3. Application in -Algebras
In this section, we take a -algebra X as the universe of discourse unless otherwise specified.
Definition 1. A neutrosophic -structure over X is called a neutrosophic -subalgebra of X if the following condition is valid: Example 1. Consider a -algebra with the following Cayley table.* | | | | |
θ | θ | θ | θ | θ |
a | a | θ | θ | a |
b | b | a | θ | b |
c | c | c | c | θ |
The neutrosophic -structureover X is a neutrosophic -subalgebra of X. Let
be a neutrosophic
-structure over
X and let
be such that
. Consider the following sets:
The set
is called the
-level set of
. Note that
Theorem 1. Let be a neutrosophic -structure over X and let be such that . If is a neutrosophic -subalgebra of X, then the nonempty -level set of is a subalgebra of X.
Proof. Let
be such that
and
. If
, then
,
,
,
,
and
. It follows from Equation (
4) that
,
, and
.
Hence, , and therefore is a subalgebra of X. ☐
Theorem 2. Let be a neutrosophic -structure over X and assume that , and are subalgebras of X for all with . Then is a neutrosophic -subalgebra of X.
Proof. Assume that there exist
such that
. Then
for some
. Hence
but
, which is a contradiction. Thus
for all
. If
for some
, then
where
. Thus
and
, which is a contradiction. Therefore
for all
. Now, suppose that there exist
and
such that
Then
and
, which is a contradiction. Hence
for all
. Therefore
is a neutrosophic
-subalgebra of
X. ☐
Because is a completely distributive lattice with respect to the usual ordering, we have the following theorem.
Theorem 3. If is a family of neutrosophic -subalgebras of X, then forms a complete distributive lattice.
Proposition 1. If a neutrosophic -structure over X is a neutrosophic -subalgebra of X, then , and for all .
Proof. Straightforward. ☐
Theorem 4. Let be a neutrosophic -subalgebra of X. If there exists a sequence in X such that , and , then , and .
Proof. By Proposition 1, we have
,
and
for all
. Hence
,
and
for every positive integer
n. It follows that
Hence , and . ☐
Proposition 2. If every neutrosophic -subalgebra of X satisfies:for all , then is constant. Proof. Using Equations (
1) and (
5), we have
,
and
for all
. It follows from Proposition 1 that
,
and
for all
. Therefore
is constant. ☐
Definition 2. A neutrosophic -structure over X is called a neutrosophic -ideal of X if the following assertion is valid: Example 2. The neutrosophic -structure over X in Example 1 is a neutrosophic -ideal of X.
Example 3. Consider a -algebra where is a -algebra and is the adjoint -algebra of the additive group of integers (see [6]). Let be a neutrosophic -structure over X given bywhere and . Then is a neutrosophic -ideal of X. Proposition 3. Every neutrosophic -ideal of X satisfies the following assertions: Proof. Let be such that . Then , and so
This completes the proof. ☐
Proposition 4. Let be a neutrosophic -ideal of X. Then
- (1)
- (2)
- (3)
for all .
Proof. Note that
for all
. Assume that
,
and
for all
. It follows from Equation (2) and Proposition 3 that
and
for all
.
Conversely, suppose
for all
. If we substitute
z for
y in Equation (
9), then
for all
by using (III) and Equation (
1). ☐
Theorem 5. Let be a neutrosophic -structure over X and let be such that . If is a neutrosophic -ideal of X, then the nonempty -level set of is an ideal of X.
Proof. Assume that
for
with
. Clearly,
. Let
be such that
and
. Then
,
,
,
,
and
. It follows from Equation (
6) that
so that
. Therefore
is an ideal of
X. ☐
Theorem 6. Let be a neutrosophic -structure over X and assume that , and are ideals of X for all with . Then is a neutrosophic -ideal of X.
Proof. If there exist
such that
,
and
, respectively, then
,
and
for some
and
. Then
,
and
. This is a contradiction. Hence,
,
and
for all
. Assume that there exist
such that
,
and
. Then there exist
and
such that
It follows that
,
,
,
,
and
. However,
,
and
. This is a contradiction, and so
for all
. Therefore
is a neutrosophic
-ideal of
X. ☐
Proposition 5. For any neutrosophic -ideal of X, we have Proof. Let
be such that
. Then
, and so
This completes the proof. ☐
Theorem 7. In a -algebra, every neutrosophic -ideal is a neutrosophic -subalgebra.
Proof. Let
be a neutrosophic
-ideal of a
-algebra
X. For any
, we have
and
Hence is a neutrosophic -subalgebra of a -algebra X. ☐
The converse of Theorem 7 may not be true in general, as seen in the following example.
Example 4. Consider a -algebra with the following Cayley table.* | | 1 | 2 | 3 | 4 |
θ | θ | θ | θ | θ | θ |
1 | 1 | θ | θ | θ | θ |
2 | 2 | 1 | θ | 1 | θ |
3 | 3 | 3 | 3 | θ | θ |
4 | 4 | 4 | 4 | 3 | θ |
Let be a neutrosophic -structure over X, which is given as follows: Then is a neutrosophic -subalgebra of X, but it is not a neutrosophic -ideal of X as , , or .
Theorem 7 is not valid in a -algebra; that is, if X is a -algebra, then there is a neutrosophic -ideal that is not a neutrosophic -subalgebra, as seen in the following example.
Example 5. Consider the neutrosophic -ideal of X in Example 3. If we take and in , then . Hence Therefore is not a neutrosophic -subalgebra of X.
For any elements
,
,
, we consider sets:
Clearly, , and .
Theorem 8. Let , and be any elements of X. If is a neutrosophic -ideal of X, then , and are ideals of X.
Proof. Clearly,
,
and
. Let
be such that
and
. Then
It follows from Equation (
6) that
Hence , and therefore , and are ideals of X. ☐
Theorem 9. Let , , and let be a neutrosophic -structure over X. Then
- (1)
If ,
and
are ideals of X, then the following assertion is valid: - (2)
If satisfies Equation (11) andthen , and are ideals of X for all , and .
Proof. (1) Assume that , and are ideals of X for , , . Let be such that , and . Then and , where . It follows from (I2) that for . Hence , and .
(2) Let
,
and
and suppose that
satisfies Equations (
11) and (
12). Clearly,
by Equation (
12). Let
be such that
and
. Then
which implies that
,
, and
. It follows from Equation (
11) that
,
and
. Thus,
, and therefore
,
and
are ideals of
X.☐
Definition 3. A neutrosophic -ideal of X is said to be closed if it is a neutrosophic -subalgebra of X.
Example 6. Consider a -algebra with the following Cayley table.* | | 1 | | | |
θ | θ | θ | a | b | c |
1 | 1 | θ | a | b | c |
a | a | a | θ | c | b |
b | b | b | c | θ | a |
c | c | c | b | a | θ |
Let be a neutrosophic -structure over X which is given as follows: Then is a closed neutrosophic -ideal of X.
Theorem 10. Let X be a -algebra, For any and with , and , let be a neutrosophic -structure over X given as follows:where . Then is a closed neutrosophic -ideal of X. Proof. Because
, we have
,
and
for all
. Let
. If
, then
Suppose that
. If
then
, and if
then
. In either case, we have
For any
, if any one of
x and
y does not belong to
, then
If
, then
. Hence
Therefore is a closed neutrosophic -ideal of X. ☐
Proposition 6. Every closed neutrosophic -ideal of a -algebra X satisfies the following condition: Proof. Straightforward. ☐
We provide conditions for a neutrosophic -ideal to be closed.
Theorem 11. Let X be a -algebra. If is a neutrosophic -ideal of X that satisfies the condition of Equation (
13),
then is a neutrosophic -subalgebra and hence is a closed neutrosophic -ideal of X. Proof. Note that
for all
. Using Equations (
10) and (
13), we have
Hence is a neutrosophic -subalgebra and is therefore a closed neutrosophic -ideal of X. ☐