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. 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., (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 , 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 =( a ₁ , a ₂ , ..., a_n ) ∈ Lⁿ and b ∈ L . We define two equations with respect to an unknown x = ( x ₁ , ..., ( x_n ) ∈ Lⁿ as Then, (1) (I) is solvable iff it has the least solution y = ( y ₁ , ..., y_n ) ∈ Lⁿ 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 ) ∈ 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 =(( 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 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 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 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: 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 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 = { 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, 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 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 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 = ( x ₁ , ..., x_n ) ∈ Lⁿ as Then, (1) (III) is solvable iff it has the least solution x = ( x ₁ , ..., x_n ) ∈ Lⁿ such that (2) (IV) is solvable iff it has the least solution x = ( x ₁ , ..., 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 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). 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.