Bandler and Kohout investigated the solvability of fuzzy relation equations with infimplication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with infimplication compositions in BLalgebras. In this paper, we investigate various solutions of fuzzy relation equations with infimplication compositions in pseudo BLalgebras.
1. Introduction
Sanchez
[1]
introduced the theory of fuzzy relation equations with various types of compositions: maxmin, minmax, and min
α
. Fuzzy relation equations with new types of compositions (continuous tnorm and residuated lattice) have been developed
[2

5]
. In particular, Bandler and Kohout
[6]
investigated the solvability of fuzzy relation equations with infimplication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with infimplication compositions in BLalgebras. In contrast, noncommutative structures play an important role in metric spaces and algebraic structures (groups, rings, quantales, and pseudo BLalgebras)
[7

15]
. Georgescu and Iorgulescu
[12]
introduced pseudo MValgebras as the generalization of MValgebras. Georgescu and Leustean
[11]
introduced generalized residuated lattice as a noncommutative structure.
In this paper, we investigate various solutions of fuzzy relation equations with infimplication compositions
A_{i}
⇒
R
=
B_{i}
and
A_{i}
→
R
=
B_{i}
in pseudo BLalgebras.
2. Preliminaries
Definition 2.1.
[11]
A structure (
L
, ˅, ˄, ⊙, →, ⇒, ⊤, ⊥) is called
apseudo BLalgebra
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 BLchain is a linear pseudo BLalgebra, i.e., a pseudo BLalgebra such that its lattice order is total.
In this paper, we assume that (
L
, ˅, ˄, ⊙, →, ⇒, ⊤, ⊥) is a pseudo BLalgebra.
Lemma 2.2.
[11]
For each
x
,
y
,
z
,
x_{i}
,
y_{i}
∈
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 BLAlgebras
Theorem 3.1.
Let
a
=(
a
_{1}
,
a
_{2}
, ...,
a_{n}
) ∈
L^{n}
and
b
∈
L
. We define two equations with respect to an unknown x = (
x
_{1}
, ..., (
x_{n}
) ∈
L^{n}
as
Then, (1) (I) is solvable iff it has the least solution y = (
y
_{1}
, ...,
y_{n}
) ∈
L^{n}
such that
y_{j}
=
b
⊙
a_{j}
,
j
=1, ...,
n
.
(2) (II) is solvable iff it has the least solution x =((
x
_{1}
, ..., (
x_{n}
) ∈
L^{n}
such that
x_{j}
=
a_{j}
⊙
b
,
j
=1, ...,
n
.
(3) If (I) is solvable, then
(4) If (II) is solvable, then
Proof
. (1) (⇒) Let x =((
x
_{1}
, ..., (
x_{n}
) be a solution of (I). Since
Moreover,
Thus, y = (
b
⊙
a
_{1}
, ...,
b
⊙
a_{n}
) is the least solution.
(⇐) It is trivial.
(3) Let x =(
x
_{1}
, ...,
x_{n}
) 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 BLchain in equations (I) and (II) of Theorem 3.1.
(1) If
b
˂ ⊤ and
with
B
= {
a_{jk}
1 ≤
k
≤
m
,
b
= (
a_{jk}
)*}, then
is a maximal solution of (II). Moreover, if
x
is a solution of (II), there exists
k
∈{
j_{k}
1 ≤
k
≤
m
} such that
x_{jk}
= 0,
j
=
k
,
x_{j}
≥
a_{j}
⊙
b
,
j
≠
k
where there exists x
_{jk}
∈
X
such that x ≤ x
_{jk}
.
(2) If
b
˂⊤ and
with
B
= {
a_{jk}
1 ≤
k
≤
m
,
b
= (
a_{jk}
)
^{0}
}, then
is a maximal solution of (I). Moreover, if
x
is a solution of (I), there exists
k
∈ {
j_{k}
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
x_{jk}
≥
a_{jk}
⊙
b
and
Since
b
˂ 1,
Since
L
is linear,
a_{jk}
>
x_{jk}
. 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}
, ...,
x_{n}
) be a solution of (II). Since
by the linearity of
L
, there exists a family
K
= {
j_{k}

a_{jk}
∈
B
,
a_{jk}
⇒⊥ =
b
, 1 ≤
k
≤
m
} such that
, because by linearity of
L
,
a_{jk}
∉
B
, (
a_{j}
)
^{∗}
>
b
implies that
For
k
∈
K
, since
a_{k}
⇒⊥ =
a_{k}
⇒
x_{k}
=
b
≠ ⊤ and
L
is linear,
a_{k}
>
x_{k}
and
a_{k}
⊙
b
=
a_{k}
⊙ (
a_{k}
⇒
x_{k}
)=
a_{k}
⊙ (
a_{k}
⇒ ⊥) = ⊥
a_{k}
∧
x_{k}
=
x_{k}
. 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 latticegroup with totally order ≤. (ref.
[1]
)
The structure
is a Pseudo BLchain 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 BLchain.
L
satisfies the right conditional cancellation law if
L
satisfies the left conditional cancellation law if
Theorem 3.5.
Let
L
denote a pseudo BLchain in two equations (I) and (II) of Theorem 3.1.
Then, (1) If
L
satisfies the right conditional cancellation law
b
˂ ⊤ and
with
B
= {
a_{jk}
1 ≤
k
≤
m
,
b
˃ (
a_{jk}
)
^{*}
}, then
is a maximal solution family of (II). Moreover, if
x
is a solution of (II), there exists a family
K
= {
j_{k}

a_{jk}
∈
B
,
a_{jk}
⇒
x_{jk}
=
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
= {
a_{jk}
 1 ≤
k
≤
m
,
b
= (
a_{jk}
)
^{0}
}, then
is a maximal solution of (I). Moreover, if x is a solution of (I), there exists
k
∈ {
j_{k}
 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
x_{jk}
≥
a_{jk}
⊙
b
and
Since
b
˂ 1,
Since
L
is linear,
a_{jk}
˃
x_{jk}
. Thus,

xjk=ajk∧xjk=ajk⊙ (ajk⇒xjk) =ajk⊙b.
Therefore, x = x
_{jk}
.
is a maximal solution of (II).
Let x =(
x
_{1}
, ...,
x_{n}
) denote a solution of (II). Since
by the linearity of
L
, there exists a family
K
= {
j_{k}

a_{jk}
∈
B
,
a_{jk}
⇒
x_{jk}
=
b
, 1 ≤
k
≤
m
} such that
because
a_{jk}
∉
B
, (
a_{j}
)
^{0}
≥
b
implies that
For
k
∈
K
, since
a_{k}
⇒
x_{k}
=
b
≠ ⊤ and
L
is linear,
a_{k}
>
x_{k}
and
a_{k}
⊙
b
=
a_{k}
⊙ (
a_{k}
⇒
x_{k}
)=
a_{k}
∧
x_{k}
=
x_{k}
. For
j
∉
K
, since
a_{j}
⇒
x_{j}
≥
b
,
x_{j}
≥
a_{j}
⊙
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
a_{i}
=(
a
_{i1}
,
a
_{i2}
, ...,
a_{in}
) ∈
L^{n}
and
b_{i}
∈
L
. We define two equations with respect to an unknown x = (
x
_{1}
, ...,
x_{n}
) ∈
L^{n}
as
Then, (1) (III) is solvable iff it has the least solution x = (
x
_{1}
, ...,
x_{n}
) ∈
L^{n}
such that
(2) (IV) is solvable iff it has the least solution x = (
x
_{1}
, ...,
x_{n}
) ∈
L^{n}
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
x_{i}
of the ith equation is maximal, any maximal solution x of (III) (resp. (IV)) is
Proof
. (1) (⇒) Let y =(
y
_{1}
, ...,
y_{n}
) denote a solution of (III). Since
Moreover,
Then,
Substitute
Thus, (
x
_{1}
, ...,
x_{n}
) is the least solution.
(⇐) It is trivial.
(3)
(2) and (4) are similarly proved as (1) and (3), respectively.
5) Let
x_{i}
=(
x
_{i1}
, ...,
x_{in}
) denote a solution of the ith equation in (III) and
Then,
Moreover,
Hence,
Therefore,
is a solution of (III).
Moreover, if
x_{i}
is a maximal solution of the ith equation in (III), then
is a solution of (III). Let y = (
y
_{1}
, ...,
y_{n}
) 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 infimplication compositions in complete lattices. Perfilieva and Noskova investigated the solvability of fuzzy relation equations with infimplication compositions in BLalgebras. In this paper, we investigated various solutions of fuzzy relation equations with infimplication compositions in pseudo BLalgebras.
In the future, we will investigate various solutions of fuzzy relation equations with supcompositions in pseudo BLalgebras 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 GangneungWonju National University.
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