Some properties of filters are studied with respect to a congruence of
BE
algebras. The notion of
θ
filters is introduced and these classes of filters are then characterized in terms of congruence classes. A bijection is obtained between the set of all
θ
filters of a
BE
algebra and the set of all filters of the respective
BE
algebra of congruences classes.
AMS Mathematics Subject Classification : 06F35, 03G25, 08A30.
1. Introduction
The notion of BEalgebras was introduced and extensively studied by H.S. Kim and Y.H. Kim in
[5]
. These classes of
BE
algebras were introduced as a generalization of the class of
BCK
algebras of K. Iseki and S. Tanaka
[4]
. Some properties of filters of
BE
algebras were studied by S.S. Ahn and Y.H. Kim in
[1]
. In
[8]
, the notion of normal filters is introduced in
BE
algebras. In
[2
,
3]
, S.S. Ahn and Y.H. So and K.S. So introduced the notion of ideals in
BE
algebras and proved several characterizations of such ideals. Also they generalized the notion of upper sets in
BE
algebras, and discussed some properties of the characterizations of generalized upper sets related to the structure of ideals in transitive and selfdistributive
BE
algebras. In 2012, S.S. Ahn, Y.H. Kim and J.M. Ko
[1]
introduced the notion of a terminal section of
BE
algebras and derived some characterizations of commutative
BE
algebras in terms of lattice ordered relations and terminal sections. Recently in 2015, J.H. Park and Y.H. Kim
[7]
studied the properties of intersectional soft implicative filters of
BE
algebras. In
[10]
, A. Walendziak discussed some relationships between congruence relations and normal filters of a BEalgebra.
In this paper, two operations are introduced one from the set of all filters of a
BE
algebra into the algebra of filters of its congruence classes and the other from the algebra of filters of the congruence classes into the set of all filters of the given
BE
algebra. Later, it is shown that their composition is a closure operator on the set of all filters of a
BE
algebra. The concept of
θ
filters is introduced in
BE
algebras with respect to a congruence. The
θ
filters are also characterized in terms of congruence classes. A set of equivalent conditions is derived for every filter of a
BE
algebra to become a
θ
filter. Finall
y, a
bijection is obtained between the set of all
θ
filters of a
BE
algebra and the set of all filters of it’s algebra of congruence classes.
2. Preliminaries
In this section, we present certain definitions and results which are taken mostly from the papers
[1]
,
[5]
and
[8]
for the ready reference of the reader.
Definition 2.1
(
[5]
). An algebra (
X
, ∗, 1) of type (2, 0) is called a
BE
algebra if it satisfies the following properties:
(1)
x
∗
x
= 1,
(2)
x
∗ 1 = 1,
(3) 1 ∗
x
=
x
,
(4)
x
∗ (
y
∗
z
) =
y
∗ (
x
∗
z
) for all
x, y, z
∈
X
.
Theorem 2.2
(
[5]
).
Let
(
X
, ∗, 1)
be a BEalgebra. Then we have the following:
(1)
x
∗ (
y
∗
x
) = 1
(2)
x
∗ ((
x
∗
y
) ∗
y
)) = 1
We introduce a relation ≤ on a
BE
algebra
X
by
x
≤
y
if and only if
x
∗
y
= 1 for all
x, y
∈
X
. A
BE
algebra
X
is called selfdistributive if
x
∗ (
y
∗
z
) = (
x
∗
y
) ∗ (
x
∗
z
) for all
x, y, z
∈
X
. In any selfdistributive
BE
algebra, the set ⟨
a
⟩ = {
x
∈
X

a
≤
x
} = {
x
∈
X

a
∗
x
= 1} is the smallest filter containing the element
a
∈
X
which is known as the principal filter of
X
generated by
a
.
Definition 2.3
(
[1]
). Let (
X
, ∗, 1) be a
BE
algebra. A nonempty subset
F
of
X
is called a filter of
X
if, for all
x, y
∈
X
, it satisfies the following properties:
(a) 1 ∈
F
,
(b)
x
∈
F
and
x
∗
y
∈
F
imply that
y
∈
F
.
Definition 2.4
(
[8]
). Let (
X
, ∗, 1) be a
BE
algebra. A nonempty subset
F
of
X
is called a normal filter of
X
if it satisfies the following properties:
(a) 1 ∈
F
,
(b)
x
∈
X
and
a
∈
F
imply that
x
∗
a
∈
F
.
Definition 2.5
(
[10]
). Let (
X
, ∗, 1) be a
BE
algebra. A binary relation
θ
on
X
is called a congruence on
X
if (
x, y
) ∈
θ
and (
z,w
) ∈
θ
imply that (
x
∗
z
,
y
∗
w
) ∈
θ
for all
x, y, z, w
∈
X
.
For any congruence
θ
on a
BE
algebra
X
, the quotient algebra
X_{/θ}
= {[
x
]
_{θ}

x
∈
X
}, where [
x
]
_{θ}
is a congruence class of
x
modulo
θ
, is a
BE
algebra with respect to the operation [
x
]
_{θ}
∗ [
y
]
_{θ}
= [
x
∗
y
]
_{θ}
for all
x, y
∈
X
.
3.θfilter inBEalgebras
In this section, the notion of
θ
filters is introduced in
BE
algebras. A bijection is obtained between the set of all
θ
filters of a
BE
algebra and the set of all filters of the
BE
algebra of all congruence classes. We first prove the following crucial result which play a vital role in the forth coming results.
Theorem 3.1.
A nonempty subset F of a BEalgebra X is a filter of X if and only if it satisfies the following conditions for all x, y
∈
X
.
(1)
x
∈
F implies y
∗
x
∈
F
.
(2)
a, b
∈
F implies
(
a
∗ (
b
∗
x
)) ∗
x
∈
F
.
Proof
. Assume that
F
is a filter of
X
. Let
x
∈
F
. We have
x
∗ (
y
∗
x
) =
y
∗ (
x
∗
x
) =
y
∗ 1 = 1 ∈
F
. Since
x
∈
F
and
F
is a filter, we get that
y
∗
x
∈
F
. Let
a, b
∈
F
. Since
a
∗((
a
∗(
b
∗
x
))∗(
b
∗
x
)) = (
a
∗(
b
∗
x
))∗(
a
∗(
b
∗
x
)) = 1, we get that
a
≤ ((
a
∗(
b
∗
x
))∗(
b
∗
x
)). Since
a
∈
F
, it yields that (
a
∗(
b
∗
x
))∗(
b
∗
x
) ∈
F
. Hence
b
∗ ((
a
∗ (
b
∗
x
)) ∗
x
) ∈
F
. Since
b
∈
F
, it implies (
a
∗ (
b
∗
x
)) ∗
x
∈
F
.
Conversely, assume that
F
satisfies the given conditions (1) and (2). By taking
x
=
y
in the condition (1), it can be seen that 1 ∈
F
. Let
x, y
∈
X
be such that
x
,
x
∗
y
∈
F
. Then by the condition (2), we get
y
= 1 ∗
y
= ((
x
∗
y
) ∗ (
x
∗
y
)) ∗
y
= (
x
∗ ((
x
∗
y
) ∗
y
)) ∗
y
∈
F
. Therefore
F
is a filter in
X
. □
For any congruence
θ
on a
BE
algebra (
X
, ∗, 1), let us recall that the set
X_{/θ}
of all congruence classes forms a
BE
algebra with respect to the operation [
x
]
_{θ}
∗ [
y
]
_{θ}
= [
x
∗
y
]
_{θ}
for all
x, y
∈
X
. It also forms a partially ordered set ordered by set inclusion. The smallest congruence on
X
is given by
θ
_{0}
= {(
a, a
) 
a
∈
X
}.
In the following, we first introduce two operations.
Definition 3.2.
Let
θ
be a congruence on a
BE
algebra X. Define operations
α
and
β
as follows:
(1) For any filter
F
of
X
, define
α
(
F
) = { [
x
]
_{θ}
 (
x, y
) ∈
θ
for some
y
∈
F
}
(2) For any filter
of
X_{/θ}
, define
β
(
) = {
x
∈
X
 (
x, y
) ∈
θ
for some [
y
]
_{θ}
∈
}.
In the following lemma, some basic properties of the above two operations
α
and
β
are observed.
Lemma 3.3.
Let θ be a congruence on a BEalgebra X. Then we have
(1)
For any filter F of X
,
α
(
F
)
is a filter of X_{/θ}
,
(2)
For any filter
of L_{/θ}, β
(
)
is a filter of X,
(3)
α and β are isotone,
(4)
For any filter F of X, x
∈
F implies
[
x
]
_{θ}
∈
α
(
F
),
(5)
For any filter
of X_{/θ},
[
x
]
_{θ}
∈
implies x
∈
β
(
).
Proof
. (1). Let [
x
]
_{θ}
∈
X_{/θ}
and [
a
]
_{θ}
∈
α
(
F
). Then (
a, y
) ∈
θ
for some
y
∈
F
. Hence (
x
∗
a
,
x
∗
y
) ∈
θ
and
x
∗
y
∈
F
because of
F
is a filer. Thus [
x
]
_{θ}
∗ [
a
]
_{θ}
= [
x
∗
a
]
_{θ}
∈
α
(
F
). Let [
a
]
_{θ}
, [
b
]
_{θ}
∈
α
(
F
). Then (
a, x
) ∈
θ
and (
b, y
) ∈
θ
for some
x, y
∈
F
. Now, for any
t
∈
X
, we get
Since
x, y
∈
F
and
F
is a filter, we get by Theorem 3.1 that (
x
∗ (
y
∗
t
)) ∗
t
∈
F
. Hence
Therefore by Theorem 3.1, it concludes that
α
(
F
) is a filter of
X_{/θ}
.
(2). Let
x
∈
X
and
a
∈
β
(
). Then (
a, y
) ∈
θ
for some [
y
]
_{θ}
∈
. Hence (
x
∗
a
,
x
∗
y
) ∈
θ
. Since
is a filter, we get [
x
∗
a
]
_{θ}
= [
x
∗
y
]
_{θ}
= [
x
]
_{θ}
∗ [
y
]
_{θ}
∈
. Thus it yields
x
∗
a
∈
β
(
). Again, let
a, b
∈
β
(
) and
t
∈
X
. Then we get (
a, x
) ∈
θ
and (
b, y
) ∈
θ
for some [
x
]
_{θ}
∈
and [
y
]
_{θ}
∈
. Since
is a filter, we get [(
x
∗(
y
∗
t
))∗
t
]
_{θ}
= ([
x
]
_{θ}
∗([
y
]
_{θ}
∗[
t
]
_{θ}
))∗[
t
]
_{θ}
∈
. Since (
a, x
) ∈
θ
and (
b, y
) ∈
θ
, it is clear that ((
a
∗ (
b
∗
t
)) ∗
t
, (
x
∗ (
y
∗
t
)) ∗
t
) ∈
θ
. Since [(
x
∗ (
y
∗
t
)) ∗
t
]
_{θ}
∈
, we get (
a
∗ (
b
∗
t
)) ∗
t
∈
β
(
). Therefore by Theorem 3.1,
β
(
) is a filter of
X
.
(3). Let
F
_{1}
,
F
_{2}
be two filters in
X
such that
F
_{1}
⊆
F
_{2}
. Let [
x
]
_{θ}
∈
α
(
F
_{1}
). Then, we get (
x, y
) ∈
θ
for some
y
∈
F
_{1}
⊆
F
_{2}
. Consequently, we get that [
x
]
_{θ}
∈
α
(
F
_{2}
). Therefore
α
(
F
_{1}
) ⊆
α
(
F
_{2}
). Again, let
be two filters of
X_{/θ}
such that
. Suppose
x
∈
. Then, it infers that (
x, y
) ∈
θ
for some [
y
]
_{θ}
∈
. Hence
y
∈
. Therefore
.
(4). For any
x
∈
F
, we have (
x, x
) ∈
θ
. Hence it concludes [
x
]
_{θ}
∈
α
(
F
).
(5). For any [
x
]
_{θ}
∈
, we have (
x, x
) ∈
θ
. Hence we get
x
∈
β
(
). □
The following corollary is a direct consequence of the above lemma.
Corollary 3.4.
Let θ be a congruence on a BEalgebra X. Then we have
(1)
For any normal filter F of X, α
(
F
)
is a normal filter of X_{/θ}
.
(2)
For any normal filter
of L_{/θ}
,
β
(
)
is a normal filter of X
.
Lemma 3.5.
Let θ be a congruence on a BEalgebra X. For any filter F of X, αβα
(
F
) =
α
(
F
).
Proof
. Let [
x
]
_{θ}
∈
α
(
F
). Then (
x, y
) ∈
θ
for some
y
∈
F
. Since
y
∈
F
, by Lemma 3.3(4), we get [
y
]
_{θ}
∈
α
(
F
). Since (
x, y
) ∈
θ
and [
y
]
_{θ}
∈
α
(
F
), we get
x
∈
βα
(
F
). Hence [
x
]
_{θ}
∈
αβα
(
F
). Thus
α
(
F
) ⊆
αβα
(
F
). Conversely, let [
x
]
_{θ}
∈
αβα
(
F
). Then (
x, y
) ∈
θ
for some
y
∈
βα
(
F
). Since
y
∈
βα
(
F
), there exists [
a
]
_{θ}
∈
α
(
F
) such that (
y, a
) ∈
θ
. Hence [
x
]
_{θ}
= [
y
]
_{θ}
= [
a
]
_{θ}
∈
α
(
F
). Therefore
αβα
(
F
) ⊆
α
(
F
).
We now intend to show that the composition
βα
is a closure operator on the set
Ƒ
(
X
) of all filters of a
BE
algebra
X
.
Proposition 3.6.
For any filter F of X, the map F → βα
(
F
)
is a closure operator on Ƒ
(
X
).
That is, for any two filters F, G of X, we have the following:
(
a
)
F
⊆
βα
(
F
).
(
b
)
βαβα
(
F
) =
βα
(
F
).
(
c
)
F
⊆
G
⇒
βα
(
F
) ⊆
βα
(
G
).
Proof
. (
a
). Let
x
∈
F
. Then by Lemma 3.3(4), we get [
x
]
_{θ}
∈
α
(
F
). Since (
x, x
) ∈
θ
and
α
(
F
) is a filter in
X_{/θ}
, we get
x
∈
βα
(
F
). Therefore
F
⊆
βα
(
F
). (
b
). Since
βα
(
F
) is a filter in
X
, by above condition (
a
), we get
βα
(
F
) ⊆
βα
[
βα
(
F
)]. Conversely, let
x
∈
βα
[
βα
(
F
)]. Then we obtain (
x, y
) ∈
θ
for some [
y
]
_{θ}
∈
αβα
(
F
). Thus by above Lemma 3.5, we get that [
y
]
_{θ}
∈
α
(
F
). Hence
x
∈
βα
(
F
). Therefore, it concludes
βα
[
βα
(
F
)] ⊆
βα
(
F
).
(
c
). Suppose
F,G
are two filters of
X
such that
F
⊆
G
. Let
x
∈
βα
(
F
). Then we get [
x
]
_{θ}
∈
α
(
F
). Hence [
x
]
_{θ}
= [
y
]
_{θ}
for some
y
∈
F
⊆
G
. Since
y
∈
G
, we get [
x
]
_{θ}
= [
y
]
_{θ}
∈
α
(
G
). Therefore
x
∈
βα
(
G
). Hence
βα
(
F
) ⊆
βα
(
G
). □ 0
Denoting by
Ƒ
(
X_{/θ}
) the set of all filters of
X_{/θ}
, we can therefore define a mapping
α
:
Ƒ
(
X
) →
Ƒ
(
X_{/θ}
) by
F
↦
α
(
F
) also another mapping
β
:
Ƒ
(
X_{/θ}
) →
Ƒ
(
X
) by
F
↦
α
(
F
). Then we have the following:
Proposition 3.7.
Let θ be congruence on a BEalgebra X. Then α is a residuated map with residual map β
.
Proof
. For every
F
∈
Ƒ
(
X
), by Proposition 3.6(a), we have that
F
⊆
βα
(
F
). Let
F
∈
Ƒ
(
X_{/θ}
). Suppose [
x
]
_{θ}
∈
F
. Then we get
x
∈
β
(
F
). Since
β
(
F
) is a filter of
X
, we get [
x
]
_{θ}
∈
αβ
(
F
). Hence, it yields
F
⊆
αβ
(
F
). Conversely, let [
x
]
_{θ}
∈
αβ
(
F
). Then [
x
]
_{θ}
= [
y
]
_{θ}
for some
y
∈
β
(
F
). Since
y
∈
β
(
F
), we get [
x
]
_{θ}
= [
y
]
_{θ}
∈
F
. Hence
αβ
(
F
) ⊆
F
. Therefore for every
F
∈
Ƒ
(
X_{/θ}
), we obtain that
αβ
(
F
) =
F
. Since
α
and
β
are isotone, it follows that
α
is residuated and that the residual of
α
is nothing but
β
. □
We now introduce the notion of
θ
filters in a
BE
algebra.
Definition 3.8.
Let
θ
be a congruence on a
BE
algebra
X
. A filter
F
of
X
is called a
θ
filter if
βα
(
F
) =
F
.
For any congruence
θ
on a
BE
algebra
X
, it can be easily observed that the filter {1} is a
θ
filter if and only if [1]
_{θ}
= {1}. Moreover, we have the following:
Lemma 3.9.
Let θ be a congruence on a bounded BEalgebra X with smallest element
0.
For any filter F of X, the following hold:
(1)
If F is a θfilter then
[1]
_{θ}
⊆
F
, (2)
If F is a proper θfilter then F
∩[0]
_{θ}
= ∅.
In the following theorem, a set of sufficient conditions is derived for a proper filter of a
BE
algebra to become a
θ
filter.
Theorem 3.10.
Let θ be a congruence on a BEalgebra X. A proper filter F of X is a θfilter if it satisfies the following conditions:
(1)
For x, y
∈
X with x
≠
y, either x
∈
F or y
∈
F
(2)
To each x
∈
F, there exists x′
∉
F such that
(
x, x′
) ∈
θ
Proof
. Let
F
be a proper filter of
X
. Clearly
F
⊆
βα
(
F
). Conversely, let
x
∈
βα
(
F
). Then (
x, y
) ∈
θ
for some [
y
]
_{θ}
∈
α
(
F
). Hence [
y
]
_{θ}
= [
a
]
_{θ}
for some
a
∈
F
. Since
a
∈
F
, there exists
a′
∉
F
such that (
a, a′
) ∈
θ
. Since (
x, y
) ∈
θ
, (
y, a
) ∈
θ
and (
a, a′
) ∈
θ
, by the transitive property, we get (
x, a′
) ∈
θ
. Since
a
′ ∉
F
, by conditions (1) and (2), we get
x
∈
F
. Therefore
βα
(
F
) =
F
. □
We now characterize
θ
filters in the following:
Theorem 3.11.
Let θ be a congruence on a BEalgebra X. For any filter F of X, the following conditions are equivalent:
(1)
F is a θfilter;
(2)
For any x, y
∈
X
, [
x
]
_{θ}
= [
y
]
_{θ}
and x
∈
F imply that y
∈
F;
(3)
(4)
x
∈
F implies
[
x
]
_{θ}
⊆
F
.
Proof
. (1) ⇒ (2): Assume that
F
is a
θ
filter of
X
. Let
x, y
∈
X
be such that [
x
]
_{θ}
= [
y
]
_{θ}
. Then (
x, y
) ∈
θ
. Suppose
x
∈
F
=
βα
(
F
). Then (
x, a
) ∈
θ
for some [
a
]
_{θ}
∈
α
(
F
). Thus (
a, y
) ∈
θ
and [
a
]
_{θ}
∈
α
(
F
). Therefore
y
∈
βα
(
F
) =
F
.
(2) ⇒ (3): Assume the condition (2). Let
x
∈
F
. Since
x
∈ [
x
]
_{θ}
, we get
F
⊆
. Conversely, let
a
∈
. Then (
a, x
) ∈
θ
for some
x
∈
F
. Hence [
a
]
_{θ}
= [
x
]
_{θ}
. By the condition (2), we get
a
∈
F
. Therefore
F
=
.
(3) ⇒ (4): Assume the condition (3). Let
a
∈
F
. Then we get that (
x, a
) ∈
θ
for some
x
∈
F
. Let
t
∈ [
a
]
_{θ}
. Then (
t, a
) ∈
θ
. Hence (
x, t
) ∈
θ
. Thus it yields
t
∈ [
x
]
_{θ}
⊆
F
. Therefore it can be concluded that [
a
]
_{θ}
⊆
F
.
(4) ⇒ (1): Assume the condition (4). Clearly
F
⊆
βα
(
F
). Conversely, let
x
∈
βα
(
F
). Then (
x, y
) ∈
θ
for some [
y
]
_{θ}
∈
α
(
F
). Hence [
y
]
_{θ}
= [
a
]
_{θ}
for some
a
∈
F
. Since
a
∈
F
, by condition (4), we get that
x
∈ [
y
]
_{θ}
= [
a
]
_{θ}
⊆
F
. Thus
βα
(
F
) ⊆
F
. Therefore
F
is a
θ
filter of
X
. □
In the following theorem, a set of equivalent conditions is obtained to characterize the smallest congruence in terms of
θ
filters of
BE
algebra.
Theorem 3.12.
Let θ be a congruence on a selfdistributive BEalgebra X. Then the following conditions are equivalent:
(1)
θ is the smallest congruence;
(2)
Every filter is a θfilter;
(3)
Every principal filter is a θfilter.
Proof
. (1) ⇒ (2): Assume that
θ
is the smallest congruence on
X
. Let
F
be a filter of
X
and
x
∈
F
. Let
t
∈ [
x
]
_{θ}
. Then (
t, x
) ∈
θ
. Hence
t
=
x
∈
F
. Thus we get [
x
]
_{θ}
⊆
F
. Therefore by above Theorem 3.11,
F
is a
θ
filter.
(2) ⇒ (3): It is obvious.
(3) ⇒ (1): Assume that every principal filter is a
θ
filter. Let
x, y
∈
X
be such that (
x, y
) ∈
θ
. Then [
x
]
_{θ}
= [
y
]
_{θ}
. Since ⟨
y
⟩ is a
θ
filter, we get
x
∈ [
x
]
_{θ}
= [
y
]
_{θ}
⊆ ⟨
y
⟩. Since
X
is selfdistributive, we get
y
∗
x
= 1. Hence
y
≤
x
. Similarly, we get
x
≤
y
. Hence
x
=
y
. Therefore
θ
is the smallest congruence. □
Finally, this article is concluded by obtaining a bijection between the set of all
θ
filters of a
BE
algebra and the set of all filters of its quotient algebra.
Theorem 3.13.
Let θ be a congruence on a BEalgebra X. Then there exists a bijection between the set Ƒ_{θ}
(
X
)
of all θfilters of X and the set of all filters of the BEalgebra X_{/θ} of all congruence classes.
Proof
. Define a mapping
ψ
:
Ƒ_{θ}
(
X
) ↦
Ƒ
(
X_{/θ}
) by
ψ
(
F
) =
α
(
F
) for all
F
∈
Ƒ_{θ}
(
X
). Let
F,G
∈
Ƒ_{θ}
(
X
). Then
ψ
(
F
) =
ψ
(
G
) ⇒
α
(
F
) =
α
(
G
) ⇒
βα
(
F
) =
βα
(
G
) ⇒
F
=
G
(since
F,G
∈
Ƒ_{θ}
(
X
)). Hence
ψ
is oneone. Again, let
be a filter of
Ƒ
(
X_{/θ}
). Then
β
(
) is a filter in
X
. We now show that
β
(
) is a
θ
filter in
X
. We have always
β
(
) ⊆
βαβ
(
). Let
x
∈
βαβ
(
). Then we get (
x, y
) ∈
θ
for some [
y
]
_{θ}
∈
αβ
(
) =
. Hence
x
∈
β
(
). Therefore
β
(
) =
βαβ
(
). Now for this
β
(
) ∈
X
, we get
ψ
[
β
(
)] =
αβ
(
) =
. Therefore
ψ
is onto. Therefore
ψ
is a bijection between
Ƒ_{θ}
(
X
) and
Ƒ
(
X_{/θ}
). □
Acknowledgements
The author is greatly thankful to the referee for his valuable comments and suggestion for the improvement of the paper.
BIO
M. Sambasiva Rao received his M.Sc. and Ph.D. degrees from Andhra University, Andhra Pradesh, India. Since 2002 he has been at M.V.G.R. College of Engineering, Vizianagaram. His research interests include abstract algebra, implication algebras and Fuzzy Mathematics.
Department of Mathematics, MVGR College of Engineering, Chintalavalasa, Vizianagaram, Andhra Pradesh, India535005.
email: mssraomaths35@rediffmail.com
Ahn S.S.
,
So Y.H.
(2008)
On ideals and upper sets in BEalgebras
Sci. Math. Jpn.
68
279 
285
Iseki K.
,
Tanaka S.
(1979)
An introduction to the theory of BCKalgebras
Math. Japon.
23
1 
26
Kim H.S.
,
Kim Y.H.
(2007)
On BEalgebras
Sci. Math. Jpn.
66
113 
116
Park J.H.
,
Kim Y.H.
(2015)
Intsoft positive implicative filters in BEalgebras
J. Appl. Math. & Informatics
33
459 
467
DOI : 10.14317/jami.2015.459
Walendziak A.
(2012)
On normal filters and congruence relations in Balgebras
Commentationes Mathematicae
52
199 
205