1. Introduction
The notion of
-algebras was formulated by
. The motivation of this notion is based on both set theory and propositional calculus (see [
1]). As a generalization of this notion,
-,
-,
-,
-algebras have been developed by many researchers (see [
2,
3,
4]). Jun et al. [
5] introduced the notion of a
-algebra, which is a generalization of
-algebras. The concept of
B-algebras was introduced by Neggers and Kim [
6]. Kim and Kim [
7] defined the notion of a
-algebra. They showed that a
-algebra is equivalent to a 0-commutative
B-algebra. Kim and Kim [
8] introduced the notion of a
-algebra, and proved that every
-algebra is 0-commutative, and obtained several algebras which are logically equivalent to the
-algebra. Walendziak [
9] introduced a
-algebra, which is a generalization of a
B-algebra, and investigated some properties of (normal) ideals in
-algebras. Kim and Kim [
10] defined the notion of a
-algebra, and showed that an algebra
A is a
-algebra if and only if it is a 0-commutative
-algebra. Kim et al. [
11] introduced the notions of (pre-)Coxeter algebras in the Smarandache setting, and Kim and Kim [
12] discussed some relations between (pre-)Coxeter algebras and its related topics.
In this paper, we introduce the notion of a -algebra consisting of 3 simple axioms, and we show it is logically equivalent to several known algebras. In particular, we investigate the role of the axiom for proving the logically (non-) equivalence to several algebras. Moreover, we show that a -algebra with is logically equivalent to several algebras, and we show some relationships between a -algebra with and several related algebras.
3. -Algebras
In this section, we define a notion of a -algebra, and we investigate some relations between -algebras and other algebras, i.e., B-algebras, 0-commutative B-algebras, -algebras, -algebras, -algebras, -algebras, -algebras, -algebras and Coxeter-algebras.
Definition 1. An algebra is said to be a -algebra if it satisfies and , where
- (BV)
for all .
Example 1. Let be a set with the following table:* | 0 | 1 | 2 | 3 | 4 |
0 | 0 | 2 | 1 | 4 | 3 |
1 | 1 | 0 | 3 | 2 | 4 |
2 | 2 | 4 | 0 | 3 | 1 |
3 | 3 | 1 | 4 | 0 | 2 |
4 | 4 | 3 | 2 | 1 | 0 |
Then it is easy to see that is a -algebra.
Proposition 3. If is a -algebra, then the followings hold:
- (i)
,
- (ii)
,
- (iii)
,
- (iv)
,
- (v)
,
- (vi)
,
- (vii)
- (viii)
,
- (ix)
,
- (x)
,
- (xi)
,
for any .
Proof. (i). If we let
in
, then
by
. (ii). If we let
in (i), then, by
and
, we obtain
. (iii). If we let
in
, then
, and hence
. (iv). If we let
in
, then
. (v). If we let
in (i), then
. By applying (ii), we obtain
. (vi). If we let
in
, then, by applying (iii) and (i), we have
. (vii). If
in (iv), then
, which implies
. It follows that
by (ii). (viii). If
, then
and hence
by (ii). (ix). If we let
in (iv), then
. By applying (ii), we obtain
. (x). Assume
. Then, by applying (iv), we have
By applying (viii), we obtain . (xi). If we assume , then, by (iii), we obtain . By applying (viii), we have . By (x), we obtain . □
By applying (iii) and (vii) of Proposition 3, we obtain the following theorem:
Theorem 1. Let be a -algebra. Then
- (i)
it is a -algebra,
- (ii)
it is a -algebra.
Remark 1. The converse of Theorem 1(i) does not hold in general.
Example 2. Let be a set. Define a binary operation “*” on A as follows:* | 0 | 1 | 2 | 3 | 4 |
0 | 0 | 1 | 2 | 4 | 3 |
1 | 1 | 0 | 4 | 1 | 4 |
2 | 2 | 3 | 0 | 3 | 4 |
3 | 3 | 1 | 4 | 0 | 0 |
4 | 4 | 3 | 3 | 0 | 0 |
Then is a -algebra, but not a -algebra, since .
Theorem 2. If is a Coxeter algebra, then it is a -algebra.
Proof. Let
be a Coxeter algebra. Since
A satisfies the conditions
and
, it is enough to show the condition
. Given
, we have
Hence A is a -algebra. □
Remark 2. The converse of Theorem 2 does not hold in general.
Example 3. Let be a -algebra as in Example 1. Then it is not a Coxeter algebra, since .
Lemma 1. If is a -algebra, then it is a 0-commutative.
Proof. It follows immediately from Proposition 3(v). □
Theorem 3. If is a -algebra, then it is a B-algebra.
Proof. Let
A be a
-algebra. Since
A satisfies the conditions
and
, it is enough to show the condition
. If
and
z be any elements of
A, then
Hence A is a B-algebra. □
Remark 3. The converse of this theorem does not hold.
Example 4. Let A be the set of all real numbers except for a negative integer . Define a binary operation “*” on A as follows:for any . Then is a B-algebra [
6],
but it is not a -algebra, since . By using Lemma 1 and Theorem 3, we obtain the following:
Corollary 1. If is a -algebra, then it is a 0-commutative B-algebra.
Theorem 4. If is a 0-commutative B-algebra, then it is a -algebra.
Proof. Let
be a 0-commutative
B-algebra. Then
for all
. Hence
is a
-algebra. □
By Corollary 1 and Theorem 4, we obtain the following corollary:
Corollary 2. An algebra is a -algebra if and only if it is a 0-commutative B-algebra.
An algebra is called a -algebra if the conditions and hold, where
- (BT)
for all
Theorem 5. An algebra is a -algebra if and only if it is a -algebra.
Proof. Assume that
is a
-algebra. We show the condition
. Given
, we have
for all
. Hence
A is a
-algebra.
Conversely, assume that
A is a
-algebra. If we let
in
, then, by (A1) and (A2), we have
for all
. To show that
A is a
-algebra, we show that the condition
holds. In fact, we have
for all
. Hence
A is a
-algebra. □
Kim and Kim defined the notion of a
-algebra. An algebra
is said to be a
-
algebra [
10] if it satisfies the conditions
,
and
, where
- (BN)
for all
. It was proved that
-algebras are equivalent to 0-commutative
-algebras [
10].
Theorem 6. If is a -algebra, then it is a -algebra.
Proof. Let
A be a
-algebra. Then
A satisfies the conditions
and
. We show that the condition
holds.
for all
. Hence
A is a
-algebra. □
Remark 4. The converse of Theorem 6 does not hold in general.
Example 5. Let be a set with the following table:* | 0 | 1 | 2 | 3 |
0 | 0 | 1 | 2 | 3 |
1 | 1 | 0 | 1 | 1 |
2 | 2 | 1 | 0 | 1 |
3 | 3 | 1 | 1 | 0 |
Then is a -algebra, but not a -algebra, since .
By Corollary 2, Theorem 5 and Proposition 2, we obtain the interesting result below:
Theorem 7. The following statements are logically equivalent:
- (1)
A is a -algebra,
- (2)
A is a 0-commutative B-algebra,
- (3)
A is a -algebra,
- (4)
A is a -algebra,
- (5)
A is a -algebra.
We summarize the above discussion, and we give a diagram describing some relations between -algebras and its related algebras as follows:
4. -Algebras with Some Conditions
In this section, we investigate some relationships between several general algebraic structures and -algebras with special conditions as follows:
- (D)
- (F)
for all . Note that Boolean groups and the Klein four group are -algebras with .
Theorem 8. If is a -algebra with , then it is a pre-Coxeter algebra.
Proof. Let be a -algebra with . If , then by Proposition 3(vii). On the other hand, by Proposition 3(iii), we obtain . Hence A is a pre-Coxeter algebra. □
Remark 5. The converse of Theorem 8 does not hold in general.
Example 6. Let be a set with the following table:* | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
0 | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
1 | 1 | 0 | 3 | 2 | 5 | 4 | 7 | 6 | 9 | 8 |
2 | 2 | 3 | 0 | 1 | 6 | 8 | 4 | 9 | 5 | 7 |
3 | 3 | 2 | 1 | 0 | 9 | 7 | 8 | 5 | 6 | 4 |
4 | 4 | 5 | 6 | 9 | 0 | 1 | 2 | 8 | 7 | 3 |
5 | 5 | 4 | 8 | 7 | 1 | 0 | 9 | 3 | 2 | 6 |
6 | 6 | 7 | 4 | 8 | 2 | 9 | 0 | 1 | 3 | 5 |
7 | 7 | 6 | 9 | 5 | 8 | 3 | 1 | 0 | 4 | 2 |
8 | 8 | 9 | 5 | 6 | 7 | 2 | 3 | 4 | 0 | 1 |
9 | 9 | 8 | 7 | 4 | 3 | 6 | 5 | 2 | 1 | 0 |
Then is a pre-Coxeter algebra, but it is not a -algebra with , since .
Theorem 9. An algebra is a -algebra with if and only if it is a Coxeter algebra.
Proof. Assume that
is a
-algebra with
. Then
A satisfies the conditions
and
. Given
, we obtain
This shows that A is a Coxeter algebra.
Conversely, assume that
is a Coxeter algebra. Then
A satisfies the conditions
and
. Given
, we have
Hence is a -algebra with . □
Note that the -algebra in Example 1 does not satisfy the condition , and hence it is not a Coxeter algebra.
Lemma 2. Let be a -algebra. Then the followings are equivalent: for any ,
- (i)
,
- (ii)
.
Proof. (i) ⇒ (ii). Suppose the condition (i) holds. Then, for any
, we have
(ii) ⇒ (i). Suppose the condition (ii) holds. If we let in (ii), then, by (A2) and Proposition 3(ii), we obtain for any . □
Example 7. In Example 1, we see that and hence the condition does not hold for some . In fact, .
Lemma 3. If is a B-algebra with , then for any .
Proof. Since A is a B-algebra with , if we let in , then for all . □
Example 8. In Example 4, we see that and hence the condition does not hold for some . In fact, .
Theorem 10. An algebra is a -algebra with if and only if it is a B-algebra with .
Proof. Let
be a
-algebra with the condition
. Then, for any
, we have
Hence is a B-algebra with .
Conversely, let
A be a
B-algebra with
. Then
for any
. Hence
A is a
-algebra with
. □
Lemma 4. If is a -algebra with , then for any .
Proof. Since is a -algebra with , by Proposition 3(iii) and , we have for any . □
Lemma 5. If is a -algebra with , then
- (i)
,
- (ii)
.
for any .
Proof. (i). Let A be a -algebra with . If we let in , then and hence for all .
(ii). If we let in , then by (i) for any . □
Theorem 11. An algebra is a -algebra with if and only if it is a -algebra with .
Proof. Assume that
A is a
-algebra with
. Then
A satisfies the conditions (A1) and (A2). By Proposition 3(iii),
A satisfies the condition
. Let
and
z be any elements of
A. Then
Hence A is a -algebra with .
Conversely, assume that
A is a
-algebra with
. Then
A satisfies the conditions (A1) and (A2). By Lemma 5(i),
A satisfies the condition
. Let
and
z be any elements of
A. Then
Hence A is a -algebra with . □
Lemma 6. Let be a -algebra with . Then
- (i)
,
- (ii)
,
- (iii)
,
- (iv)
,
for any .
Proof. Let A be a -algebra with . (i). If we let in , then . By applying and , we obtain .
(ii). If we let in , then . By applying and , we obtain .
(iii). Let in . Then . By applying (ii) and , we obtain for all .
(iv). If we let in , then . By applying (iii) and , we obtain for all . □
Theorem 12. An algebra is a -algebra with if and only if it is a -algebra with .
Proof. Let be a -algebra with . Then A satisfies the conditions (A1) and (A2). By Lemma 2, A satisfies the condition . By applying Proposition 3(vi), we obtain for all , which shows that the condition holds. Hence is a -algebra with .
Conversely, let
A be a
-algebra with
. Then
A satisfies the condition (A1) and (A2). By Lemma 6(iii),
A satisfies the condition
. Given
, we have
Hence A is a -algebra with . □
From Lemma 2, we have the following corollary:
Corollary 3. An algebra is a -algebra with if and only if it is a -algebra with .
Theorem 13. Let be a B-algebra. Then the followings are equivalent: for any ,
- (i)
,
- (ii)
.
Proof. (i) ⇒ (ii). Let
A be a
B-algebra. Assume
A satisfies the condition (i). Then
for any
.
(ii) ⇒ (i). Assume
A satisfies the condition (ii). Then
for any
. □
Lemma 7. Let be a -algebra with . Then, for any , we have
- (i)
,
- (ii)
.
Proof. (i). Since
A is a
-algebra with
, we have
for any
.
(ii). By (i), is a Coxeter algebra, and hence for all . □
By Theorem 7, we know that a -algebra is logically equivalent to a -algebra, but there is no direct proof of its equivalence. Using the condition , we describe its relationship as below:
Theorem 14. An algebra is a -algebra with if and only if it is a -algebra with .
Proof. Let
be a
-algebra with
. We show that the condition
holds. If we take
and
z in
A, then
Hence A is a -algebra with .
Conversely, assume that
A is a
-algebra with
. We show the condition
. Given
, by using Lemma 7(i) and
, we obtain:
Hence A is a -algebra with . □
Theorem 15. If is a -algebra with , then it is a -algebra with .
Proof. Assume that
is a
-algebra with
. We prove the condition
. Given
, we have
Hence A is a -algebra with . □
Remark 6. The converse of Theorem 15 does not hold in general.
Example 9. Let be a set with the following table: Then it is easy to see that A is a -algebra with , but not a -algebra with , since .
From the above theorems, we obtain the following equivalent statements:
Corollary 4. The followings are equivalent:
- (i)
is a -algebra with ,
- (ii)
is a B-algebra with ,
- (iii)
is a -algebra with ,
- (iv)
is a -algebra with ,
- (v)
is a -algebra with ,
- (vi)
is a -algebra with ,
- (vii)
is a Coxeter algebra,
- (viii)
is a Boolean group.
We provide a diagram describing some relations between -algebras with and its related algebras as follows: