The notion of quasilattice implication algebras is a generalization of lattice implication algebras. In this paper, we give an optimized definition of quasilattice implication algebra and show that this algebra is a distributive lattice and that this algebra is a lattice implication algebra. Also, we define a congruence relation Φ
_{F}
induced by a filter
F
and show that every congruence relation on a quasilattice implication algebra is a congruence relation Φ
_{F}
induced by a filter
F
.
AMS Mathematics Subject Classification : 03G10, 06B10.
1. Introduction
The notion of lattice implication algebras was introduced in
[8]
to research a latticevalued logic, which is a logical system equipped with a logical implication and an involution unary operation on a lattice. This logical system was studied from the algebraic viewpoint in many literature
[5
,
7
,
8
,
9]
, and some operators on this algebra was studied in
[4
,
11]
A
lattice implication algebra
is a bounded lattice (
L
, ∧, ∨, 0, 1) with a binary operation “ → ” and an orderreversing involution “ ′ ” satisfying the following axioms: for all
x, y, z
∈
L
,

(I1)x→ (y→z) =y→ (x→z),

(I2)x→x= 1,

(I3) (x→y) →y= (y→x) →x,

(I4)x→y=y→x= 1 ⇒x=y,

(I5)x→y=y'→x',

(L1) (x∨y) →z= (x→z) ∧ (y→z),

(L2) (x∧y) →z= (x→z) ∨ (y→z).
The notion of filters on algebras with implication was introduced and studied in
[3
,
6]
. This filter is known as deductive filter and different from the notion of filters on lattices. This filter was proposed and studied as the notion of filters on lattice implication algebras in
[1
,
2
,
10
,
12]
. On a lattice implication algebra, the filter of lattice is the generalized concept of the filter.
A quasilattice implication algebra was introduced in
[5
,
7]
as an algebraic system (
L
,∧,∨,→, ′, 0, 1) satisfying the axioms (I1)(I5). This algebra is a generalization of lattice implication algebras and has the binary operation → and the involution ′ in the axioms for definition.
In this paper, the quasilattice implication algebra will be more clearly defined, and we show that this algebra is a distributive lattice, and hence this algebra is a lattice implication algebra. Also, an alternative definition of quasilattice implication algebra will be given. In section 3, we define a congruence relation Φ
_{F}
induced by a filter
F
and show that every congruence relation on a quasilattice implication algebra is a congruence relation Φ
_{F}
induced by a filter
F
.
2. Quasilattice implication algebra
We will define the notion of quasilattice implication algebras by the following optimized type, and
x
→
y
will be denoted by
xy
.
Definition 2.1.
A
quasilattice implication algebra
is an algebraic system (
L
, ·, ′, 1) with a binary operation “ · ”, an involution “ ′ ” and an element 1 satisfying the following axioms: for all
x, y, z
∈
L
,

(Q1)x(yz) =y(xz),

(Q2)xx= 1,

(Q3) (xy)y= (yx)x,

(Q4)xy= 1 andyx= 1 implyx=y,

(Q5)xy=y′x′.
In the definition of quasilattice implication algebra
L
, the involution ′ is an unary operation on
L
such that
x′′
=
x
for every
x
∈
L
.
Lemma 2.2.
Let L be a quasilattice implication algebra. Then L satisfies the following: for all x
∈
L
,
Proof
. (1) Let
x
∈
L
. Then
x
(1
x
) = 1(
xx
) = 11 = 1 by (Q1) and (Q2). Also, we have (1
x
)
x
= (
x
1)1 = (
x
1)(11) = 1((
x
1)1) = 1((1
x
)
x
) = (1
x
)(1
x
) = 1 by (Q3), (Q2) and (Q1). Hence
x
(1
x
) = (1
x
)
x
= 1. This implies
x
= 1
x
by (Q4).
(2) Let
x
∈
L
. Then
x
1 = (1
x
)1 = (1
x
)(
xx
) =
x
((1
x
)
x
) =
x
(
x
1)1) = (
x
1)(
x
1) = 1 by (1) of this lemma, (Q2), (Q1) and (Q3). □
Lemma 2.3.
Let L be a quasilattice implication algebra. If we define a binary relation
" ≤ "
by
for any x, y
∈
L, then
(
L
,≤)
is a poset with the greatest element
1
and the smallest element
1′.
Proof
. For every
x
∈
L
,
x
≤
x
by (Q2), and for any
x, y
∈
L, x
≤
y
and
y
≤
x
imply
x
=
y
by (Q4).
To show the transitivity, let
x
≤
y
and
y
≤
z
. Then
xy
= 1 and
yz
= 1, and we have
xz
=
x
(1
z
) =
x
((
yz
)
z
) =
x
((
zy
)
y
) = (
zy
)(
xy
) = (
zy
)1 = 1 by 2.2, (Q3) and (Q1). This implies
x
≤
z
. Hence (
L
,≤) is a poset.
Also, 1 is the greatest element in
L
by 2.2(2). Let
x
∈
L
. Then 1 =
x′
1 = 1′
x′′
= 1′
x
by 2.2(2), (Q5) and the definition of involution ′. This implies 1′ ≤
x
for every
x
∈
L
. Hence 1′ is the smallest element in
L
. □
We will denote the smallest element 1′ in
L
by 0.
Lemma 2.4.
Let L be a quasilattice implication algebra. Then L satisfies the following: for every x, y, z
∈
L
,

(1) 1′ = 0 and 0′ = 1,

(2)x′=x0,

(3)x≤yz implies y≤xz,

(4) (xy) ≤ (yz)(xz),

(5) (xy) ≤ (zx)(zy),

(6)x≤y implies yz≤xz and zx≤zy,

(7)x≤ (xy)y,

(8)y≤xy,

(9)x≤y implies y′≤x′,

(10) ((xy)y)y=xy.
Proof
. (1) It is clear that 1′ = 0 by 2.3, and 0′ = 1′′ = 1.
(2) For every
x
∈
L, x′
= 1
x′
=
x′′
1′ =
x
0 by 2.2(1), (Q5) and (1) of this lemma.
(3) Let
x
≤
yz
. Then
y
(
xz
) =
x
(
yz
) = 1. Hence
y
≤
xz
.
(4) Let
x, y
∈
L
. Then we have
Hence
xy
≤ (
yz
)(
xz
).
(5) Let
x, y, z
∈
L
. Then
zx
≤ (
xy
)(
zy
) by (4) of this lemma. Hence
xy
≤ (
zx
)(
zy
) by (3) of this lemma.
(6) Let
x
≤
y
. Then 1 =
xy
≤ (
yz
)(
xz
) by (4) of this lemma. This implies (
yz
)(
xz
) = 1. Hence
yz
≤
xz
. Also, 1 =
xy
≤ (
zx
)(
zy
) by (5) of this lemma. This implies (
zx
)(
zy
) = 1, and hence
zx
≤
zy
.
(7) Let
x, y
∈
L
. Then
x
((
xy
)
y
) = (
xy
)(
xy
) = 1. Hence
x
≤ (
xy
)
y
.
(8) Let
x, y
∈
L
. Then
y
(
xy
) =
x
(
yy
) =
x
1 = 1. Hence
y
≤
xy
.
(9) Let
x
≤
y
, Then 1 =
xy
=
y′x′
. Hence
y′
≤ x′.
(10) Let
x, y
∈
L
. Then
xy
≤ ((
xy
)
y
)
y
by (7) of this lemma. Also, since
x
≤ (
xy
)
y
, ((
xy
)
y
)
y
≤
xy
by (6) of this lemma. Thus
xy
= ((
xy
)
y
)
y
. □
Theorem 2.5.
A quasilattice implication algebra L is a lattice with
for every x, y
∈
L
.
Proof
. Let
x, y
∈
L
. Then (
xy
)
y
is an upper bound of
x
and
y
by (7) and (8) of 2.4. Suppose that
u
is an upper bound of
x
and
y
. Then
x
≤
u
and
y
≤
u
, and
yu
= 1. This implies
uy
≤
xy
, and
by 2.4(6) and (Q3). Hence (
xy
)
y
is the least upper bound of
x
and
y
, and
x
∨
y
= (
xy
)
y
.
Also, since
x′
≤
x′
∨
y′
and
y′
≤
x′
∨
y′
, (
x′
∨
y′
)′ ≤
x′′
=
x
and (
x′
∨
y′
)′ ≤
y′′
=
y
by 2.4(9). Hence (
x′
∨
y′
)′ is a lower bound of
x
and
y
. Suppose that
l
≤
x
and
l
≤
y
. Then
x′
≤
l′
and
y′
≤
l′
. This implies
x′
∨
y′
≤
l′
. Hence
l
=
l′′
≤ (
x′
∨
y′
)′. This means (
x′
∨
y′
)′ is the greatest lower bound of
x
and
y
, and
x
∧
y
= (
x′
∨
y′
)′. □
Lemma 2.6.
Let L be a quasilattice implication algebra. Then L satisfies the following: for every x, y, z
∈
L
,

(1) (x∨y)′ =x′∧y′,

(2) (x∧y)′ =x′∨y′,

(3) (x∨y)z= (xz) ∧ (yz),

(4)z(x∧y) = (zx) ∧ (zy).
Proof
. (1) Let
x, y
∈
L
. Then
x′
∧
y′
= (
x′′
∨
y′′
)′ = (
x
∨
y
)′ by 2.5.
(2) Let
x, y
∈
L
. Then (
x
∧
y
)′ = (
x′
∨
y′
)′′ =
x′
∨
y′
by 2.5.
(3) Let
x, y, z
∈
L
. Then
x
≤
x
∨
y
and
y
≤
x
∨
y
. This implies (
x
∨
y
)
z
≤
xz
and (
x
∨
y
)
z
≤
yz
by 2.4(6). Hence (
x
∨
y
)
z
≤ (
xz
) ∧ (
yz
). Also, since (
xz
) ∧ (
yz
) ≤
xz
and (
xz
) ∧ (
yz
) ≤
yz
, we have
by 2.4(3). This implies
x
∨
y
≤ ((
xz
) ∧ (
yz
))
z
, and (
xz
) ∧ (
yz
) ≤ (
x
∨
y
)
z
by 2.4(3). Hence (
x
∨
y
)
z
= (
xz
) ∧ (
yz
).
(4) Let
x, y, z
∈
L
. Then we have
z
(
x
∧
y
) = (
x
∧
y
)′
z′
= (
x′
∨
y′
)
z′
= (
x′z′
) ∧ (
y′z′
) = (
zx
) ∧ (
zy
) by (Q5) and (3) of this lemma. □
Theorem 2.7.
Let L be a quasilattice implication algebra. Then L is distributive.
Proof
. Let
x, y, z
∈
L
. Then it is clear that (
x
∧
y
) ∨ (
x
∧
z
) ≤
x
∧ (
y
∨
z
) since
x
∧
y
≤
x
∧ (
y
∨
z
) and
x
∧
z
≤
x
∧ (
y
∨
z
). Conversely, we have
since (
x′y′
) ∧ (
x′z′
) ≤
x′y′
and (
x′y′
) ∧ (
x′z′
) ≤
x′z′
. This implies
Hence
x
∧ (
y
∨
z
) = (
x
∧
y
) ∨ (
x
∧
z
). □
In a lattice
L
, it is well known that the following are equivalent:

(1)x∧ (y∨z) = (x∧y) ∨ (x∧z) for everyx, y, z∈L,

(2)x∨ (y∧z) = (x∨y) ∧ (x∨z) for everyx, y, z∈L.
This mean that a lattice
L
is a distributive if and only if
x
∨(
y
∧
z
) = (
x
∨
y
)∧(
x
∨
z
) for every
x, y, z
∈
L
.
Theorem 2.8.
Every quasilattice implication algebra is a lattice implication algebra.
Proof
. Let (
L
,→, ′, 1) be a quasilattice implication algebra. Then it satisfies the axioms (I1)(I5) from the definition. Also
L
is a lattice and satisfies the axiom (L1) by 2.6(3). So we need only to show that it satisfies the axiom (L2).
Let
x, y, z
∈
L
. Then we have
This implies
since
z
≤ (
xz
) ∨ (
yz
). Thus
L
satisfies the axiom (L2) and so it is a lattice implication algebra. □
It is clear that a lattice implication algebra is a quasilattice implication algebra. From the above theorem, the notion of quasilattice implication algebras is equivalent to that of lattice implication algebras.
Theorem 2.9.
A set L is a quasilattice implication algebra if and only if there are a binary operation · on L and two elements
0, 1
in L satisfying the following: for every x, y, z
∈
L
,

(Q1)x(yz) =y(xz),

(Q2)xx= 1,

(Q3) (xy)y= (yx)x,

(Q4)xy= 1and yx= 1imply x=y,

(B) 0x= 1.
Proof
. Let
L
be a quasilattice implication algebra. Then it satisfies the properties (Q1)(Q4) and (B) by the definition of quasilattice implication algebra and 2.3.
Conversely, suppose that
L
be a set with a binary operation · and two elements 0, 1 satisfying the properties (Q1)(Q4) and (B). Then we need to show that there is an involution ′ on
L
satisfying (Q5) :
xy
=
y′x′
for every
x, y
∈
L
.
It can be proved that 1
x
=
x
for every
x
∈
L
in the same way as the proof of 2.2(1). Let
x′
=
x
0. Then we have
x′′
= (
x
0)0 = (0
x
)
x
= 1
x
=
x
by (Q3) and (B). So ′ is an involution, and it satisfies the following.
by (Q1). Hence (
L
, ·,′ , 1) is a quasilattice implication algebra. □
3. Congruence relations on quasilattice implication algebras
A subset
F
of a quasilattice implication algebra
L
is called a
filter
of
L
if it satisfies the following: for any
x, y
∈
L
,

(F1) 1 ∈F,

(F2)x∈Fandxy∈Fimplyy∈F.
Lemma 3.1.
If F is a filter of a quasilattice implication algebra L, then F is a lattice filter of lattice L, i.e., F satisfies the following:

(1)x∈F and x≤y imply y∈F,

(2)x, y∈F implies x∧y∈F.
Proof
. (1) Let
x
∈
F
and
x
≤
y
. Then
xy
= 1 ∈
F
. Hence
y
∈
F
since
x
∈
F
.
(2) Let
x, y
∈
F
. Then
y
≤
xy
. This implies
xy
∈
F
by (1) of this lemma, and
x
(
x
∧
y
) = (
xx
) ∧ (
xy
) = 1 ∧ (
xy
) =
xy
∈
F
by 2.6(4). Since
x
(
x
∧
y
) ∈
F
and
x
∈
F
,
x
∧
y
∈
F
. □
The converse of Lemma 3.1 is not true in general, as the following example shows.
Example 3.2.
Let
Q
= {0,
a, b, c, d
, 1} be a set with a binary operation · defined by the following Cayley table:
If we define
x′
=
x
0 for every
x
∈
Q
, then (
Q
, ·, ′, 1) is a quasilattice implication algebra. Also, it is a lattice with
x
∨
y
= (
xy
)
y
and
x
∧
y
= (
x′
∨
y′
)′ = ((
x′y′
)
y′
)′. This lattice is depicted by Hasse diagram of
Figure 1
. Let
F
= {
a, c, d
, 1}. Then
Hasse diagram of a lattice Q
F
is a lattice filter of lattice
Q
, but it is not filter of
Q
, because
a
∈
F
and
ab
=
d
∈
F
but
b
≠
F
.
For any filter
F
of a quasilattice implication algebra
L
, we can define a binary relation Φ
_{F}
on
L
by
for any
x, y
∈
L
.
Lemma 3.3.
Let F be a filter of L. Then
Φ
_{F} is a congruence relation.
Proof
. For any
x
∈
L
, it is clear that
x
Φ
_{F}x
, since
xx
= 1 ∈
F
, and that
x
Φ
_{F}y
implies
y
Φ
_{F}x
.
To show the transitivity of Φ
_{F}
, let
x
Φ
_{F}y
and
y
Φ
_{F}z
. Then
xy, yx
∈
F
and
yz, zy
∈
F
. Since
xy
≤ (
yz
)(
xz
) by 2.4(4), (
yz
) (
xz
) ∈
F
by 3.1(1). This implies
xz
∈
F
since
yz
∈
F
. Also, we can see
zx
∈
F
in the similar way. Hence
x
Φ
_{F}z
. Thus Φ
_{F}
is an equivalence relation in
L
.
To show that Φ
_{F}
is a congruence relation on
L
, let
x
Φ
_{F}y
and
z
∈
L
. Then
xy, yx
∈
F
. Since
xy
∈
F
and
xy
≤ (
zx
)(
zy
) by 2.4(5), (
zx
)(
zy
) ∈
F
. Also, since
yx
∈
F
and
yx
≤ (
zy
)(
zx
), (
zy
)(
zx
) ∈
F
. Thus (
zx
)Φ
_{F}
(
zy
), and Φ
_{F}
is left compatible. In the similar way, we can show (
xz
)(
yz
), (
yz
)(
xz
) ∈
F
by 2.4(4). That is (
xz
)Φ
_{F}
(
yz
), and Φ
_{F}
is right compatible. Hence Φ
_{F}
is a congruence relation on
L
. □
We will call this relation Φ
_{F}
as a
congruence relation induced by a filter F
.
For any equivalence relation Θ on a quasilattice implication algebra
L
, we will write [
x
]
_{Θ}
for the equivalence classes.
Lemma 3.4.
Let L be a quasilattice implication algebra. If
Θ
is a congruence relation on L, then
[1]
_{Θ}
is a filter of L
.
Proof
. Let Θ be a congruence relation on
L
. Then it is trivial that 1 ∈ [1]
_{Θ}
. If
x
∈ [1]
_{Θ}
and
xy
∈ [1]
_{Θ}
, then
x
Θ1 and
xy
Θ1 imply
xy
Θ1
y
and
xy
Θ1, hence
y
Θ1 since 1
y
=
y
. This implies
y
∈ [1]
_{Θ}
. Thus [1]
_{Θ}
is a filter of
L
. □
Theorem 3.5.
Every congruence relation on a quasilattice implication algebra L is a congruence relation induced by a filter.
Proof
. Suppose that Θ is a congruence relation on
L
. Then [1]
_{Θ}
is a filter by 3.4. Set
F
= [1]
_{Θ}
, and we will show that Θ = Φ
_{F}
.
Let
x
Θ
y
. Then
xy
Θ
yy
and
yx
Θ
yy
, since Θ is a congruence relation. This implies
xy
Θ1 and
yx
Θ1, and
xy, yx
∈ [1]
_{Θ}
=
F
. Thus
x
Φ
_{F}y
. Also, let
x
Φ
_{F}y
. Then
xy
,
yx
∈
F
= [1]
_{Θ}
. This implies
xy
Θ1 and
yx
Θ1, and (
xy
)
y
Θ1
y
and (
yx
)
x
Θ1
x
, i.e., (
xy
)
y
Θ
y
and (
yx
)
x
Θ
x
. Hence
x
Θ
y
since (
xy
)
y
= (
yx
)
x
.
This mean Θ = Φ
_{F}
, and Θ is a congruence relation induced by the filter
F
= [1]
_{Θ}
. □
Let
L
be a quasilattice implication algebra. Then the family
Fil
(
L
) (resp.
Con
(
L
)) of all filters of
L
(resp. all congruence relations on
L
) is partially ordered by set inclusion, and it is a complete lattice with
for arbitrary subset {
F_{α}

α
∈ Λ} of
Fil
(
L
) (resp. {Θ
_{α}

α
∈ Λ} of
Con
(
L
)), where ⟨
X
⟩ is the filter (resp. the congruence relation ) generated by a subset
X
of
L
(resp. of
L
×
L
).
Lemma 3.6.
Let L be a quasilattice implication algebra. Then it satisfies the following:

(1) [1]ΦF=F for every F∈Fil(L),

(2) Φ[1]Θ= Θfor everyΘ ∈Con(L),

(3)F⊆G if and only ifΦF⊆ ΦGfor any F,G∈FIl(L),

(4) Θ ⊆ Ψif and only if[1]Θ⊆ [1]Ψfor anyΘ,Ψ ∈Con(L).
Proof
. (1) Let
F
∈
Fil
(
L
). Then we have
Hence [1]
_{ΦF}
=
F
.
(2) It was proved in the proof of 3.5.
(3) Let
F
⊆
G
in
Fil
(
L
) and
x
Φ
_{F}y
. Then
xy, yx
∈
F
, and
xy, yx
∈
G
since
F
⊆
G
. This implies
x
Φ
_{G}y
. Hence Φ
_{F}
⊆ Φ
_{G}
. Conversely, let Φ
_{F}
⊆ Φ
_{G}
and
x
∈
F
. Then
x
1 = 1 ∈
F
, since
F
is a filter, and 1
x
=
x
∈
F
. This implies
x
Φ
_{F}
1, and
x
Φ
_{G}
1 since Φ
_{F}
⊆ Φ
_{G}
. This implies 1
x
=
x
∈
G
. Hence
F
⊆
G
.
(4) Let Θ,Ψ ∈
Con
(
L
). Then since Θ = Φ
_{[1]Θ}
and Ψ = Φ
_{[1]Ψ}
by (2) of this lemma, we have
by (3) of this lemma. □
Theorem 3.7.
Let L be a quasilattice implication algebra and ϕ
:
Fil
(
L
) →
Con
(
L
)
a map defined by ϕ
(
F
) = Φ
_{F} for every F
∈
Fil
(
L
).
Then it satisfies the following:

(1)ϕ is orderisomorphism,

(2)ϕ preserves arbitrary join and arbitrary meet.
Proof
. (1) Let
ϕ
:
Fil
(
L
) →
Con
(
L
) be a map defined by
ϕ
(
F
) = Φ
_{F}
for every
F
∈
Fil
(
L
). Then
F
⊆
G
if and only if ϕ(
F
) = Φ
_{F}
⊆ Φ
_{G}
=
ϕ
(
G
) by 3.6(3). Hence
ϕ
is orderembedding. Also let Θ ∈
Con
(
L
). Then there exists
F
= [1]
_{Θ}
∈
Fil
(
L
) such that
ϕ
(
F
) = Φ
_{[1]Θ}
= Θ by 3.6(2). Hence
ϕ
is onto.
(2) Let {
F_{α}

α
∈ Λ} be an arbitrary subset of
Fil
(
L
). Then
ϕ
(
F_{β}
) ⊆
ϕ
(∨
_{α}
_{∈Λ}
F_{α}
) for every
β
∈ Λ, since
ϕ
is orderpreserving by (1) of this theorem. Hence
ϕ
(∨
_{α}
_{∈Λ}
F_{α}
) is an upper bound of the set {
ϕ
(
F_{α}
) 
α
∈ Λ}
Suppose that
ϕ
(
F_{β}
) = Φ
_{Fβ}
⊆ Θ for every
β
∈ Λ. Then by (1) and (4) of 3.6,
F_{β}
= [1]
_{ΦFβ}
⊆ [1]
_{Θ}
for every
β
∈ Λ. This implies ∨
_{α}
_{∈}
_{Λ}
F_{α}
⊆ [1]
_{Θ}
, and hence
Thus
ϕ
(∨
_{α}
_{∈Λ}
F_{α}
) is the least upper bound of the set {
ϕ
(
F_{α}
) 
α
∈ Λ}. Hence
ϕ
(∨
_{α}
_{∈Λ}
F_{α}
) = ∨
_{α}
_{∈Λ}
ϕ
(
F_{α}
). Also, we can show
ϕ
(∧
_{α}
_{∈Λ}
F_{α}
) = ∧
_{α}
_{∈Λ}
ϕ
(
F_{α}
) in the similar way. □
From the above theorem,
Fil
(
L
) and
Con
(
L
) of a quasilattice implication algebra
L
have the same structure as complete lattices.
BIO
Yong Ho Yon received M.Sc. from Chungbuk National University, and Ph.D. from Chungbuk National University. He is currently a professor at Mokwon University since 2011. His research interests are lattice theory and algebras with implication.
Division of Information and Communication Convergence Engineering, Mokwon University, Daejeon 302729, Korea.
email: yhyon@mokwon.ac.kr
Borzooei R.A.
,
Hosseiny S.F.
(2013)
Finite lattice implication algebras
Jordan J. Math. Stat.
6
265 
283
Kang S.E.
,
Ahn S.S.
(2012)
Rough filters of BEalgebras
J. Appl. Math. & Informatics
30
1023 
1030
Lee S.D.
,
Kim K.H.
(2013)
On derivations of lattice implication algebras
Ars Combin.
108
279 
288
Liu J.
,
Xu Y.
(1999)
Lattice implication algebras and MValgebras
Chinese Quart. J. Math.
14
17 
23
Xu Y.
(1993)
Lattice implication algebras
J. Southwest Jiaotong Univ.
1
20 
27
Xu Y.
,
Qin K.Y.
(1992)
Lattice H implication algebras and lattice implication algebra classes
J. Hebei Mining and Civil Engineering Institute
3
139 
143
Xu Y.
,
Qin K.Y.
(1993)
On filters of lattice implication algebras
J. Fuzzy Math.
1
251 
260
Yon Y.H.
,
Kim K.H.
(2013)
On fderivations of lattice implication algebras
Ars Combin.
110
205 
215
Zhu H.
,
Zhao J.B.
,
Xu Y.
(2008)
The nfold prime filter of residuated lattice implication algebra
J. Zhengzhou Univ. Nat. Sci. Ed.
40
19 
22