Bandler and Kohout investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions in pseudo BL-algebras.
1. Introduction
Sanchez
[1]
introduced the theory of fuzzy relation equations with various types of compositions: max-min, min-max, and min-
α
. Fuzzy relation equations with new types of compositions (continuous t-norm and residuated lattice) have been developed
[2
-
5]
. In particular, Bandler and Kohout
[6]
investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In contrast, noncommutative structures play an important role in metric spaces and algebraic structures (groups, rings, quantales, and pseudo BL-algebras)
[7
-
15]
. Georgescu and Iorgulescu
[12]
introduced pseudo MV-algebras as the generalization of MV-algebras. Georgescu and Leustean
[11]
introduced generalized residuated lattice as a noncommutative structure.
In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions
Ai
⇒
R
=
Bi
and
Ai
→
R
=
Bi
in pseudo BL-algebras.
2. Preliminaries
Definition 2.1.
[11]
A structure (
L
, ˅, ˄, ⊙, →, ⇒, ⊤, ⊥) is called
apseudo BL-algebra
if it satisfies the following conditions:
(A1) (
L
, ˅, ˄, →, ⊤, ⊥) is bounded where ⊤ is the universal upper bound and ⊥ denotes the universal lower bound;
(A2) (
L
, ⊙, ⊤) is a monoid;
(A3) it satisfies a residuation, i.e.,
(A4)
a
˄
b
= (
a
→
b
) ⊙
a
=
a
⊙ (
a
⇒
b
).
(A5) (
a
→
b
) ˅ (
b
→
a
) = ⊤ and (
a
⇒
b
) ˅ (
b
⇒
a
) = ⊤.
We denote
a
0
=
a
→⊥ and
a
∗
=
a
⇒⊥.
A pseudo BL-chain is a linear pseudo BL-algebra, i.e., a pseudo BL-algebra such that its lattice order is total.
In this paper, we assume that (
L
, ˅, ˄, ⊙, →, ⇒, ⊤, ⊥) is a pseudo BL-algebra.
Lemma 2.2.
[11]
For each
x
,
y
,
z
,
xi
,
yi
∈
L
, we have the following properties:
(1) If
y
≤
z
, (
x
⊙
y
) ≤ (
x
⊙
z
),
x
→
y
≤
x
→
z
, and
z
→
x
≤
y
→
x
for →∈ {→, ⇒}.
(2)
x
⊙
y
≤
x
∧
y
≤
x
∨
y
.
(3) (
x
⊙
y
) →
z
=
x
→ (
y
→
z
) and (
x
⊙
y
) ⇒
z
=
y
⇒ (
x
⇒
z
).
(4)
x
→ (
y
⇒
z
)=
y
⇒ (
x
→
z
) and
x
⇒ (
y
→
z
)=
y
→ (
x
⇒
z
).
(5)
x
⊙ (
x
⇒
y
) ≤
y
and (
x
→
y
) ⊙
x
≤
y
.
(6)
x
⊙ (
y
∨
z
)=(
x
⊙
y
) ∨ (
x
⊙
z
) and (
x
∨
y
) ⊙
z
= (
x
⊙
z
) ∨ (
y
⊙
z
).
(7)
x
→
y
= ⊤ iff
x
≤
y
iff
x
⇒
y
= ⊤
3. Fuzzy Relation Equations in Pseudo BL-Algebras
Theorem 3.1.
Let
a
=(
a
1
,
a
2
, ...,
an
) ∈
Ln
and
b
∈
L
. We define two equations with respect to an unknown x = (
x
1
, ..., (
xn
) ∈
Ln
as
Then, (1) (I) is solvable iff it has the least solution y = (
y
1
, ...,
yn
) ∈
Ln
such that
yj
=
b
⊙
aj
,
j
=1, ...,
n
.
(2) (II) is solvable iff it has the least solution x =((
x
1
, ..., (
xn
) ∈
Ln
such that
xj
=
aj
⊙
b
,
j
=1, ...,
n
.
(3) If (I) is solvable, then
(4) If (II) is solvable, then
Proof
. (1) (⇒) Let x =((
x
1
, ..., (
xn
) be a solution of (I). Since
Moreover,
Thus, y = (
b
⊙
a
1
, ...,
b
⊙
an
) is the least solution.
(⇐) It is trivial.
(3) Let x =(
x
1
, ...,
xn
) denote a solution of (I). Then,
b
=
(2) and (4) are similarly proved as (1) and (3), respectively.
Theorem 3.2.
Let
L
denote a pseudo BL-chain in equations (I) and (II) of Theorem 3.1.
(1) If
b
˂ ⊤ and
with
B
= {
ajk
|1 ≤
k
≤
m
,
b
= (
ajk
)*}, then
is a maximal solution of (II). Moreover, if
x
is a solution of (II), there exists
k
∈{
jk
|1 ≤
k
≤
m
} such that
xjk
= 0,
j
=
k
,
xj
≥
aj
⊙
b
,
j
≠
k
where there exists x
jk
∈
X
such that x ≤ x
jk
.
(2) If
b
˂⊤ and
with
B
= {
ajk
|1 ≤
k
≤
m
,
b
= (
ajk
)
0
}, then
is a maximal solution of (I). Moreover, if
x
is a solution of (I), there exists
k
∈ {
jk
|1 ≤
k
≤
m
} such that
where there exists x
jk
∈
X
such that x ≤ x
jk
.
Proof
. (1) (⇒)
is a solution of (II) because
Let x ≥ x
jk
be a solution of (II). Then,
with
xjk
≥
ajk
⊙
b
and
Since
b
˂ 1,
Since
L
is linear,
ajk
>
xjk
. Since
we have
-
xjk=ajk∧xjk=ajk⊙ (ajk⇒xjk) =ajk⊙b=ajk⊙ (ajk⇒⊥)= ⊥.
Thus, x = x
jk
.
is a maximal solution of (II).
Let x =(
x
1
, ...,
xn
) be a solution of (II). Since
by the linearity of
L
, there exists a family
K
= {
jk
|
ajk
∈
B
,
ajk
⇒⊥ =
b
, 1 ≤
k
≤
m
} such that
, because by linearity of
L
,
ajk
∉
B
, (
aj
)
∗
>
b
implies that
For
k
∈
K
, since
ak
⇒⊥ =
ak
⇒
xk
=
b
≠ ⊤ and
L
is linear,
ak
>
xk
and
ak
⊙
b
=
ak
⊙ (
ak
⇒
xk
)=
ak
⊙ (
ak
⇒ ⊥) = ⊥
ak
∧
xk
=
xk
. Then,
(⇐) It is trivial.
(2) It is similarly proved as (1).
Example 3.3.
Let
K
= {(
x
,
y
) ∈
R
2
|
x
> 0} denote a set, and we define an operation ⊗ :
K
×
K
→
K
as follows:
-
(x1,y1) ⊗ (x2,y2)=(x1x2,x1y2+y1).
Then, (
K
, ⊗) is a group with
e
= (1, 0),
We have a positive cone
P
= {(
a
,
b
) ∈
R
2
|
a
=1,
b
≥ 0, or
a
> 1} because
P
∩
P
−1
= {(1, 0)},
P
⊙
P
⊂
P
, (
a
,
b
)
−1
⊙
P
⊙ (
a
,
b
)=
P
, and
P
∪
P
−1
=
K
. For (
x
1
,
y
1
), (
x
2
,
y
2
) ∈
K
, we define
(
x
1
,
y
1
) ≤ (
x
2
,
y
2
) ⇔ (
x
1
,
y
1
)
−1
⊙ (
x
2
,
y
2
) ∈
P
, (
x
2
,
y
2
) ⊙ (
x
1
,
y
1
)
−1
∈
P
⇔
x
1
˂
x
2
or
x
1
=
x
2
,
y
1
≤
y
2
.
Then, (
K
, ≤⊗) is a lattice-group with totally order ≤. (ref.
[1]
)
The structure
is a Pseudo BL-chain where
is the least element and ⊤ = (1, 0) is the greatest element from the following statements:
Furthermore, we have (
x
,
y
)=(
x
,
y
)
∗◦
=(
x
,
y
)
◦∗
from:
(1) An equation is defined as
Since
by Theorem 3.1(3), it is not solvable.
(2) An equation is defined as
Since
-
X= {x = ((x1,y1), (x2,y2), ⊥) or x = ((x1,y1), ⊥, (x3,y3)) | (x1,y1), (x2,y2), (x3,y3) ≥ ⊥}
is a solution set of (I).
M
= {(⊤, ⊤, ⊥), (⊤, ⊥, ⊤)} is a maximal solution family of (I).
(3) An equation is defined as
Since
by Theorem 3.1(3), it is not solvable.
(4) An equation is defined as
Since
X
= {x = ((
x
1
,
y
1
), (
x
2
,
y
2
), ⊥) | (
x
1
,
y
1
), (
x
2
,
y
2
) ≥ ⊥} is a solution family of (II). (⊤, ⊤, ⊥) is a maximal solution of (II).
Definition 3.4.
Let
L
denote a pseudo BL-chain.
L
satisfies the right conditional cancellation law if
L
satisfies the left conditional cancellation law if
Theorem 3.5.
Let
L
denote a pseudo BL-chain in two equations (I) and (II) of Theorem 3.1.
Then, (1) If
L
satisfies the right conditional cancellation law
b
˂ ⊤ and
with
B
= {
ajk
|1 ≤
k
≤
m
,
b
˃ (
ajk
)
*
}, then
is a maximal solution family of (II). Moreover, if
x
is a solution of (II), there exists a family
K
= {
jk
|
ajk
∈
B
,
ajk
⇒
xjk
=
b
, 1 ≤
k
≤
m
} such that
where there exists x
jk
∈
X
such that x ≤ x
jk
.
(2) If
L
satisfies the left conditional cancellation law
b
˂ ⊤ and
with
B
= {
ajk
| 1 ≤
k
≤
m
,
b
= (
ajk
)
0
}, then
is a maximal solution of (I). Moreover, if x is a solution of (I), there exists
k
∈ {
jk
| 1 ≤
k
≤
m
} such that
where there exists x
jk
∈
X
such that x ≤ x
jk
.
Proof
. (1)
is a solution of (II) because
Let x ≥ x
jk
denote a solution of (II). Then,
with
xjk
≥
ajk
⊙
b
and
Since
b
˂ 1,
Since
L
is linear,
ajk
˃
xjk
. Thus,
-
xjk=ajk∧xjk=ajk⊙ (ajk⇒xjk) =ajk⊙b.
Therefore, x = x
jk
.
is a maximal solution of (II).
Let x =(
x
1
, ...,
xn
) denote a solution of (II). Since
by the linearity of
L
, there exists a family
K
= {
jk
|
ajk
∈
B
,
ajk
⇒
xjk
=
b
, 1 ≤
k
≤
m
} such that
because
ajk
∉
B
, (
aj
)
0
≥
b
implies that
For
k
∈
K
, since
ak
⇒
xk
=
b
≠ ⊤ and
L
is linear,
ak
>
xk
and
ak
⊙
b
=
ak
⊙ (
ak
⇒
xk
)=
ak
∧
xk
=
xk
. For
j
∉
K
, since
aj
⇒
xj
≥
b
,
xj
≥
aj
⊙
b
. Hence,
(⇐) It is trivial.
(2) It is similarly proved as (1).
Example 3.6.
The structure
is defined as that in Example 3.3. Then,
L
satisfies the right conditional cancellation law because
-
⊥ ˂ (a,b) ⊙ (x1,y1) ≤ (a,b) ⊙ (x2,y2)
-
(⇔)⊥ ˂ (ax1,ay1+b) ≤ (ax2,ay2+b)
-
(⇒)ax1=ax2,ay1+b≤ay1+b, orax1˂ax2
-
(⇒)x1=x2,y1≤y1, orx1˂x2
-
(⇒)(x1,y1)
-
≤ (x2,y2).
Similarly,
L
satisfies the left conditional cancellation law.
(1) An equation is defined as
Since
is a maximal solution of (II) because
X
= {x = ((
x
1
,
y
1
), (
x
2
,
y
2
), ⊥ | (
x
1
,
y
1
), (
x
2
,
y
2
) ≥ ⊥} is a solution set of (II).
(2) An equation is defined as
Since
and
and
are maximal solutions of (II) because
is a solution set of (II).
(3) An equation is defined as
Since
-
X= {x = ((x1,y1), (x2,y2), ⊥) or x = (x1,y1), ⊥, (x3,y3)) | (x1,y1), (x2,y2), (x3,y3) ≥ ⊥}
is a solution set of (I).
Theorem 3.7.
Let
ai
=(
a
i1
,
a
i2
, ...,
ain
) ∈
Ln
and
bi
∈
L
. We define two equations with respect to an unknown x = (
x
1
, ...,
xn
) ∈
Ln
as
Then, (1) (III) is solvable iff it has the least solution x = (
x
1
, ...,
xn
) ∈
Ln
such that
(2) (IV) is solvable iff it has the least solution x = (
x
1
, ...,
xn
) ∈
Ln
such that
(3) If (III) is solvable, then
(4) If (IV) is solvable, then
(5) If (III) (resp. (IV)) is solvable and x
1
, ..., x
m
is a solution of each ith equation,
i
=1, 2, ...,
m
, then
is a solution of (III) (resp. (IV)). Moreover, if each solution
xi
of the ith equation is maximal, any maximal solution x of (III) (resp. (IV)) is
Proof
. (1) (⇒) Let y =(
y
1
, ...,
yn
) denote a solution of (III). Since
Moreover,
Then,
Substitute
Thus, (
x
1
, ...,
xn
) is the least solution.
(⇐) It is trivial.
(3)
(2) and (4) are similarly proved as (1) and (3), respectively.
5) Let
xi
=(
x
i1
, ...,
xin
) denote a solution of the ith equation in (III) and
Then,
Moreover,
Hence,
Therefore,
is a solution of (III).
Moreover, if
xi
is a maximal solution of the ith equation in (III), then
is a solution of (III). Let y = (
y
1
, ...,
yn
) denote a solution of (III). Then, y ≤ x
i
for each
i
= 1, ...
m
. Then,
Hence, x is a maximal solution of (III).
Example 3.8.
The structure
is defined as that in Example 3.3.
(1) An equation is defined as
is a solution set.
is a maximal solution set.
(2) An equation is defined as
is a solution set.
is a maximal solution set.
is a solution set of (1) and (2).
is a maximal solution set of (1) and (2).
(3) An equation is defined as
is a solution set.
is a maximal solution set.
(4) An equation is defined as
is a solution set.
is a maximal solution set.
is a solution set of (3) and (4).
is a maximal solution set of (3) and (4).
4. Conclusion
Bandler and Kohout
[6]
investigated the solvability of fuzzy relation equations with inf-implication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with inf-implication compositions in BL-algebras. In this paper, we investigated various solutions of fuzzy relation equations with inf-implication compositions in pseudo BL-algebras.
In the future, we will investigate various solutions of fuzzy relation equations with sup-compositions in pseudo BL-algebras and other algebraic structures.
- Conflict of Interest
No potential conflict of interest relevant to this article was reported.
Acknowledgements
This work was supported by the Research Institute of NaturalScience of Gangneung-Wonju National University.
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