if and only if,
- 2.
if and only if, and
- 3.
are not comparable if and only ifare not comparable then A and B are equivalent anti-fuzzy multi-subnear rings (ideals) of R.
Example 6. Letbe any non-trivial near ring with identity. Thenandare equivalent anti-fuzzy multi-ideals of R. In what follows, when we say that a near ring has a certain number of anti-fuzzy multi-ideals we mean that it has a certain number of non-equivalent fuzzy multi-ideals.
Corollary 8. Letbe a near ring. Then R has at least one anti-fuzzy multi-ideal beside the constant anti-fuzzy multi-ideal.
Proof. Let
be a near ring. Then
R has at least two ideals:
R and
. With
as an ideal of
R and using Proposition 15, we get that
for some
. One can define the anti-fuzzy multi-ideal corresponding to
as:
where
. □
Corollary 9. Letbe a simple near ring. Then R has only two non-equivalent anti-fuzzy multi-ideals.
Example 7. Let p be a prime natural number andbe the near ring of integers modulo p under standard addition “+" modulo p and “·" defined byfor all. Sinceis a simple near ring, it follows by means of Corollary 9 that it has only two non-equivalent anti-fuzzy multi-ideals.