### Fuzzy relation equations in pseudo BL-algebras

• • #### ABSTRACT

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.

• #### KEYWORD

Pseudo BL-algebras , inf-implication compositions , fuzzy relation equations

• ### 1. Introduction

Sanchez  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  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  introduced pseudo MV-algebras as the generalization of MV-algebras. Georgescu and Leustean  introduced generalized residuated lattice as a noncommutative structure.

In this paper, we investigate various solutions of fuzzy relation equations with inf-implication compositions AiR = Bi and AiR = Bi in pseudo BL-algebras.

### 2. Preliminaries

Definition 2.1.  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) ab = (ab) ⊙ a = a ⊙ (ab).

(A5) (ab) ？ (ba) = ？ and (ab) ？ (ba) = ？.

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.  For each x, y, z, xi, yiL, we have the following properties:

(1) If yz, (xy) ≤ (xz), xyxz, and zxyx for →∈ {→, ⇒}.

(2) xyxyxy.

(3) (xy) → z = x → (yz) and (xy) ⇒ z = y ⇒ (xz).

(4) x → (yz)= y ⇒ (xz) and x ⇒ (yz)= y → (xz).

(5) x ⊙ (xy) ≤ y and (xy) ⊙ xy.

(6) x ⊙ (yz)=(xy) ∨ (xz) and (xy) ⊙ z = (xz) ∨ (yz).

(7) xy = ？ iff xy iff xy = ？

### 3. Fuzzy Relation Equations in Pseudo BL-Algebras

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

Then, (1) (I) is solvable iff it has the least solution y = (y1, ..., yn) ∈ Ln such that yj = baj, j =1, ..., n.

(2) (II) is solvable iff it has the least solution x =((x1, ..., (xn) ∈ Ln such that xj = ajb, j =1, ..., n.

(3) If (I) is solvable, then

(4) If (II) is solvable, then

Proof. (1) (⇒) Let x =((x1, ..., (xn) be a solution of (I). Since

Moreover,

Thus, y = (ba1, ..., ban) is the least solution.

(？) It is trivial.

(3) Let x =(x1, ..., 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 ≤ km, b = (ajk)*}, then

is a maximal solution of (II). Moreover, if x is a solution of (II), there exists k∈{jk |1 ≤ km} such that

xjk = 0, j = k, xjajb, jk

where there exists xjkX such that x ≤ xjk.

(2) If b？？ and

with B = {ajk |1 ≤ km, b = (ajk)0}, then

is a maximal solution of (I). Moreover, if x is a solution of (I), there exists k ∈ {jk |1 ≤ km} such that

xjk = 0, j = k, xj ≥ b ⊙aj, j ≠ k

where there exists xjkX such that x ≤ xjk.

Proof. (1) (⇒)

is a solution of (II) because

Let x ≥ xjk be a solution of (II). Then,

with xjkajkb 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 = xjk.

is a maximal solution of (II).

Let x =(x1, ..., xn) be a solution of (II). Since

by the linearity of L, there exists a family K = {jk | ajkB, ajk ⇒⊥ = b, 1 ≤ km} such that

, because by linearity of L, ajkB, (aj) >b implies that

For kK, since ak ⇒⊥ = akxk = b ≠ ？ and L is linear, ak >xk and akb = ak ⊙ (akxk)= ak ⊙ (ak ⇒ ⊥) = ⊥ akxk = xk. 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 × KK 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 PP？1 = {(1, 0)}, PPP, (a, b)？1P ⊙ (a, b)= P, and PP？1 = K. For (x1,y1), (x2,y2) ∈ K, we define

(x1,y1) ≤ (x2,y2) ⇔ (x1,y1)？1 ⊙ (x2,y2) ∈ P, (x2,y2) ⊙ (x1,y1)？1Px1x2 or x1 = x2,y1y2.

Then, (K, ≤？) is a lattice-group with totally order ≤. (ref. )

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 = {ajk |1 ≤ km, b ？ (ajk)*}, then

is a maximal solution family of (II). Moreover, if x is a solution of (II), there exists a family K = {jk | ajkB, ajkxjk = b, 1 ≤ km} such that

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

where there exists xjkX such that x ≤ xjk.

(2) If L satisfies the left conditional cancellation law b ？ ？ and

with B = {ajk | 1 ≤ km, b= (ajk)0}, then

is a maximal solution of (I). Moreover, if x is a solution of (I), there exists k ∈ {jk | 1 ≤ km} such that

xk = b⊙ak, j = k, xj ≥ b⊙aj, j ≠ k

where there exists xjkX such that x ≤ xjk.

Proof. (1)

is a solution of (II) because

Let x ≥ xjk denote a solution of (II). Then,

with xjkajkb and

Since b ？ 1,

Since L is linear, ajkxjk. Thus,

xjk = ajk ∧ xjk = ajk ⊙ (ajk ⇒ xjk ) = ajk ⊙ b.

Therefore, x = xjk.

is a maximal solution of (II).

Let x =(x1, ..., xn) denote a solution of (II). Since

by the linearity of L, there exists a family K = {jk | ajkB, ajkxjk = b, 1 ≤ km} such that

because ajkB, (aj)0b implies that

For kK, since akxk = b ≠ ？ and L is linear, ak >xk and akb = ak ⊙ (akxk)= akxk = xk. For jK, since ajxjb, xjajb. 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 ai =(ai1,ai2, ..., ain) ∈ Ln and biL. We define two equations with respect to an unknown x = (x1, ..., xn) ∈ Ln as

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

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

(3) If (III) is solvable, then

(4) If (IV) is solvable, then

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

Moreover,

Then,

Substitute

Thus, (x1, ..., xn) is the least solution.

(？) It is trivial.

(3)

(2) and (4) are similarly proved as (1) and (3), respectively.

5) Let xi =(xi1, ..., 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 = (y1, ..., yn) denote a solution of (III). Then, y ≤ xi 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  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.