3. Neutrosophic Permeable Values
Given a NS
in a set
X,
and
, we consider the following sets:
We say , , and are upper neutrosophic -subsets of X, and , , and are lower neutrosophic -subsets of X, where . We say , , and are strong upper neutrosophic -subsets of X, and , , and are strong lower neutrosophic -subsets of X, where .
Definition 1 ([
7])
. A NS in a -algebra X is called an - neutrosophic subalgebra of X if the following assertions are valid:for all , and . Lemma 1 ([
7])
. A NS in a -algebra X is an -neutrosophic subalgebra of X if and only if satisfies Proposition 1. Every -neutrosophic subalgebra of a -algebra X satisfies Proof. Straightforward. ☐
Theorem 1. If is an -neutrosophic subalgebra of a -algebra X, then the lower neutrosophic -subsets of X are S-energetic subsets of X, where .
Proof. Let
and
be such that
. Then
and thus
or
; that is,
or
. Thus
. Therefore
is an
S-energetic subset of
X. Similarly, we can verify that
is an
S-energetic subset of
X. We let
and
be such that
. Then
It follows that or ; that is, or . Hence , and therefore is an S-energetic subset of X. ☐
Corollary 1. If is an -neutrosophic subalgebra of a -algebra X, then the strong lower neutrosophic -subsets of X are S-energetic subsets of X, where .
Proof. Straightforward. ☐
The converse of Theorem 1 is not true, as seen in the following example.
Example 1. Consider a -algebra with the binary operation * that is given in Table 1 (see [14]). Let be a NS in X that is given in Table 2. If , , and , then , , and are S-energetic subsets of X. Becauseand/orit follows from Lemma 1 that is not an -neutrosophic subalgebra of X. Definition 2. Let be a NS in a -algebra X and , where , and are subsets of . Then is called a neutrosophic permeable S-value for if the following assertion is valid: Example 2. Let be a set with the binary operation * that is given in Table 3. Then is a -algebra (see [14]). Let be a NS in X that is given in Table 4. It is routine to verify that is a neutrosophic permeable S-value for .
Theorem 2. Let be a NS in a -algebra X and , where , and are subsets of . If satisfies the following condition:then is a neutrosophic permeable S-value for . Proof. Let
be such that
. Then
Similarly, if
for
, then
. Now, let
be such that
. Then
Therefore is a neutrosophic permeable S-value for . ☐
Theorem 3. Let be a NS in a -algebra X and , where , and are subsets of . If satisfies the following conditions:andthen is a neutrosophic permeable S-value for . Proof. Let
be such that
,
, and
. Then
and
by Equations (3), (V), (
15), and (
16). It follows that
Therefore is a neutrosophic permeable S-value for . ☐
Theorem 4. Let be a NS in a -algebra X and , where , and are subsets of . If is a neutrosophic permeable S-value for , then upper neutrosophic -subsets of X are S-energetic where .
Proof. Let
be such that
,
, and
. Using Equation (
13), we have
,
, and
. It follows that
and
Hence , , and . Therefore , , and are S-energetic subsets of X. ☐
Definition 3. Let be a NS in a -algebra X and , where , and are subsets of . Then is called a neutrosophic anti-permeable S-value for if the following assertion is valid: Example 3. Let be a set with the binary operation * that is given in Table 5. Then is a -algebra (see [14]). Let be a NS in X that is given in Table 6. It is routine to verify that is a neutrosophic anti-permeable S-value for .
Theorem 5. Let be a NS in a -algebra X and , where , and are subsets of . If is an -neutrosophic subalgebra of X, then is a neutrosophic anti-permeable S-value for .
Proof. Let
be such that
,
, and
. Using Lemma 1, we have
and thus
is a neutrosophic anti-permeable
S-value for
. ☐
Theorem 6. Let be a NS in a -algebra X and , where , and are subsets of . If is a neutrosophic anti-permeable S-value for , then lower neutrosophic -subsets of X are S-energetic where .
Proof. Let
be such that
,
, and
. Using Equation (
17), we have
,
, and
, which imply that
and
Hence , , and . Therefore , , and are S-energetic subsets of X. ☐
Definition 4. A NS in a -algebra X is called an - neutrosophic ideal of X if the following assertions are valid:for all and . Theorem 7. A NS in a -algebra X is an -neutrosophic ideal of X if and only if satisfies Proof. Assume that Equation (
20) is valid, and let
,
, and
for any
,
and
. Then
,
, and
. Hence
,
, and
, and thus Equation (
18) is valid. Let
be such that
,
,
,
,
, and
for all
and
. Then
,
,
,
,
, and
. It follows from Equation (
20) that
Hence , , and . Therefore is an -neutrosophic ideal of X.
Conversely, let
be an
-neutrosophic ideal of
X. If there exists
such that
, then
and
, where
. This is a contradiction, and thus
for all
. Assume that
for some
. Taking
implies that
and
; but
. This is a contradiction, and thus
for all
. Similarly, we can verify that
for all
. Now, suppose that
for some
. Then
and
by taking
. This is impossible, and thus
for all
. Suppose there exist
such that
, and take
. Then
,
, and
, which is a contradiction. Thus
for all
. Therefore
satisfies Equation (
20). ☐
Lemma 2. Every -neutrosophic ideal of a -algebra X satisfies Proof. Let
be such that
. Then
, and thus
by Equation (
20). This completes the proof. ☐
Theorem 8. A NS in a -algebra X is an -neutrosophic ideal of X if and only if satisfies Proof. Let
be an
-neutrosophic ideal of
X, and let
be such that
. Using Theorem 7 and Lemma 2, we have
Conversely, assume that
satisfies Equation (
22). Because
for all
, it follows from Equation (
22) that
for all
. Because
for all
, we have
for all
by Equation (
22). It follows from Theorem 7 that
is an
-neutrosophic ideal of
X. ☐
Theorem 9. If is an -neutrosophic ideal of a -algebra X, then the lower neutrosophic -subsets of X are I-energetic subsets of X where .
Proof. Let
,
, and
be such that
,
, and
. Using Theorem 7, we have
for all
. It follows that
and
Hence , , and are nonempty, and therefore , and are I-energetic subsets of X. ☐
Corollary 2. If is an -neutrosophic ideal of a -algebra X, then the strong lower neutrosophic -subsets of X are I-energetic subsets of X where .
Proof. Straightforward. ☐
Theorem 10. Let , where , and are subsets of . If is an -neutrosophic ideal of a -algebra X, then
- (1)
the (strong) upper neutrosophic -subsets of X are right stable where ;
- (2)
the (strong) lower neutrosophic -subsets of X are right vanished where .
Proof. (1) Let , , , and . Then , , and . Because , , and , it follows from Lemma 2 that , , and ; that is, , , and . Hence the upper neutrosophic -subsets of X are right stable where . Similarly, the strong upper neutrosophic -subsets of X are right stable where .
(2) Assume that , , and for any . Then , , and . Because , , and , it follows from Lemma 2 that , , and ; that is, , , and . Therefore the lower neutrosophic -subsets of X are right vanished where . In a similar way, we know that the strong lower neutrosophic -subsets of X are right vanished where . ☐
Definition 5. Let be a NS in a -algebra X and , where , and are subsets of . Then is called a neutrosophic permeable I-value for if the following assertion is valid: Example 4. (1) In Example 2, is a neutrosophic permeable I-value for .
(2) Consider a -algebra with the binary operation * that is given in Table 7 (see [14]). Let be a NS in X that is given in Table 8. It is routine to check that is a neutrosophic permeable I-value for .
Lemma 3. If a NS in a -algebra X satisfies the condition of Equation (
14)
, then Proof. Straightforward. ☐
Theorem 11. If a NS in a -algebra X satisfies the condition of Equation (
14)
, then every neutrosophic permeable I-value for is a neutrosophic permeable S-value for . Proof. Let
be a neutrosophic permeable
I-value for
. Let
be such that
,
, and
. It follows from Equations (
23), (3), (III), and (V) and Lemma 3 that
and
Hence , , and . Therefore is a neutrosophic permeable S-value for . ☐
Given a NS in a -algebra X, any upper neutrosophic -subsets of X may not be I-energetic where , as seen in the following example.
Example 5. Consider a -algebra with the binary operation * that is given in Table 9 (see [14]). Let be a NS in X that is given in Table 10. Then , , and . Because and , we know that is not an I-energetic subset of X.
We now provide conditions for the upper neutrosophic -subsets to be I-energetic where .
Theorem 12. Let be a NS in a -algebra X and , where , and are subsets of . If is a neutrosophic permeable I-value for , then the upper neutrosophic -subsets of X are I-energetic subsets of X where .
Proof. Let
and
, where
, and
are subsets of
such that
,
, and
. Because
is a neutrosophic permeable
I-value for
, it follows from Equation (
23) that
for all
. Hence
and
Hence , , and are nonempty, and therefore the upper neutrosophic -subsets of X are I-energetic subsets of X where . ☐
Theorem 13. Let be a NS in a -algebra X and , where , and are subsets of . If satisfies the following condition:then is a neutrosophic permeable I-value for . Proof. Let
and
, where
, and
are subsets of
such that
,
, and
. Using Equation (
25), we obtain
for all
. Therefore
is a neutrosophic permeable
I-value for
. ☐
Combining Theorems 12 and 13, we have the following corollary.
Corollary 3. Let be a NS in a -algebra X and , where , and are subsets of . If satisfies the condition of Equation (
25)
, then the upper neutrosophic -subsets of X are I-energetic subsets of X where . Definition 6. Let be a NS in a -algebra X and , where , and are subsets of . Then is called a neutrosophic anti-permeable I-value for if the following assertion is valid: Theorem 14. Let be a NS in a -algebra X and , where , and are subsets of . If satisfies the condition of Equation (19), then is a neutrosophic anti-permeable I-value for .
Proof. Let
be such that
,
, and
. Then
for all
by Equation (
20). Hence
is a neutrosophic anti-permeable
I-value for
. ☐
Theorem 15. Let be a NS in a -algebra X and , where , and are subsets of . If is a neutrosophic anti-permeable I-value for , then the lower neutrosophic -subsets of X are I-energetic where .
Proof. Let
,
, and
. Then
,
, and
for all
by Equation (
26). It follows that
and
Hence , and are nonempty, and therefore the lower neutrosophic -subsets of X are I-energetic where . ☐
Combining Theorems 14 and 15, we obtain the following corollary.
Corollary 4. Let be a NS in a -algebra X and , where , and are subsets of . If satisfies the condition of Equation (19), then the lower neutrosophic -subsets of X are I-energetic where .
Theorem 16. If is an -neutrosophic subalgebra of a -algebra X, then every neutrosophic anti-permeable I-value for is a neutrosophic anti-permeable S-value for .
Proof. Let
be a neutrosophic anti-permeable
I-value for
. Let
be such that
,
, and
. It follows from Equations (
26), (3), (III), and (V) and Proposition 1 that
and
Hence , , and . Therefore is a neutrosophic anti-permeable S-value for . ☐