2. Main Discussion
Definition 1. Let
be a ring; the symbolic 4-plithogenic ring is:
.
Operations on
:
Addition:
.
Multiplication:
.
It is clear that (
is a ring.
If is commutative, then
is commutative, and if
has a unity, then
has the same unity.
Example 1. Consider the ring
, the corresponding
is:
.
If
, then:
.
Theorem 1. Let
be a 4-plithogenic symbolic ring, with unity.
Let
be an arbitrary element; then:
is invertible if and only if
,, are invertible.
+ [
.
Proof.
Assume that
is invertible, then there exists
such that
; hence:
From
,
is invertible.
By adding to , we obtain ; thus, is invertible.
By adding to , ; hence, is invertible.
By adding to to to ,
; hence, is invertible.
Adding all equations gives:
; hence, is invertible.
From the first part, we have:
; then:
. □
Example 2. Take
,
is the corresponding symbolic 4-plithogenic ring; consider
; then:
is invertible with
, is invertible with
,
is invertible with
,
hence:
.
Definition 2. If
, then is idempotent if and only if
.
Theorem 2. If
, then
is idempotent if and only if
are idempotent.
Proof.
.
Equation
implies that
is idempotent.
By adding to , we obtain ; hence, is idempotent.
By adding to to , we obtain ; hence, is idempotent.
By adding to to to , we obtain ; thus, is idempotent. By adding all equations to each other, we obtain:
, thus is idempotent. Thus the proof is complete. □
Example 3. Take
,
is the corresponding symbolic 4-plithogenic ring, and consider
; thus, we have:
.
Theorem 3. Let
be a commutative symbolic 4-plithogenic ring; hence, if
, then
+
for every
.
Proof. For , it holds easily. Assume that it is true for , and prove it for .
+
.
So, this proof is complete by induction. □
Definition 3. Let
,, be ideals of the ring
and define the symbolic 4-plithogenic AH-ideal:
.
If
, then
is called an AHS-ideal.
Example 4. Let
be the ring of integers; then,
are ideals of
.
is an AHS-ideal of
.
is an AHS-ideal of
.
Theorem 4. Let
be an AHS- ideal of
; thus, is an ideal with an ordinary meaning.
Proof. can be written as , where is an ideal of .
It is clear that is a subgroup of .
Let ,
Then if , we have:
; thus, G is an ideal. □
Definition 4. Let
be two rings;
are the corresponding symbolic 4-plithogenic rings. Let
be ring homomorphisms; thus, we define the AH-homomorphism:
such that:
+
If
, then
is called an AHS-homomorphism.
Remark 1. If
are isomorphisms, then
is called an AH-isomorphism.
Example 5. Take
,
,
such that:
. It is clear that
are homomorphisms.
We define
, where:
+,
Which is an AH-homomorphism.
Theorem 5. Let
be a mapping, then:
If
is an AHS-homomorphism, then
is a ring homomorphism.
If
is an AHS-homomorphism, then it is an isomorphism.
Proof.
Assume that is an AHS-homomorphism, then, are homomorphisms.
Let , and we have:
.
This implies the proof.
Using a similar discussion, we obtain the desired proof. □
Theorem 6. Let
be ideals of the ring
; then:
is an ideal of
.
Proof.
It is clear that is non-empty set.
Let
For two arbitrary elements of , then:
, which is because:
Let ; then:
; thus, is ideal of . □
Example 6. Take
, the ring of integers, and consider the ideals
; then:
Also,
is another ideal of
.
Definition 5. Let
be a ring; the symbolic 5-plithogenic ring is:
.
Operations on
:
Addition:
.
Multiplication:
+ (
).
It is clear that (
is a ring.
If
is commutative, then
is commutative, and if
has a unity, then
has the same unity.
Example 7. Consider the ring
; the corresponding is:
.
If
, then:
.
Theorem 7. Let
be a 5-plithogenic symbolic ring, with unity.
Let
be an arbitrary element; then:
is invertible if and only if
,, are invertible.
+ [
.
Proof.
Assume that
is invertible, then there exists
such that
; hence:
From
,
is invertible.
By adding to , we obtain ; thus, is invertible.
By adding to , ; hence, is invertible.
By adding to to to , ; hence, is invertible.
Adding all Equations to gives:
; hence, is invertible.
Adding all Equations to gives:
;
Hence, is invertible.
From the first part, we have:
; then:
. □
Example 8. Take
,
is the corresponding symbolic 5-plithogenic ring, and consider
; then:
is invertible with
, is invertible with
,
is invertible with
,
hence,
.
Definition 6. Let
, then,
is idempotent if and only if
.
Theorem 8. Let
; then,
is idempotent if and only if
,
,
, are idempotent.
Proof.
.
Equation
implies that
is idempotent.
By adding to , we obtain ; hence, is idempotent.
By adding to to , we obtain ; hence, is idempotent.
By adding to to to , we obtain ; thus, is idempotent.
By adding all equations from to to each other, we obtain:
, thus is idempotent.
By adding all equations from to , we obtain:
, thus is idempotent.
Thus, the proof is complete. □
Example 9. Take
,
is the corresponding symbolic 5-plithogenic ring, and consider
; thus, we have:
.
Theorem 9. Let
be a commutative symbolic 5-plithogenic ring; hence, if
, then
+
for every
.
Proof.
For , it holds easily. Assume that it is true for and prove it for .
+
.
So, this proof is complete by induction. □
Definition 7. Let
,,, be ideals of the ring
; define the symbolic 5-plithogenic AH-ideal:
.
If
, then
is called an AHS-ideal.
Example 10. Let
be the ring of integers; then,
are ideals of
.
is an AHS-ideal of
.
is an AHS-ideal of
.
Theorem 10. Let
be an AHS- ideal of
; then,
is an ideal with an ordinary meaning.
Proof.
can be written as , where is an ideal of .
It is clear that is a subgroup of .
Let ,
Then if , we have:
; thus, G is an ideal. □
Definition 8. Let
be two rings,
are the corresponding symbolic 5-plithogenic rings, and let
be ring homomorphisms; we define the AH-homomorphism:
such that:
+
If
, then
is called an AHS-homomorphism.
Remark 2. If
are isomorphisms, then
is called an AH-isomorphism.
Example 11. Take
,
,
such that:
. It is clear that
are homomorphisms.
We define
, where:
+,
Which is an AH-homomorphism.
Theorem 11. Let
be a mapping; then:
If
is an AHS-homomorphism, then
is a ring homomorphism.
If is an AHS-homomorphism, then it is an isomorphism.
Proof.
Assume that is an AHS-homomorphism; then, are homomorphisms.
Let
; we have:
.
This implies the proof.
Using a similar discussion, we obtain the desired proof. □
The following table shows the number of units in the ring R, symbolic 2-plithogenic ring
, symbolic 3-plithogenic ring
, and symbolic 4-plithogenic ring
, and symbolic 5-plithogenic ring
Classical Ring | Symbolic 5-Plithogenic Ring | Symbolic 2-Plithogenic Ring | Symbolic 3-Plithogenic Ring | Symbolic 4-Plithogenic Ring |
Z
(2 units) | Z(I) (64 units) |
(8 units) |
(16 units) |
(32 units) |
(1 unit) | (I) (1 unit) |
(1 unit) |
(1 unit) |
(1 unit) |
(2 units) | (I) (64 units) |
(8 units) |
(16 units) |
(32 units) |
(2 units) | (I) (64 units) |
(8 units) |
(16 units) |
(32 units) |
(4 units) | (I) (4096 units) |
(64 units) |
(256 units) |
(1024 units) |
(2 units) | (I) (64 units) |
(8 units) |
(16 units) |
(32 units) |
(6 units) | (I)
units) |
(216 units) |
(1296 units) |
(7776 units) |
(4 units) | (I) (4096 units) |
(64 units) |
(256 units) |
(1024 units) |
(6 units) | (I)
units) |
(216 units) |
(1296 units) |
(7776 units) |
(4 units) | (I) (4096 units) |
(64 units) |
(256 units) |
(1024 units) |