Sanchez [1] introduced the theory of fuzzy relation equations with various types of compositions: max-min, min-max, and min-
In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions
Definition 2.1. [11] A structure (
(A1) (
(A2) (
(A3) it satisfies a residuation, i.e.,
a ⊙ b ≤ c iff a ≤ b → c iff b ≤ a ⇒ c.
(A4)
(A5) (
We denote
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 (
Lemma 2.2. [11] For each
(1) If
(2)
(3) (
(4)
(5)
(6)
(7)
Theorem 3.1. Let a =(
Then, (1) (I) is solvable iff it has the least solution y = (
(2) (II) is solvable iff it has the least solution x =((
(3) If (I) is solvable, then
(4) If (II) is solvable, then
Moreover,
Thus, y = (
(？) It is trivial.
(3) Let x =(
(2) and (4) are similarly proved as (1) and (3), respectively.
Theorem 3.2. Let
(1) If
with
is a maximal solution of (II). Moreover, if x is a solution of (II), there exists
where there exists x_{jk} ∈
(2) If
with
is a maximal solution of (I). Moreover, if x is a solution of (I), there exists
xjk = 0, j = k, xj ≥ b ⊙aj, j ≠ k
where there exists x_{jk} ∈
is a solution of (II) because
Let x ≥ x_{jk} be a solution of (II). Then,
with
Since
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 =(
by the linearity of
, because by linearity of
For
(？) It is trivial.
(2) It is similarly proved as (1).
Example 3.3. Let
(x1,y1) ？ (x2,y2)=(x1x2,x1y2 + y1).
Then, (
We have a positive cone
(
Then, (
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 (
(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).
(3) An equation is defined as
Since
by Theorem 3.1(3), it is not solvable.
(4) An equation is defined as
Since
Definition 3.4. Let
？ ？ a ⊙ x ≤ a ⊙ y ⇒ x ≤ y.
？ ？ x ⊙ a ≤ y ⊙ a ⇒ x ≤ y.
Theorem 3.5. Let
Then, (1) If
with
is a maximal solution family of (II). Moreover, if x is a solution of (II), there exists a family
xk = ak ⊙ b, k ∈ K, xj ≥ aj ⊙ b, j ？ K
where there exists x_{jk} ∈
(2) If
with
is a maximal solution of (I). Moreover, if x is a solution of (I), there exists
xk = b⊙ak, j = k, xj ≥ b⊙aj, j ≠ k
where there exists x_{jk} ∈
is a solution of (II) because
Let x ≥ x_{jk} denote a solution of (II). Then,
with
Since
Since
xjk = ajk ∧ xjk = ajk ⊙ (ajk ⇒ xjk ) = ajk ⊙ b.
Therefore, x = x_{jk}.
is a maximal solution of (II).
Let x =(
by the linearity of
because
For
xk = ak ⊙ b, k ∈ K, xj ≥ aj ⊙ b, j ？ K
(？) It is trivial.
(2) It is similarly proved as (1).
Example 3.6. The structure
is defined as that in Example 3.3. Then,
⊥ ？ (a, b) ⊙ (x1,y1) ≤ (a, b) ⊙ (x2,y2)
(⇔)⊥ ？ (ax1, ay1 + b) ≤ (ax2, ay2 + b)
(⇒)ax1 = ax2, ay1 + b ≤ ay1 + b, or ax1 ？ ax2
(⇒)x1 = x2,y1 ≤ y1, or x1 ？ x2
(⇒)(x1,y1)
≤ (x2,y2).
Similarly,
(1) An equation is defined as
Since
is a maximal solution of (II) because
(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
Then, (1) (III) is solvable iff it has the least solution x = (
(2) (IV) is solvable iff it has the least solution x = (
(3) If (III) is solvable, then
(4) If (IV) is solvable, then
(5) If (III) (resp. (IV)) is solvable and x_{1}, ..., x
is a solution of (III) (resp. (IV)). Moreover, if each solution
Moreover,
Then,
Substitute
Thus, (
(？) It is trivial.
(3)
(2) and (4) are similarly proved as (1) and (3), respectively.
5) Let
Then,
Moreover,
Hence,
Therefore,
is a solution of (III).
Moreover, if
is a solution of (III). Let y = (
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).
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.
No potential conflict of interest relevant to this article was reported.