In this paper we introduce the notion of medial
B
algebras, and we obtain a fundamental theorem of
B
homomorphism for
B
algebras.
AMS Mathematics Subject Classification : 06F35.
1. Introduction
Y. Imai and K. Iséki introduced two classes of abstract algebras:
BCK
algebras and
BCI
algebras
[4
,
5]
. It is known that the class of
BCK
algebras is a proper subclass of the class of
BCI
algebras. In
[2
,
3]
Q. P. Hu and X. Li introduced a wide class of abstract algebras:
BCH
algebras. They have shown that the class of
BCI
algebras is a proper subclass of the class of
BCH
algebras. J. Neggers and H. S. Kim
[8]
introduced the notion of
d
algebras, i.e., (I)
x
∗
x
= 0;(V) 0 ∗
x
= 0; (VI)
x
∗
y
= 0 and
y
∗
x
= 0 implay
x
=
y
, which is another useful generalization of
BCK
algebras, and then they investigated several relations between
d
algebras and oriented digraphs. Recently, Y. B. Jun, E. H. Roh and H. S. Kim
[6]
introduced a new notion, called an
BH
algebra, i.e., (I),(II)
x
∗ 0 = 0 and (IV), which is a generalization of
BCH
/
BCI
/
BCK
algebras. They also defined the notions of ideals and boundedness in
BH
algebras, and showed that there is a maximal ideal in bounded
BH
algebras. J. Neggers and H. S. Kim
[9]
introduced and investigated a class of algebras, i.e., the class of
B
algebras, which is related to several classed of algebras of interest such as
BCH
/
BCI
/
BCK
algebras and which seems to have rather nice properties without being excessively complicated otherwise. Furthermore, a digraph on algebras defined below demonstrates a rather interesting connection between
B
algebras and groups. J. R. Cho and H. S. Kim
[1]
discussed further relations between
B
algebras and other classed of algebras, such as quasigroups. J. Neggers and H. S. Kim
[10]
introduced the notion of normality in
B
algebras and obtained a fundamental theorem of
B
homomorphism for
B
algebras.
In this paper we introduce the notion of medial
B
algebras, and we obtain a fundamental theorem of
B
homomorphism for
B
algebras.
2. Preliminaries
In this section, we introduce some notions and results which have also been discussed in
[1
,
9]
. A
B
algebra is a nonempty set
X
with a constant 0 and a binary operation "∗" satisfying the following axioms:

(I)x∗x= 0,

(II)x∗ 0 =x,

(III) (x∗y) ∗z=x∗ (z∗ (0 ∗y))
for all
x
,
y
,
z
in
X
.
Example 2.1.
Let
X
:= {0, 1, 2} be a set with the following table:
Then (
X
; ∗, 0) is a
B
algebra.
Example 2.2
(
[9]
). Let
X
be the set of all real numbers except for a negative integer −
n
. Define a binary operation ∗ on
X
by
Then (
X
; ∗, 0) is a
B
algebra.
Example 2.3.
Let
X
:= {0, 1, 2, 3, 4, 5} be a set with the following table:
Then (
X
; ∗, 0) is a
B
algebra (see
[10]
).
Example 2.4
(
[9]
). Let
F
<
x
,
y
,
z
> be the free group on three elements. Define
u
∗
v
:=
vuv
^{−2}
. Thus
u
∗
u
=
e
and
u
∗
e
=
u
. Also
e
∗
u
=
u
^{−1}
. Now, given
a
,
b
,
c
, ∈
F
<
x
,
y
,
z
>, let
Let
N
(∗) be the normal subgroup of
F
<
x
,
y
,
z
> generated by the elements
w
(
a
,
b
,
c
). Let
G
=
F
<
x
,
y
,
z
> /
N
(∗). On
G
define the operation "·" as usual and define
It follows that (
uN
(∗)) ∗ (
uN
(∗)) =
eN
(∗), (
uN
(∗)) ∗ (
eN
(∗)) =
uN
(∗) and
Hence (
G
; ∗,
eN
(∗)) is a
B
algebra.
Lemma 2.5
(
[9]
).
If
(
X
; ∗, 0)
is a B

algebra
,
then y
∗
z
=
y
∗ (0 ∗ (0 ∗
z
))
for any y
,
z
∈
X
.
Proposition 2.6
(
[9]
).
If
(
X
; ∗; 0)
is a B

algebra
,
then
for any x
,
y
,
z
∈
X
.
Lemma 2.7
(
[1]
).
Let
(
X
; ∗, 0)
be a B

algebra
.
Then we have the following statements
.

(i)if x∗y= 0then x=y for any x,y∈X;

(ii)if0 ∗x= 0 ∗y then x=y for any x,y∈X;

(iii) 0 ∗ (0 ∗x) =x for any x∈X.
Let (
X
; ∗, 0
_{X}
) and (
Y
; •, 0
_{Y}
) be
B
algebras. A mapping
φ
:
X
→
Y
is called a
B

homomorphism
[10]
if
φ
(
x
∗
y
) =
φ
(
x
) •
φ
(
y
) for any
x
,
y
∈
X
.
Example 2.8
(
[10]
). Let
X
:= {0, 1, 2, 3} be a set with the following table:
Then (
X
; ∗, 0) is a
B
algebra
[1]
. If we define
φ
(0) = 0,
φ
(1) = 3,
φ
(2) = 3 and
φ
(3) = 0, then
φ
:
X
→
Y
is a
B
homomorphism.
A
B
homomorphism
φ
:
X
→
Y
is called a
B

isomorphism
[10]
if
φ
is a bijection, and denote it by
X
≅
Y
. Note that if
φ
:
X
→
Y
is a
B
isomorphism then
φ
^{−1}
:
Y
→
X
is also a
B
isomorphism. If we define
φ
(0) = 0,
φ
(1) = 2,
φ
(2) = 1 and
φ
(3) = 3 in Example 2.8, then
φ
:
X
→
Y
is a
B
isomorphism. Let
φ
:
X
→
Y
be a
B
homomorphism. Then the subset {
x
∈
X

φ
(
x
) = 0
_{Y}
} of
X
is called the
kernel
of the
B
homomorphism
φ
, and denote it by
Kerφ
Definition 2.9
(
[10]
). Let (
X
; ∗, 0) be a
B
algebra. A nonempty subset
N
of
X
is called a
subalgebra
of
X
if
x
∗
y
∈
N
, for any
x
,
y
∈
N
.
In Example 2.8,
N
_{1}
:= {0, 3} is a subalgebra of
X
, while
N
_{2}
:= {0, 1} is not a subalgebra of
X
, since 0 ∗ 1 = 2 ∉
N
_{2}
. Note that any subalgebra of a
B
algebra is also a
B
algebra.
Theorem 2.10
(
[10]
).
Let
(
X
; ∗, 0)
be a B

algebra and
Then the following are equivalent
:

(a)N is a subalgebra of X.

(b)x∗ (0 ∗y), 0 ∗y∈N,for any x,y∈N.
Note that any kernel of a
B
homomorphism is a subalgebra of
X
.
3. Medial Balgebras
Let (
X
; ∗, 0) be a
B
algebra and let
N
be a subalgebra of
X
. The set
X
(resp.,
N
) is said to be
medial
if (
x
∗
n
_{1}
) ∗ (
y
∗
n
_{2}
) = (
x
∗
y
) ∗ (
n
_{1}
∗
n
_{2}
) for any
x
,
y
,
n
_{1}
,
n
_{2}
∈
X
(resp., for any
x
,
y
,
n
_{1}
,
n
_{2}
∈
N
).
Example 3.1.
The
B
algebra in Example 2.8, is medial. The
B
algebra in Example 2.3, is not medial, since (5∗2)∗(4∗3) = 4∗1 = 5 ≠ 3 = 1∗5 = (5∗4)∗(2∗3).
J. Neggers and H. S. Kim
[10]
introduced the notion of a normal subalgebra in
B
algebras. A nonempty subset
N
of
X
is said to be
normal
(or
normal subalgebra
) of
X
if (
x
∗
a
) ∗ (
y
∗
b
) ∈
N
for any
x
∗
a
,
y
∗
b
∈
N
.
Example 3.2.
The subalgebra
N
_{1}
= {0, 3} is both a normal and a medial subalgebra of
X
in Example 2.8, while the subalgebra
N
_{2}
= {0, 3} in Example 2.3 is medial, but not normal.
Example 3.3.
Let
X
:= {0, 1, 2, 3} be a set with the following table:
Then (
X
; ∗, 0) is a
B
algebra and the subalgebra
N
_{3}
= {0, 2} is a medial subalgebra of
X
.
Let (
X
; ∗, 0) be a
B
algebra and let
N
be a subalgebra of
X
. Define a relation ∼
_{N}
on
X
by
x
∼
_{N} y
if and only if
x
∗
N
=
y
∗
N
, where
x
,
y
∈
X
. Then it is easy to show that ∼
_{N}
is an equivalence relation on
X
. Assume
X
is medial (or
N
is a medial subalgebra of
X
). If
x
∼
_{N} y
and
a
∼
_{N} b
, where
x
,
y
,
a
,
b
∈
N
, then
x
∗
N
=
y
∗
N
and
a
∗
N
=
b
∗
N
and hence
x
=
y
∗
n
_{1}
,
a
=
b
∗
n
_{2}
for some
n
_{1}
,
n
_{2}
∈
N
. Hence
x
∗
a
= (
y
∗
n
_{1}
) ∗ (
b
∗
n
_{2}
) = (
y
∗
b
) ∗ (
n
_{1}
∗
n
_{2}
) ∈ (
y
∗
b
) ∗
N
, since
X
(resp.,
N
) is medial. For any (
x
∗
a
) ∗
n
_{3}
∈ (
x
∗
a
) ∗
N
, we have
Hence (
x
∗
a
) ∗
N
⊆ (
y
∗
b
) ∗
N
. Similarly, we obtain (
y
∗
b
) ∗
N
⊆ (
x
∗
a
) ∗
N
. This means that
x
∗
a
∼
_{N} y
∗
b
, i.e., ∼
_{N}
is a congruence relation on
X
. Denote the equivalence class containing
x
by [
x
]
_{N}
, i.e., [
x
]
_{N}
= {
y
∈
X

x
∼
_{N} y
} and let
X
/
N
:= {[
x
]
_{N}

x
∈
X
}. We show that
X
/
N
is a
B
algebra.
Theorem 3.4.
Let X be a medial B

algebra and let N be a subalgebra of X
.
Then X
/
N is a medial B

algebra with N
= [0]
_{N}
.
Proof
. If we define [
x
]
_{N}
∗ [
y
]
_{N}
:= [
x
∗
y
]
_{N}
then the operation "∗" is welldefined, since ∼
_{N}
is a congruence relation on
X
. We claim that [0]
_{N}
=
N
. If
x
∈ [0]
_{N}
, then
x
∗
N
= 0 ∗
N
, and hence by (II)
x
=
x
∗ 0 ∈
x
∗
N
= 0 ∗
N
, i.e.,
x
= 0 ∗
n
for some
n
∈
N
. Since
N
is a subalgebra and 0 ∈
N
,
x
= 0 ∗
n
∈
N
. Hence [0]
_{N}
⊆
N
.
For any
x
∈
N
, since
N
is subalgebra of
X
, 0 ∗
x
∈
N
, say
n
_{1}
= 0 ∗
x
. By applying Lemma 2.7(iii),
x
= 0 ∗ (0 ∗
x
) ∈ 0 ∗
N
. We show that
x
∗
N
= 0 ∗
N
. For any
x
∗
n
∈
x
∗
N
,
Hence
x
∗
N
⊆ 0 ∗
N
. If
y
∈ 0 ∗
N
, then
y
= 0 ∗
n
_{2}
for some
n
_{2}
∈
N
. Hence
y
= 0 ∗
n
_{2}
= (
x
∗
x
) ∗
n
_{2}
=
x
∗ (
n
_{2}
∗ (0 ∗
x
)). Since
x
∈
N
, by Theorem 2.10,
n
_{2}
∗ (0 ∗
x
) ∈
N
. Hence
y
∈
x
∗
N
, i.e., 0 ∗
N
⊆ x ∗
N
. Thus
x
∗
N
= 0 ∗
N
, i.e., x ∼
_{N}
0. Hence
x
∈ [0]
_{N}
, proving
N
⊆ [0]
_{N}
. Checking three axioms and mediality is trivial and we omit the proof.
Theorem 3.4 can be replaced by the following statement:
Theorem 3.4′.
Let X be a B

algebra and N be a medial subalgebra of X
.
Then X
/
N is a medial B

algebra with N
= [0]
_{N}
.
The
B
algebra
X
/
N
discussed in Theorems 3.4 and 3.4′ is called the
quotient B
algebra of
X
by
N
.
Proposition 3.5.
Let N be a medial subalgebra of the B

algebra
(
X
; ∗, 0).
Then the mapping γ
:
X
→
X
/
N
,
given by γ
(
x
) := [
x
]
_{N}
,
is a surjective B

homomorphism
,
and Kerγ
=
N
.
Proof
. The mapping
γ
is obviously surjective. For all
x
,
y
∈
X
,
γ
(
x
∗
y
) = [
x
∗
y
]
_{N}
= [
x
]
_{N}
∗ [
y
]
_{N}
=
γ
(
x
) ∗
γ
(
y
). Hence
γ
is a
B
homomorphism. We claim that {
x
∈
X
 [
x
]
_{N}
= [0]
_{N}
} =
N
. For any
n
∈
N
, we show that
n
∗
N
= 0 ∗
N
. If
n
_{1}
∈
N
, by Lemma 2.7(iii),
n
∗
n
_{1}
= (0 ∗ (0 ∗
n
)) ∗
n
_{1}
= 0 ∗ (
n
_{1}
∗ (0 ∗ (0 ∗
n
))) = 0 ∗ (
n
_{1}
∗
n
) ∈ 0 ∗
N
, i.e.,
n
∗
N
⊆ 0 ∗
N
. For any 0 ∗
n
_{2}
∈ 0 ∗
N
, 0 ∗
n
_{2}
= (
n
∗
n
) ∗
n
_{2}
=
n
∗ (
n
_{2}
∗ (0 ∗
n
)) ∈
n
∗
N
, i.e., 0 ∗
N
⊆
n
∗
N
. This proves 0 ∗
N
=
n
∗
N
, i.e., [
n
]
_{N}
= [0]
_{N}
. If [
x
]
_{N}
= [0]
_{N}
, then
x
∗
N
= 0 ∗
N
, i.e.,
x
= 0 ∗
n
_{1}
for some
n
_{1}
∈
N
. Since
N
is a subalgebra of
X
,
x
= 0 ∗
n
_{1}
∈
N
. Hence
proving the proposition.
The mapping
γ
discussed in Proposition 3.5 is called the
natural
(or
canonical
)
B
homomorphism of
X
onto
X
/
N
.
Proposition 3.6.
Let X be a medial B

algebra
.
If φ
:
X
→
Y is a B

homomorphism
,
then the kernel Kerφ is a medial subalgebra of X
.
Proof
. Straightforward.
By Theorem 3.4 and Proposition 3.6, if
φ
:
X
→
Y
is a
B
homomorphism, then
X
/
Kerφ
is a
B
algebra.
A
B
algebra (
X
; ∗, 0) is said to be
commutative
[9]
if
a
∗ (0 ∗
b
) =
b
∗ (0 ∗
a
) for any
a
,
b
∈
X
. The
B
algebra in Example 2.1 is commutative, while the
B
algebra in Example 2.3 is not commutative, since 3 ∗ (0 ∗ 4) = 2 ≠ 1 = 4 ∗ (0 ∗ 3).
Theorem 3.7.
Let X be a commutative medial B

algebra and let φ
:
X
→
Y be a B

homomorphism
.
Then X
/
Kerφ
≅
Imφ
.
In particular
,
if φ is surjective
,
then X
/
Kerφ
≅
Y
.
Proof
. Let
K
:=
Kerφ
. If we define Ψ :
X
/
K
→
Imφ
by Ψ([
x
]
_{K}
) :=
φ
(
x
), then Ψ is welldefined. In fact, suppose that [
x
]
_{K}
= [
y
]
_{K}
. Then
x
∼
_{K} y
and
x
∗
K
=
y
∗
K
, i.e.,
x
=
y
∗
k
_{1}
,
y
=
x
∗
k
_{2}
for some
k
_{1}
,
k
_{2}
∈
K
. Hence
φ
(
x
) =
φ
(
y
∗
k
_{1}
) =
φ
(
y
) ∗
φ
(
k
_{1}
) =
φ
(
y
) ∗ 0 =
φ
(
y
), i.e., Ψ([
x
]
_{K}
) = Ψ([
y
]
_{K}
). Suppose that Ψ([
x
]
_{K}
) = Ψ([
y
]
_{K}
), where [
x
]
_{K}
, [
y
]
_{K}
∈
X
/
K
. Then
φ
(
x
) =
φ
(
y
). If
α
∈ [
x
]
_{K}
, then
α
∼
_{K} x
and
α
∗
K
=
x
∗
K
. This means that
α
=
x
∗
k
_{1}
,
x
=
α
∗
k
_{2}
for some
k
_{1}
,
k
_{2}
∈
K
. Hence
φ
(
α
) =
φ
(
x
∗
k
_{1}
) =
φ
(
x
) ∗
φ
(
k
_{1}
) =
φ
(
x
) =
φ
(
y
), which implies
φ
(
α
∗
y
) =
φ
(
α
) ∗
φ
(
y
) = 0. Hence
α
∗
y
∈
Kerφ
=
K
, i.e.,
α
∗
y
=
k
_{3}
for some
k
_{3}
∈
K
. Similarly,
φ
(
y
) ∗
φ
(
α
) = 0 implies
y
∗
α
=
k
_{4}
for some
k
_{4}
∈
K
. Sice
X
is commutative,
For any
α
∗
k
_{4}
∈
α
∗
K
,
α
∗
k
= (
y
∗
k
_{4}
) ∗
k
=
y
∗ (
k
∗ (0 ∗
k
_{4}
)) ∈
y
∗
K
. Hence
α
∗
K
⊆
y
∗
K
. Conversely, we have
proving
y
∗
K
⊆
α
∗
K
. Hence
α
∗
K
=
y
∗
K
, i.e.,
α
∼
_{K} y
. This proves
α
∈ [
y
]
_{K}
. Similarly, [
y
]
_{K}
⊆ [
x
]
_{K}
. Thus [
x
]
_{K}
= [
y
]
_{K}
, proving that Ψ is injective. Obviously Ψ is surjective. Since Ψ([
x
]
_{K}
∗ [
y
]
_{K}
) = Ψ([
x
∗
y
]
_{K}
) =
φ
(
x
∗
y
) =
φ
(
x
) ∗
φ
(
y
) = Ψ([
x
]
_{K}
) ∗ Ψ([
y
]
_{K}
), Ψ is a
B
homomorphism. Hence
X
/
Kerφ
≅
Imφ
.
Example 3.8.
In Example 2.8, since
K
=
Kerφ
= {0, 3}, we have [0]
_{K}
= {0, 3} and [1]
_{K}
= {
x
∈
X

x
∗ 1 ∈
K
} = {1, 2}. Hence
X
/
Kerφ
= {[0]
_{K}
, [1]
_{K}
} and
X
/
Kerφ
≅
Imφ
by defining Ψ([0]
_{K}
) =
φ
(0) and Ψ([1]
_{K}
) =
φ
(1).
BIO
Young Hee Kim is working as a professor in Department of Mathematics and is interested in BEalgebras.
Department of Mathematics, Chungbuk National University, Cheongju 361763, Korea.
email:yhkim@chungbuk.ac.kr
Kim J.R.
,
Cho H.S.
(2001)
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