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 A_i ⇒ R = B_i and A_i → R = B_i in pseudo BL-algebras.

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.,

a ⊙ b ≤ c iff a ≤ b → c iff b ≤ a ⇒ c.

(A4) a ？ b = (a → b) ⊙ a = a ⊙ (a ⇒ b).

(A5) (a → b) ？ (b → a) = ？ and (a ⇒ b) ？ (b ⇒ a) = ？.

We denote a⁰ = 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, 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 = ？

Theorem 3.1. Let a =(a₁,a₂, ..., a_n) ∈ Lⁿ and b ∈ L. We define two equations with respect to an unknown x = (x₁, ..., (x_n) ∈ Lⁿ as

Then, (1) (I) is solvable iff it has the least solution y = (y₁, ..., y_n) ∈ Lⁿ such that y_j = b ⊙ a_j, j =1, ..., n.

(2) (II) is solvable iff it has the least solution x =((x₁, ..., (x_n) ∈ Lⁿ 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 =((x₁, ..., (x_n) be a solution of (I). Since

Moreover,

Thus, y = (b ⊙ a₁, ..., b ⊙ a_n) is the least solution.

(？) It is trivial.

(3) Let x =(x₁, ..., 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 BL-chain 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

xjk = 0, j = k, xj ≥ b ⊙aj, j ≠ k

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 =(x₁, ..., 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) ∈ R² | 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) ∈ R² | 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 (x₁,y₁), (x₂,y₂) ∈ K, we define

(x₁,y₁) ≤ (x₂,y₂) ⇔ (x₁,y₁)^？1 ⊙ (x₂,y₂) ∈ P, (x₂,y₂) ⊙ (x₁,y₁)^？1 ∈ P ⇔ x₁ ？ x₂ or x₁ = x₂,y₁ ≤ y₂.

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 = ((x₁,y₁), (x₂,y₂), ⊥) | (x₁,y₁), (x₂,y₂) ≥ ⊥} 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

？ ？ a ⊙ x ≤ a ⊙ y ⇒ x ≤ y.

L satisfies the left conditional cancellation law if

？ ？ x ⊙ a ≤ y ⊙ a ⇒ x ≤ y.

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 = {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

xk = ak ⊙ b, k ∈ K, xj ≥ aj ⊙ b, j ？ K

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

xk = b⊙ak, j = k, xj ≥ b⊙aj, j ≠ k

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 =(x₁, ..., 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,

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, 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, or ax1 ？ ax2

(⇒)x1 = x2,y1 ≤ y1, or x1 ？ 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 = ((x₁, y₁), (x₂, y₂), ⊥ | (x₁, y₁), (x₂, y₂) ≥ ⊥} 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) ∈ Lⁿ and b_i ∈ L. We define two equations with respect to an unknown x = (x₁, ..., x_n) ∈ Lⁿ as

Then, (1) (III) is solvable iff it has the least solution x = (x₁, ..., x_n) ∈ Lⁿ such that

(2) (IV) is solvable iff it has the least solution x = (x₁, ..., x_n) ∈ Lⁿ such that

(3) If (III) is solvable, then

(4) If (IV) is solvable, then

(5) If (III) (resp. (IV)) is solvable and x₁, ..., 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 =(y₁, ..., y_n) denote a solution of (III). Since

Moreover,

Then,

Substitute

Thus, (x₁, ..., 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 = (y₁, ..., 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).

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.