Folding theory applied to pseudo-hoops

1 Introduction

Non-classical logic has become a considerable formal tool for computer science and artificial intelligence to deal with fuzzy information and uncertainty information. Many-valued logic, a great extension and development of classical logic, has always been a crucial direction in non-classical logic. In [11], Georgescu, Leustean, and Preoteasa presented pseudo-hoops were initially been introduced by Bosbach in ([3, 4]) under the name “residuated integral monoids”. The prefix “pseudo” stands for non-commutative or not necessarily the commutative type of algebra. It followed naturally after the introduction of pseudo-MV algebras [10], pseudo-Wajsberg algebras ([6, 7]) and pseudo-BL algebras ([8, 9]). All the above are non-commutative generalizations of algebras for many-valued logics. Pseudo-hoops are weaker structures, and pseudo-MV, pseudo-Wajsberg, and pseudo-BL algebras arise as particular cases of them. Pseudo-hoops are monoids endowed with orders. Moreover, the orders are canonical (actually inverse canonical) they are given by divisibility relations with respect to the monoid operation, and the orders have residuals. Moreover, if a pseudo-hoop is commutative it is called hoop and some mathematicians study this algebraic structure in different fileds, for example, in [5], the authors defined the concepts of local and perfect semihoops, then they introduced primary and perfect filters on it and investigated the relation between perfect filters with perfect semihoops and primary filter with the local semihoops. In [1] the authors characterized the notions of (implicative, maximal, prime) ideals in hoops and investigated the relation between them defined a congruence relation on hoops by ideals and studied the quotient that is made by it. By using this congruence relation, they showed that an ideal is maximal if and only if the quotient hoop is a simple MV-algebra. In [15, 16] the authors defined node and nodal filter in hoops and proved that the sets of all nodes are a bounded distributive lattice. Moreover, they investigated that under which conditions the set of all nodal filters in hoop is a Hertz algebra, Heyting algebra, Kleene algebra, semi-De Morgan algebra, Hilbert algebra and BCK-algebra. In [2], they defined fuzzy point of filter in hoops and found some equivalent definitions of them. Then they defined a congruence relation on hoops by an (∈ , ∈)-fuzzy filter and show that the quotient structure of this relation is a hoop.

The algebraic structures corresponding to Hájek’s propositional (fuzzy) basic logic, BL-algebras, are particular cases of hoops. The main example of BL-algebras in interval [0, 1] endowed with the structure induced by a t-norm. Hàjek introduced his Basic Fuzzy Logics, (BL-logics, in short) in 1998 [12], as logics of continuous t-norms, a multitude of research papers related to algebraic counterparts of BL-logics, has been published. In [13, 17–19] the authors defined the notion of different kinds of n-fold filters in BL-algebras. They studied the relation among of them.

Now, in this paper, we extend different types of n-fold (implicative, positive implicative, fantastic) filters to pseudo-hoops and investigate definitions that are equivalent to those and the relation among them.

2 Preliminaries

In this section, we recollect some definitions and results which will be used in this paper.

Definition 2.1. [11] A pseudo-hoop is an algebra (H, ∧ , ⊙ , → , ⇝ , 1) of type (2, 2, 2, 2, 0) such that, for all a, b, c ∈ H,

(PH-1) (H, ∧ , 1) is a meet-semilattice with the greatest element 1,

(PH-2) (H, ⊙ , 1) is a monoid with unit 1,

(PH-3) a ⊙ b ≤ c if and only if a ≤ b → c if and only if b ≤ a ⇝ c,

(PH-4) a ∧ b = (a → b) ⊙ a = a ⊙ (a ⇝ b). (divisibility condition)

On pseudo-hoop H, we define a ≤ b if and only if a → b = 1 (or equivalently a ⇝ b = 1), then (H, ≤) is a poset. If ⊙ is commutative (or equivalently → =⇝), then H is said to be a hoop. We say that a pseudo-hoop H is bounded if it has an element 0 ∈ H such that 0 ≤ a, for all a ∈ H. We let a⁰ = 1, a ⁿ  = a^n-1 ⊙ a, for any n ∈ ℕ . We define two unary operations, a^- = a → 0 and a^∼ = a ⇝ 0, for all a ∈ H. If (a^-) ^∼ = (a^∼) ^- = a, for all a ∈ H, then the bounded pseudo-hoop H is said to have the pseudo double negation property, (PDN) for short. (See [11]).

Definition 2.2. [11] A lattice-ordered group(ℓ-group) is an algebra (G, + , - , 0, ∧ , ∨) such that (G, + , - , 0) is a group and it is also a lattice that satisfies the identities x + (y ∧ z) + t = (x + y + t) ∧ (x + z + t) and x + (y ∨ z) + t = (x + y + t) ∨ (x + z + t). Throughout, we write x ≤ y as a shorthand for x ∨ y = y or x ∧ y = x, and ℓ-group as an abbreviation for lattice-ordered group.

Example 2.3. [11] Let (G, + , - , 0, ∧ , ∨) be an arbitrary ℓ-group. For an arbitrary element 0 ≤ u ∈ G define the following operations on the set [0, u]:

x ⊙ y = (x - u + y) ∨0,   x → y = (y - x + u) ∧ u   {and  x ⇝ y = (u - x + y) ∧ u, for any x, y ∈ [0, u]. By routine calculation, we can see that ([0, u] , ⊙ , → , ⇝ , u) is a bounded pseudo-hoop.

Example 2.4. [11] Let G = (G, + , - , 0, ∨ , ∧) be an arbitrary ℓ-group and N (G) be the negative cone of G, that is N (G) = {a ∈ G ∣ a ≤ 0}. On N (G), we define the following operations:

a ⊙ b = a + b,   a → b = (b - a) ∧0  and  a ⇝ b = (- a + b) ∧0 . Then N (G) = (N (G) , ⊙ , → , ⇝ , 0) is an unbounded pseudo-hoop. The following proposition provide some properties of pseudo-hoops.

Proposition 2.5. [3, 4] Let (H, ⊙ , → , ⇝ , 1) be a pseudo-hoop. Then the following conditions hold, for all a, b, c ∈ H:

(i) 1 → a = a, 1 ⇝ a = a,

(ii) c → a ≤ (b → c) → (b → a), c ⇝ a ≤ (b ⇝ c) ⇝ (b ⇝ a),

(iii) a → b ≤ (b → c) ⇝ (a → c), a ⇝ b ≤ (b ⇝ c) → (a ⇝ c),

(iv) a ⇝ (b → c) = b → (a ⇝ c), a → (b ⇝ c) = b ⇝ (a → c),

(v) (a ⊙ b) → c = a → (b → c), (a ⊙ b) ⇝ c = b ⇝ (a ⇝ c),

(vi) a ≤ b if and only if a → b = 1 if and only if a ⇝ b = 1,

(vii) a → a = 1, a ⇝ a = 1,

(viii) a ⊙ b ≤ a ∧ b,

(ix) a ≤ b → a, a ≤ b ⇝ a,

(x) a → 1 =1, a ⇝ 1 =1,

(xi) a ≤ (a → b) ⇝ b, a ≤ (a ⇝ b) → b,

(xii) a ≤ b implies c → a ≤ c → b, c ⇝ a ≤ c ⇝ b,

(xiii) a ≤ b implies b → c ≤ a → c, b ⇝ c ≤ a ⇝ c.

Definition 2.6. [11] Let H be a pseudo-hoop. A non-empty subset F of H is called a filter if it satisfies,

(F1) a, b ∈ F implies a ⊙ b ∈ F,

(F2) a ∈ F and a ≤ b imply b ∈ F, for any b ∈ H.

A filter F of H is called proper if and only if F ≠ H.

Proposition 2.7. [11] Let H be a pseudo-hoop. If F is a non-empty subset of H such that 1 ∈ F, then the following statements are equivalent, for any a, b, c ∈ H,

(i) F is a filter,

(ii) if a, a → b ∈ F, then b ∈ F,

(iii) if a, a ⇝ b ∈ F, then b ∈ F.

Definition 2.8. [11] A filter F of pseudo-hoop H is called normal if a → b ∈ F if and only if a ⇝ b ∈ F, for all a, b ∈ H.

Definition 2.9. [11] Let F be a normal filter of a pseudo-hoop H and ≡ _F be a congruence relation on H and quotient of it is denoted by H/F, H/F = {[a] |a ∈ H}, where [a] = {b ∈ H ∣ a ≡  _F b} = {b ∈ H ∣ a → b, b → a ∈ F}. Define the operations ⊙, → , ⇝ on H/F as follows: [ a ] ⊙ [ b ] = [ a ⊙ b ] , [ a ] → [ b ] = [ a → b ] and [ a ] ⇝ [ b ] = [ a ⇝ b ] .

Note. From now one in this paper, we let H = ( H , ⊙ , → , ⇝ , 1 ) be a pseudo-hoop, unless otherwise state.

3 n-Fold positive implicative filters

In this section, we introduce the notion of n-fold positive implicative filters in pseudo-hoops and investigate some properties of them.

Definition 3.1. H is called an n-fold positive implicative pseudo-hoop, if for all a ∈ H and n ∈ ℕ , aⁿ⁺¹ = a ⁿ .

Example 3.2. Let H = {0, a, b, 1}. We define ⊙, → and ⇝ on H as follows:

Then (H, ⊙ , → , ⇝ , 0, 1) is a bounded pseudo-hoop and H is a 3-fold positive implicative pseudo-hoop, but since b³ ≠ b², H is not a 2-fold positive implicative pseudo-hoop.

Proposition 3.3. If H is an n-fold positive implicative pseudo-hoop, then H is an (n + 1)-fold positive implicative pseudo-hoop.

Proof. Let H be an n-fold positive implicative pseudo-hoop. Then aⁿ⁺¹ = a ⁿ , for all a ∈ H. Since aⁿ⁺² = aⁿ⁺¹ ⊙ a = a ⁿ  ⊙ a = aⁿ⁺¹, we get H is an (n + 1)-fold positive implicative pseudo-hoop. □

Definition 3.4. A non-empty subset F of H is called an n-fold positive implicative filter of H if, for all a, b, c ∈ H,

(NPI1) 1 ∈ F,

(NPI2) a ⁿ  → (b → c) ∈ F and a ⁿ  ⇝ b ∈ F imply a ⁿ  → c ∈ F,

(NPI3) a ⁿ  ⇝ (b ⇝ c) ∈ F and a ⁿ  → b ∈ F imply a ⁿ  ⇝ c ∈ F.

The set of all n-fold positive implicative filters of H is denoted by PI F n ( H ) .

Example 3.5. Let H be the pseudo-hoop as in Example 3.2. Then { 1 } ∈ PI F 3 ( H ) but since b³ → 0 =1 ∈ {1} and b² → 0 = b ≠ 1, we get { 1 } ∉ PI F 2 ( H ) .

Example 3.6. According to Example 2.3, (G = [0, 1/2] , + , - , ∧ , ∨) is an ℓ-group, (G = [0, 1/2] , ⊙ , → , ⇝ , 1/2) is a pseudo-hoop and F = {1/2} is a filter of G and by routine calculations we can see that F ∈ PI F 2 ( G ) . But F ∉ PI F 1 ( G ) , because since (1/4) → (1/3 →1/5) =1/2 ∈ F and 1/4 ⇝1/3 =1/2 ∈ F, we get 1/4 →1/5 =9/20 ∉ F. Hence F ∉ PI F 1 ( G ) .

Proposition 3.7. Let F ∈ PI F n ( H ) . Then F is a filter of H .

Proof. Let F ∈ PI F n ( H ) . By (NPI1), 1 ∈ F, and so (F1) holds. Suppose a ∈ F and a ⇝ b ∈ F. Then, by Proposition 2.5(i), 1 ⁿ  → a = 1 → a = a ∈ F and 1 ⁿ  ⇝ (a ⇝ b) = a ⇝ b ∈ F. Thus, by (NPI3), 1 ⁿ  ⇝ b = b ∈ F. Therefore, F is a filter of H . □

Proposition 3.8. Let F ∈ PI F n ( H ) . Then, for all a, b ∈ H,

(i) a ⁿ  → a²ⁿ ∈ F and a ⁿ  ⇝ a²ⁿ ∈ F,

(ii) if aⁿ⁺¹ → b ∈ F, then a ⁿ  → b ∈ F and if aⁿ⁺¹ ⇝ b ∈ F, then a ⁿ  ⇝ b ∈ F.

Proof. (i) Let F ∈ PI F n ( H ) . Then a ⁿ  ⇝ (a ⁿ  ⇝ a²ⁿ) = a²ⁿ ⇝ a²ⁿ = 1, for any a ∈ H. By Proposition 3.7, F is a filter of H , and so a ⁿ  ⇝ (a ⁿ  ⇝ a²ⁿ) ∈ F. Since a ⁿ  → a ⁿ  = 1 ∈ F and F ∈ PI F n ( H ) , by (NPI2), a ⁿ  ⇝ a²ⁿ ∈ F. The proof of other cases is similar.

(ii) Let F ∈ PI F n ( H ) and aⁿ⁺¹ → b ∈ F. By Proposition 3.7(v), a ⁿ  → (a → b) = aⁿ⁺¹ → b ∈ F. Since a ⁿ  ≤ a, by Proposition 2.5(vi), a ⁿ  ⇝ a = 1 ∈ F and so by (NPI2), a ⁿ  → b ∈ F. The proof of other cases, is similar. □

Proposition 3.9. Let F be a filter of H . If F ∈ PI F n ( H ) , then F ∈ PI F n + 1 ( H ) .

Proof. Let F be a filter of H such that aⁿ⁺¹ → (b → c) ∈ F and aⁿ⁺¹ ⇝ b ∈ F. By Proposition 3.8(ii), a ⁿ  → (b → c) ∈ F and a ⁿ  ⇝ b ∈ F. Since F ∈ PI F n ( H ) , we have a ⁿ  → c ∈ F. Also, since aⁿ⁺¹ ≤ a ⁿ , by Proposition 2.5(xiii), a ⁿ  → c ≤ aⁿ⁺¹ → c. Since a ⁿ  → c ∈ F, by (F2), aⁿ⁺¹ → c ∈ F. Therefore, F ∈ PI F n + 1 ( H ) . □

Corollary 3.10. Let F be a positive implicative filter of H . Then F ∈ PI F n ( H ) .

Proof. By Proposition 3.9, the proof is clear. □

Lemma 3.11. Let for a, b, c, d ∈ H, a ≤ b and c ≤ d. Then a ⊙ c ≤ b ⊙ d.

Proof. Since b ⊙ c ≤ b ⊙ c, by (PH-3), b ≤ c → (b ⊙ c). Since a ≤ b, we get a ≤ c → (b ⊙ c), then a ⊙ c ≤ b ⊙ c. □

Theorem 3.12. Let F be a normal filter of H . Then F ∈ PI F n ( H ) if and only if

(i) if a ⁿ  → (b → c) ∈ F, then (a ⁿ  ⇝ b) ⇝ (a ⁿ  ⇝ c) ∈ F,

(ii) if a ⁿ  ⇝ (b ⇝ c) ∈ F, then (a ⁿ  → b) → (a ⁿ  → c) ∈ F.

Proof. (⇒) Let F ∈ PI F n ( H ) and be a normal filter of H such that a ⁿ  → (b → c) ∈ F, for a, b, c ∈ H. By (PH-4), ( a n ∧ ( b → c ) ) ⊙ ( a n ∧ b ) = ( a n → ( b → c ) ) ⊙ a n ⊙ a n ⊙ ( a n ⇝ b ) = ( a n → ( b → c ) ) ⊙ a 2 n ⊙ ( a n ⇝ b ) . Since a ⁿ  ∧ (b → c) ≤ b → c and a ⁿ  ∧ b ≤ b, by Lemma 3.11, (a ⁿ  ∧ (b → c)) ⊙ (a ⁿ  ∧ b) ≤ (b → c) ⊙ b = b ∧ c ≤ c. Thus, (a ⁿ  → (b → c)) ⊙ a²ⁿ ⊙ (a ⁿ  ⇝ b) ≤ c, and so by (PH-3), a ⁿ  → (b → c) ≤ (a²ⁿ ⊙ (a ⁿ  ⇝ b)) → c. By assumption, a ⁿ  → (b → c) ∈ F, then by (F2), (a²ⁿ ⊙ (a ⁿ  ⇝ b)) → c ∈ F. Thus, by Proposition 2.5(v), a²ⁿ → ((a ⁿ  ⇝ b) → c) ∈ F. Hence, by Proposition 2.5(ii), (a ⁿ  → a²ⁿ) → (a ⁿ  → ((a ⁿ  ⇝ b) → c))) ∈ F. Since F ∈ PI F n ( H ) , by Proposition 3.8(i), a ⁿ  → a²ⁿ ∈ F, for any a ∈ H and so by Proposition 2.7(ii), a ⁿ  → ((a ⁿ  ⇝ b) → c) ∈ F. Since F is normal, we get a ⁿ  ⇝ ((a ⁿ  ⇝ b) → c) ∈ F. By Proposition 2.5(iv), (a ⁿ  ⇝ b) → (a ⁿ  ⇝ c) ∈ F, and so (a ⁿ  ⇝ b) ⇝ (a ⁿ  ⇝ c) ∈ F. The proof of (ii) is similar.

(⇐) Let F be a normal filter of H such that a ⁿ  → (b → c) ∈ F and a ⁿ  ⇝ b ∈ F. By (i), (a ⁿ  ⇝ b) ⇝ (a ⁿ  ⇝ c) ∈ F. Since a ⁿ  ⇝ b ∈ F, by Proposition 2.7(iii), a ⁿ  ⇝ c ∈ F. Since F is normal, a ⁿ  → c ∈ F. Then (NPI2) holds. The proof of (NPI3), is similar. Therefore, F ∈ PI F n ( H ) . □

Proposition 3.13. Let H be an n-fold positive implicative pseudo-hoop. If F is a normal filter of H , then F ∈ PI F n ( H ) .

Proof. Let F be a normal filter of H such that a ⁿ  → (b → c) ∈ F and a ⁿ  ⇝ b ∈ F, for a, b, c ∈ H. Since F is normal, a ⁿ  → b ∈ F. By Proposition 2.5(iii), b → c ≤ (a ⁿ  → b) → (a ⁿ  → c), and so by Proposition 2.5(xii), a ⁿ  → (b → c) ≤ a ⁿ  → ((a ⁿ  → b) → (a ⁿ  → c)). Since a ⁿ  → (b → c) ∈ F, by Proposition 3.7 and (F2), a ⁿ  → ((a ⁿ  → b) → (a ⁿ  → c)) ∈ F. Since F is normal, we get a ⁿ  ⇝ ((a ⁿ  → b) → (a ⁿ  → c)) ∈ F. By Proposition 2.5(iv), (a ⁿ  → b) → (a ⁿ  ⇝ (a ⁿ  → c)) ∈ F. Since a ⁿ  → b ∈ F, by Proposition 2.7(ii), a ⁿ  ⇝ (a ⁿ  → c) ∈ F. Hence, a ⁿ  → (a ⁿ  → c) ∈ F. Then by Proposition 2.5(v), a²ⁿ → c ∈ F. Also, from H is an n-fold positive implicative pseudo-hoop, aⁿ⁺¹ = a ⁿ , for all a ∈ H, thus, we have a 2 n = a n + 1 ⊙ a n - 1 = a n ⊙ a n - 1 = a 2 n - 1 = a n + 1 ⊙ a n - 2 = a n ⊙ a n - 2 = a 2 n - 2 = a n + 1 ⊙ a n - 3 . By continuing this method, we get a²ⁿ = a ⁿ . Hence a ⁿ  → c ∈ F. Therefore F ∈ PI F n ( H ) . □

Remark 3.14.[11] {1} and H are normal filters of H .

Theorem 3.15. H is an n-fold positive implicative pseudo-hoop if and only if { 1 } ∈ PI F n ( H ) .

Proof. (⇒) Let H be an n-fold positive implicative pseudo-hoop. Since {1} is normal, by Proposition 3.13, { 1 } ∈ PI F n ( H ) .

(⇐) Let { 1 } ∈ PI F n ( H ) . By Proposition 2.5(viii) and (vi), a²ⁿ ≤ aⁿ⁺¹ and so a²ⁿ → aⁿ⁺¹ = 1 ∈ {1}. Thus, by Proposition 2.5(v) and (vii), a ⁿ  → (a ⁿ  → aⁿ⁺¹) = a²ⁿ → aⁿ⁺¹ = 1 ∈ {1} and a ⁿ  ⇝ a ⁿ  = 1 ∈ {1}. Since { 1 } ∈ PI F n ( H ) , by (NPI2), a ⁿ  → aⁿ⁺¹ ∈ {1}, and so a ⁿ  ≤ aⁿ⁺¹. Hence, aⁿ⁺¹ = a ⁿ . Therefore, H is an n-fold positive implicative pseudo-hoop. □

Proposition 3.16. Let F and G be two filters of H such that F ⊆ G. If F ∈ PI F n ( H ) and G is a normal filter of H , then G ∈ PI F n ( H ) .

Proof. Let F and G be two filters of H such that F ⊆ G, F ∈ PI F n ( H ) and G be normal. Suppose a ⁿ  → (b → c) ∈ G and a ⁿ  ⇝ b ∈ G. Since F ∈ PI F n ( H ) , by Proposition 3.8(i), a ⁿ  → a²ⁿ ∈ F, for all a ∈ H. Thus a ⁿ  → a²ⁿ ∈ G, for all a ∈ H. Also, by Proposition 2.5(ii), b → c ≤ (a ⁿ  → b) → (a ⁿ  → c). Then by Proposition 2.5(xii), a ⁿ  → (b → c) ≤ a ⁿ  → ((a ⁿ  → b) → (a ⁿ  → c)). Since a ⁿ  → (b → c) ∈ G, by (F2), a ⁿ  → ((a ⁿ  → b) → (a ⁿ  → c)) ∈ G. Since G is normal, we get a ⁿ  ⇝ ((a ⁿ  → b) → (a ⁿ  → c)) ∈ G. Then by Proposition 2.5(iv), (a ⁿ  → b) → (a ⁿ  ⇝ (a ⁿ  → c)) ∈ G. Since a ⁿ  ⇝ b ∈ G and G is normal, a ⁿ  → b ∈ G. Thus, by Proposition 2.7(ii), a ⁿ  ⇝ (a ⁿ  → c) ∈ G, and so a ⁿ  → (a ⁿ  → c) ∈ G. Hence, by Proposition 2.5(v), a²ⁿ → c ∈ G. By Proposition 2.5(iii), a ⁿ  → a²ⁿ ≤ (a²ⁿ → c) ⇝ (a ⁿ  → c). Since a ⁿ  → a²ⁿ ∈ G, by (F2), (a²ⁿ → c) ⇝ (a ⁿ  → c) ∈ G. Since a²ⁿ → c ∈ G, by Proposition 2.7(iii), a ⁿ  → c ∈ G. Therefore, G ∈ PI F n ( H ) □ .

Proposition 3.17. Let F be a normal filter of H . If for all a ∈ H, a ⁿ  → a²ⁿ ∈ F or a ⁿ  ⇝ a²ⁿ ∈ F, then F ∈ PI F n ( H ) .

Proof. Let F be a normal filter of H and a ⁿ  → a²ⁿ ∈ F, for any a ∈ H. Suppose a ⁿ  → (b → c) ∈ F and a ⁿ  ⇝ b ∈ F. Since F is normal, a ⁿ  → b ∈ F. By Proposition 2.5(ii), b → c ≤ (a ⁿ  → b) → (a ⁿ  → c). Thus, by Proposition 2.5(xii), a ⁿ  → (b → c) ≤ a ⁿ  → ((a ⁿ  → b) → (a ⁿ  → c)) and so by (F2), a ⁿ  → ((a ⁿ  → b) → (a ⁿ  → c)) ∈ F. Since F is normal, a ⁿ  ⇝ ((a ⁿ  → b) → (a ⁿ  → c)) ∈ F, and by Proposition 2.5(iv), (a ⁿ  → b) → (a ⁿ  ⇝ (a ⁿ  → c)) ∈ F. Since a ⁿ  → b ∈ F, by Proposition 2.7(ii), a ⁿ  ⇝ (a ⁿ  → c) ∈ F, and so a ⁿ  → (a ⁿ  → c) ∈ F. Then by Proposition 2.5(v), a²ⁿ → c ∈ F. Also, by Proposition 2.5(iii), a ⁿ  → a²ⁿ ≤ (a²ⁿ → c) ⇝ (a ⁿ  → c). Since a ⁿ  → a²ⁿ ∈ F, by (F2), (a²ⁿ → c) ⇝ (a ⁿ  → c) ∈ F. Since a²ⁿ → c ∈ F, by Proposition 2.7(iii), a ⁿ  → c ∈ F. Therefore, F ∈ PI F n ( H ) □ .

Theorem 3.18. Let F be a normal filter of H . Then F ∈ PI F n ( H ) if and only if H / F is an n-fold positive implicative pseudo-hoop.

Proof. (⇒) Let F ∈ PI F n ( H ) . Then by Proposition 3.8(i), a ⁿ  → a²ⁿ ∈ F, for all a ∈ H. Then [a]  ⁿ  → [a] ²ⁿ = [a ⁿ  → a²ⁿ] = [1] ∈ {[1]}. Since {[1]} is a normal filter of H / F , by Proposition 3.17, { [ 1 ] } ∈ PI F n ( H / F ) . By Theorem 3.15, H / F is an n-fold positive implicative pseudo-hoop.

(⇐) Let H / F be an n-fold positive implicative pseudo-hoop and F be a normal filter of H . Then [a] ²ⁿ = [a]  ⁿ , for any [a] ∈ H/F. Since [a ⁿ  → a²ⁿ] = [a]  ⁿ  → [a] ²ⁿ = [1], we have a ⁿ  → a²ⁿ ∈ F, for any a ∈ H. Hence, by Proposition 3.17, F ∈ PI F n ( H ) . □

4 n-Fold implicative filters

In this section, we introduce the notion of n-fold implicative filters in pseudo-hoops and investigate their properties, and we study the relationship between them and n-fold positive implicative filters.

Definition 4.1. Let H be a bounded pseudo-hoop. H is called an n-fold implicative pseudo-hoop, if for all a ∈ H,

(NIPH-1) (a ⁿ  → 0) ⇝ a = a,

(NIPH-2) (a ⁿ  ⇝ 0) → a = a.

Example 4.2. In Example 3.2, H is a 3-fold implicative pseudo-hoop, but since (b² → 0) ⇝ b ≠ b, H is not a 2-fold implicative pseudo-hoop.

Proposition 4.3. Every n-fold implicative pseudo-hoop is an (n + 1)-fold implicative pseudo-hoop.

Proof. Let H be an n-fold implicative pseudo-hoop. Then (a ⁿ  → 0) ⇝ a = a, for all a ∈ H. Since aⁿ⁺¹ ≤ a ⁿ , then by Proposition 2.5(xiii), a ⁿ  → 0 ≤ aⁿ⁺¹ → 0, and so (aⁿ⁺¹ → 0) ⇝ a ≤ (a ⁿ  → 0) ⇝ a = a. By Proposition 2.5(ix), a ≤ (aⁿ⁺¹ → 0) ⇝ a. Hence, a = (aⁿ⁺¹ → 0) ⇝ a. Therefore, H is an (n + 1)-fold implicative pseudo-hoop. □

Definition 4.4. A non-empty subset F of pseudo-hoop H is called an n-fold implicative filter if, for all a, b, c ∈ H,

(NI1) 1 ∈ F,

(NI2) a → ((b ⁿ  → c) ⇝ b) ∈ F and a ∈ F imply b ∈ F,

(NI3) a ⇝ ((b ⁿ  ⇝ c) → b) ∈ F and a ∈ F imply b ∈ F.

The set of all n-fold implicative filters of H is denoted by I F n ( H ) .

Example 4.5. In Example 3.2, { 1 } ∈ I F 3 ( H ) , but (b² → 0) ⇝ b = 1 and b ≠ 1, hence { 1 } ∉ I F 2 ( H ) .

Proposition 4.6. Let F ∈ I F n ( H ) . Then F is a filter of H .

Proof. Let F ∈ I F n ( H ) and a,  a ⇝ b ∈ F, for a, b ∈ H. By Proposition 2.5(x) and (i), a ⇝ ((b ⁿ  ⇝ 1) → b) = a ⇝ b ∈ F. Since a ∈ F, by (NI3), b ∈ F. Therefore, F is a filter of H . □

Proposition 4.7. Let F be a filter of H . Then F ∈ I F n ( H ) if and only if (a ⁿ  → b) ⇝ a ∈ F implies a ∈ F and if (a ⁿ  ⇝ b) → a ∈ F implies a ∈ F, for any a, b ∈ H.

Proof. (⇒) Let F ∈ I F n ( H ) and (a ⁿ  → b) ⇝ a ∈ F. By Proposition 2.5(i), 1 → ((a ⁿ  → b) ⇝ a) = (a ⁿ  → b) ⇝ a ∈ F. Thus, by (NI2), a ∈ F. The proof of other cases is similar.

(⇐) Suppose c → ((a ⁿ  → b) ⇝ a) ∈ F and c ∈ F. Then by Proposition 2.7(ii), (a ⁿ  → b) ⇝ a ∈ F. Thus, by assumption, a ∈ F, and so (NI2) holds. The proof of (NI3) is similar. Therefore, F ∈ I F n ( H ) □

Proposition 4.8. If F ∈ I F n ( H ) , then F ∈ I F n + 1 ( H ) .

Proof. Let F ∈ I F n ( H ) and (aⁿ⁺¹ → b) ⇝ a ∈ F. By Proposition 2.5(viii), aⁿ⁺¹ ≤ a ⁿ , then by Proposition 2.5(xiii), a ⁿ  → b ≤ aⁿ⁺¹ → b, and so (aⁿ⁺¹ → b) ⇝ a ≤ (a ⁿ  → b) ⇝ a. Since (aⁿ⁺¹ → b) ⇝ a ∈ F, by (F2), (a ⁿ  → b) ⇝ a ∈ F. Since F ∈ I F n ( H ) , by Proposition 4.7, a ∈ F. Therefore, F ∈ I F n + 1 ( H ) □ .

Corollary 4.9. Let H be a bounded n-fold implicative pseudo-hoop. Then F is a filter of H if and only if F ∈ I F n ( H ) .

Proof. (⇒) Let F be a filter of H and (a ⁿ  ⇝ b) → a ∈ F, for any a, b ∈ H. Since H is a bounded pseudo-hoop, we have 0 ≤ b, for all b ∈ H. By Proposition 2.5(xii), a ⁿ  ⇝ 0 ≤ a ⁿ  ⇝ b, and so (a ⁿ  ⇝ b) → a ≤ (a ⁿ  ⇝ 0) → a. Since (a ⁿ  ⇝ b) → a ∈ F, by (F2), (a ⁿ  ⇝ 0) → a ∈ F. Since H is an n-fold implicative pseudo-hoop, we have (a ⁿ  ⇝ 0) → a = a, for any a ∈ H, and so a ∈ F. Hence, by Proposition 4.7, F ∈ I F n ( H ) .

(⇐) Let F ∈ I F n ( H ) . Then by Proposition 4.6, F is a filter of H . □

Proposition 4.10. H is a bounded n-fold implicative pseudo-hoop if and only if { 1 } ∈ I F n ( H ) .

Proof. (⇒) Let H be an n-fold implicative pseudo-hoop. Then by Corollary 4.9, { 1 } ∈ I F n ( H ) .

(⇐) Let { 1 } ∈ I F n ( H ) and u = ((a ⁿ  → 0) ⇝ a) → a, for any a ∈ H. By Proposition 2.5(iv) and (iii), we have ( u n → 0 ) ⇝ u = ( u n → 0 ) ⇝ ( ( ( a n → 0 ) ⇝ a ) → a ) = ( ( a n → 0 ) ⇝ a ) → ( ( u n → 0 ) ⇝ a ) ≥ ( u n → 0 ) ⇝ ( a n → 0 ) ≥ a n → u n . Now, by Proposition 2.5(ix), a ≤ u, and this implies that a ⁿ  ≤ u ⁿ . Thus, by Proposition 2.5(vi), a ⁿ  → u ⁿ  = 1, and so (u ⁿ  → 0) ⇝ u = 1 ∈ {1}. Since { 1 } ∈ I F n ( H ) , we get u ∈ {1}. Then ((a ⁿ  → 0) ⇝ a) → a = 1. Hence, ((a ⁿ  → 0) ⇝ a) ≤ a. On other hand, by Proposition 2.5(ix), a ≤ (a ⁿ  → 0) ⇝ a, then (a ⁿ  → 0) ⇝ a = a. Therefore, H is an n-fold implicative pseudo-hoop. □

Theorem 4.11. Let H be bounded and F be a filter of H . Then F ∈ I F n ( H ) if and only if H / F is an n-fold implicative pseudo-hoop.

Proof. (⇒) Let F ∈ I F n ( H ) and ([a]  ⁿ  → [0]) ⇝ [a] = [1], for a ∈ H. Then [(a ⁿ  → 0) ⇝ a] = ([a]  ⁿ  → [0]) ⇝ [a] = [1] and thus (a ⁿ  → 0) ⇝ a ∈ F. Then by Proposition 4.7, a ∈ F and so, [a] = [1]. Hence { [ 1 ] } ∈ I F n ( H / F ) and by Proposition 4.7, H / F is an n-fold implicative pseudo-hoop.

(⇐) Suppose (a ⁿ  → 0) ⇝ a ∈ F, for a ∈ H. Then ([a]  ⁿ  → [0]) ⇝ [a] = [(a ⁿ  → 0) ⇝ a] = [1]. Since H / F is an n-fold implicative pseudo-hoop, we get ([a]  ⁿ  → [0]) ⇝ [a] = [a]. Then [a] = [1], and so a ∈ F. Hence, by Proposition 4.7, F ∈ I F n ( H ) . □

Proposition 4.12. Let F and G be two filters of bounded pseudo-hoop H such that F ⊆ G. If F ∈ I F n ( H ) , then G ∈ I F n ( H ) .

Proof. Let F and G be two filters of H such that F ⊆ G, F ∈ I F n ( H ) and (a ⁿ  → 0) ⇝ a ∈ G, for a ∈ H. Since F ∈ I F n ( H ) , by Theorem 4.11, H / F is an n-fold implicative pseudo-hoop. Hence, [(a ⁿ  → 0) ⇝ a] = ([a]  ⁿ  → [0]) ⇝ [a] = [a], for a ∈ H, and so ((a ⁿ  → 0) ⇝ a) → a ∈ F. Since F ⊆ G, we get, ((a ⁿ  → 0) ⇝ a) → a ∈ G. Since (a ⁿ  → 0) ⇝ a ∈ G, and by Proposition 2.7(ii), a ∈ G. Then by Proposition 4.7, G ∈ I F n ( H ) . □

Proposition 4.13. Let F ∈ I F n ( H ) . Then, for a, b ∈ H, the following statements hold,

(i) if (a ⁿ  → b) ⇝ b ∈ F, then (b → a) ⇝ a ∈ F,

(ii) if (a ⁿ  ⇝ b) → b ∈ F, then (b ⇝ a) → a ∈ F.

Proof. (i) Let F ∈ I F n ( H ) and (a ⁿ  → b) ⇝ b ∈ F, for a, b ∈ H. By Proposition 2.5(xi) and (xii), b ≤ (b → a) ⇝ a and then (a ⁿ  → b) ⇝ b ≤ (a ⁿ  → b) ⇝ ((b → a) ⇝ a). Since (a ⁿ  → b) ⇝ b ∈ F, by (F2), (a ⁿ  → b) ⇝ ((b → a) ⇝ a) ∈ F. By Proposition 2.5(ix), a ≤ (b → a) ⇝ a, we get a ⁿ  ≤ ((b → a) ⇝ a)  ⁿ and so ((b → a) ⇝ a)  ⁿ  → b ≤ a ⁿ  → b. By Proposition 2.5(xiii), (a ⁿ  → b) ⇝ ((b → a) ⇝ a) ≤ (((b → a) ⇝ a)  ⁿ  → b) ⇝ ((b → a) ⇝ a). Since F ∈ I F n ( H ) , by Proposition 4.6, F is filter and (a ⁿ  → b) ⇝ ((b → a) ⇝ a) ∈ F, by (F2), (((b → a) ⇝ a)  ⁿ  → b) ⇝ ((b → a) ⇝ a) ∈ F. Hence, by (NI2), (b → a) ⇝ a ∈ F.

(ii) The proof is similar to (i). □

Theorem 4.14. Let H be bounded. If F ∈ I F n ( H ) and for all a ∈ H, (a ⁿ ) ^-∼ ∈ F or (a ⁿ ) ^∼- ∈ F, then F ∈ PI F n ( H ) .

Proof. Let F ∈ I F n ( H ) and (a ⁿ ) ^-∼ ∈ F, for all a ∈ H. Suppose, a ⁿ  → (b → c) ∈ F and a ⁿ  ⇝ b ∈ F, for a, b, c ∈ H. Since (c ⁿ ) ^-∼ ∈ F, for all c ∈ H, we have (c ⁿ  → 0) ⇝0 ∈ F. Since F ∈ I F n ( H ) , by Proposition 4.13, (0 → c) ⇝ c ∈ F. By Proposition 2.5(vi) and (i), 1 ⇝ c = c ∈ F. By Proposition 2.5(ix), c ≤ a ⁿ  → c and so by (F2), a ⁿ  → c ∈ F. Therefore, F ∈ PI F n ( H ) . □

5 n-Fold fantastic filters

In this section, we introduce the notion of an n-fold fantastic filter in pseudo-hoops, and investigate some properties of it and we get the relation between n-fold filters.

Definition 5.1. H is called an n-fold fantastic pseudo-hoop if, for all a, b ∈ H,

(NFPH-1) ((a ⁿ  → b) ⇝ b) → a = b → a,

(NFPH-2) ((a ⁿ  ⇝ b) → b) ⇝ a = b ⇝ a.

Example 5.2. Let H = {0, a, b, c, d, 1}. We define ⊙, → and ⇝ on H as follows:

Then H = ( H , ⊙ , → , ⇝ , 0 , 1 ) is a 2-fold fantastic pseudo-hoop. But ((a → 0) ⇝0) → a = 1 ≠ b = b → c, so H is not a 1-fold fantastic pseudo-hoop.

Proposition 5.3. If H is an n-fold fantastic pseudo-hoop, then H is an (n + 1)-fold fantastic pseudo-hoop.

Proof. Let H be an n-fold fantastic pseudo-hoop. Then ((a ⁿ  → b) ⇝ b) → a = b → a, for a, b ∈ H. By Proposition 2.5(viii) and (xiii), aⁿ⁺¹ ≤ a ⁿ and a ⁿ  → b ≤ aⁿ⁺¹ → b, thus, (aⁿ⁺¹ → b) ⇝ b ≤ (a ⁿ  → b) ⇝ b, and so ((a ⁿ  → b) ⇝ b) → a ≤ ((aⁿ⁺¹ → b) ⇝ b) → a. Since ((a ⁿ  → b) ⇝ b) → a = b → a, we have b → a ≤ ((aⁿ⁺¹ → b) ⇝ b) → a. By Proposition 2.5(ix), b ≤ (aⁿ⁺¹ → b) ⇝ b, then ((aⁿ⁺¹ → b) ⇝ b) → a ≤ b → a. Hence, ((aⁿ⁺¹ → b) ⇝ b) → a = b → a. Therefore, H is an (n + 1)-fold fantastic pseudo-hoop. □

Definition 5.4. Let H be a pseudo-hoop. Then F is called an n-fold fantastic filter of H , if for all a, b, c ∈ H,

(NF1) 1 ∈ F,

(NF2) c → (b → a) ∈ F and c ∈ F imply ((a ⁿ  → b) ⇝ b) → a ∈ F,

(NF3) c ⇝ (b ⇝ a) ∈ F and c ∈ F imply ((a ⁿ  ⇝ b) → b) ⇝ a ∈ F.

The set of all n-fold fantastic filters of H is denoted by F F n ( H )

Example 5.5. Let H = {0, a, b, 1}. We define ⊙, → and ⇝ on H as follows:

Then H = ( H , ⊙ , → , ⇝ , 0 , 1 ) is a pseudo-hoop and F = { b , 1 } ∈ F F 2 ( H ) .

Theorem 5.6. Let F ∈ F F n ( H ) . Then F is a filter of H .

Proof. Let F ∈ F F n ( H ) and a, a → b ∈ F, for a, b ∈ H. Then by Proposition 2.5(i), a → (1 → b) = a → b ∈ F. Since a ∈ F and F ∈ F F n ( H ) , by (NF2), b = ((b ⁿ  → 1) ⇝1) → b ∈ F. Therefore, F is a filter of H . □

Proposition 5.7. Let F be a filter of H . Then F ∈ F F n ( H ) if and only if b → a ∈ F implies ((a ⁿ  → b) ⇝ b) → a ∈ F, for any a, b ∈ H.

Proof. (⇒) Let F ∈ F F n ( H ) and b → a ∈ F. Then by Proposition 2.5(i), 1 → (b → a) = b → a ∈ F and 1 ∈ F, thus by (NF2), ((a ⁿ  → b) ⇝ b) → a ∈ F.

(⇐) Suppose c → (b → a) ∈ F and c ∈ F. Then by Proposition 2.7(ii), b → a ∈ F. Thus, by assumption, ((a ⁿ  → b) ⇝ b) → a ∈ F. Therefore, F ∈ F F n ( H ) . □

Proposition 5.8. Let F and G be two filters of H such that F ⊆ G. If F ∈ F F n ( H ) , then G ∈ F F n ( H ) .

Proof. Let F and G be two filters of H such that F ⊆ G and F ∈ F F n ( H ) . Suppose b → a ∈ G, for a, b ∈ H. Let u = (b → a) ⇝ a. Then by Proposition 2.5(iv), b → u = b → ((b → a) ⇝ a) = (b → a) ⇝ (b → a) =1 ∈ F. Since F ∈ F F n ( H ) and b → u ∈ F, by Proposition 5.7, ((u ⁿ  → b) ⇝ b) → u ∈ F. Since F ⊆ G, ((u ⁿ  → b) ⇝ b) → u ∈ G, and so ((u ⁿ  → b) ⇝ b) → ((b → a) ⇝ a) ∈ G. Then by Proposition 2.5(iv), (b → a) ⇝ (((u ⁿ  → b) ⇝ b) → a) ∈ G. Since b → a ∈ G, by Proposition 2.7(ii), ((u ⁿ  → b) ⇝ b) → a ∈ G. By Proposition 2.5(ix), a ≤ (b → a) ⇝ a, and so a ≤ u. Then a ⁿ  ≤ u ⁿ . Thus, by Proposition 2.5(xiii), u ⁿ  → b ≤ a ⁿ  → b, and so (a ⁿ  → b) ⇝ b ≤ (u ⁿ  → b) ⇝ b and ((u ⁿ  → b) ⇝ b) → a ≤ (a ⁿ  → b) ⇝ b) → a. Since ((u ⁿ  → b) ⇝ b) → a ∈ G, by (F2), ((a ⁿ  → b) ⇝ b) → a ∈ G. Therefore, G ∈ F F n ( H ) . □

Proposition 5.9. If F ∈ F F n ( H ) , then F ∈ F F n + 1 ( H ) .

Proof. Let b → a ∈ F, for a, b ∈ H. Then by (NF2), ((a ⁿ  → b) ⇝ b) → a ∈ F. Also, by Proposition 2.5(viii) and (xiii), aⁿ⁺¹ ≤ a ⁿ , then a ⁿ  → b ≤ aⁿ⁺¹ → b, and so (aⁿ⁺¹ → b) ⇝ b ≤ (a ⁿ  → b) ⇝ b. Thus, ((a ⁿ  → b) ⇝ b) → a ≤ ((aⁿ⁺¹ → b) ⇝ b) → a. Since ((a ⁿ  → b) ⇝ b) → a ∈ F, by (F2), ((aⁿ⁺¹ → b) ⇝ b) → a ∈ F. Therefore, F ∈ F F n + 1 ( H ) . □

Proposition 5.10. Let H be an n-fold fantastic pseudo-hoop and F be a filter of H . Then F ∈ F F n ( H ) .

Proof. Let F be a filter of n-fold fantastic pseudo-hoop H and b → a ∈ F. Since H is an n-fold fantastic pseudo-hoop, we get ((a ⁿ  → b) ⇝ b) → a = b → a ∈ F. Therefore, F ∈ F F n ( H ) . □

Proposition 5.11. H is an n-fold fantastic pseudo-hoop if and only if { 1 } ∈ F F n ( H ) .

Proof. (⇒) Let b → a ∈ {1}, for a, b ∈ H. Since H is an n-fold fantastic pseudo-hoop, ((a ⁿ  → b) ⇝ b) → a = b → a ∈ {1}. Therefore, { 1 } ∈ F F n ( H ) .

(⇐) Let b → a ∈ {1}, for a, b ∈ H. Since { 1 } ∈ F F n ( H ) , we get ((a ⁿ  → b) ⇝ b) → a ∈ {1}, and so (a ⁿ  → b) ⇝ b ≤ a. Also, by Proposition 2.5(ix), (viii) and (xiii), a ≤ (b → a) ⇝ a, then a ⁿ  ≤ ((b → a) ⇝ a)  ⁿ and ((b → a) ⇝ a)  ⁿ  → b ≤ a ⁿ  → b, and so (a ⁿ  → b) ⇝ b ≤ (((b → a) ⇝ a)  ⁿ ) → b) ⇝ b. By Proposition 2.5(iv) and (vii), b → ((b → a) ⇝ a) = (b → a) ⇝ (b → a) ∈ {1}, since { 1 } ∈ F F n ( H ) , we have ((((b → a) ⇝ a)  ⁿ ) → b) ⇝ b) → ((b → a) ⇝ a) ∈ {1}, ((((b → a) ⇝ a)  ⁿ ) → b) ⇝ b) ≤ ((b → a) ⇝ a). Hence, (a ⁿ  → b) ⇝ b ≤ (b → a) ⇝ a, and so ((a ⁿ  → b) ⇝ b) → ((b → a) ⇝ a) =1. Thus, (b → a) ⇝ (((a ⁿ  → b) ⇝ b) → a) =1. So, b → a ≤ ((a ⁿ  → b) ⇝ b) → a. By Proposition 2.5(iii),(iv),(vi) and (x), ( ( ( a n → b ) ⇝ b ) → a ) ⇝ ( b → a ) ≥ b → ( ( a n → b ) ⇝ b ) = ( a n → b ) ⇝ 1 = 1 . Then ((a ⁿ  → b) ⇝ b) → a ≤ b → a. Hence ((a ⁿ  → b) ⇝ b) → a = b → a. Therefore H is an n-fold fantastic pseudo-hoop. □

Theorem 5.12. Let F be a filter of H . Then F ∈ F F n ( H ) if and only if H / F is an n-fold fantastic pseudo-hoop.

Proof. (⇒) Let [b] → [a] ∈ {[1]}, for a, b ∈ H. Then [b → a] = [b] → [a] = [1] and so, b → a ∈ F. Since F ∈ F F n ( H ) , ((a ⁿ  → b) ⇝ b) → a ∈ F, thus, [((a ⁿ  → b) ⇝ b) → a] = [1]. Then (([a]  ⁿ  → [b]) ⇝ [b]) → [a] ∈ {[1]}, and so { [ 1 ] } ∈ F F n ( H ) . Hence, by Proposition 5.11, H / F is an n-fold fantastic pseudo-hoop.

(⇐) Let b → a ∈ F, for a, b ∈ H. Then [b] → [a] = [b → a] = [1] ∈ {[1]}. By Proposition 5.10, { [ 1 ] } ∈ F F n ( H / F ) . Then (([a]  ⁿ  → [b]) ⇝ [b]) → [a] ∈ {[1]}, and so [((a ⁿ  → b) ⇝ b) → a] = [1]. Thus, ((a ⁿ  → b) ⇝ b) → a ∈ F. Therefore, F ∈ F F n ( H ) . □

Theorem 5.13. If F ∈ I F n ( H ) and is a normal filter of H , then F ∈ F F n ( H ) .

Proof. Let F ∈ I F n ( H ) , be a normal filter of H and a → b ∈ F. Let u = ((b ⁿ  → a) ⇝ a) → b. By Proposition 2.5(iv) and (x), b ⇝ u = b ⇝ (((b ⁿ  → a) ⇝ a) → b) = ((b ⁿ  → a) ⇝ a) → (b ⇝ b) =1. Then b ⇝ u = 1 and b ≤ u. Thus b ⁿ  ≤ u ⁿ . By Proposition 2.5(xiii), u ⁿ  → a ≤ b ⁿ  → a. Hence, (b ⁿ  → a) ⇝ u ≤ (u ⁿ  → a) ⇝ u. Also, since a → b ∈ F and F is a normal filter of H , then a ⇝ b ∈ F. By Proposition 2.5(ii), a ⇝ b ≤ ((b ⁿ  → a) ⇝ a) ⇝ ((b ⁿ  → a) ⇝ b), and by (F2), ((b ⁿ  → a) ⇝ a) ⇝ ((b ⁿ  → a) ⇝ b) ∈ F. Then ((b ⁿ  → a) ⇝ a) → ((b ⁿ  → a) ⇝ b) ∈ F, thus by Proposition 2.5(iv), (b ⁿ  → a) ⇝ (((b ⁿ  → a) ⇝ a) → b) ∈ F. Hence, (b ⁿ  → a) ⇝ u ∈ F. Since (b ⁿ  → a) ⇝ u ≤ (u ⁿ  → a) ⇝ u, we get (u ⁿ  → a) ⇝ u ∈ F. From F ∈ I F n ( H ) , by Proposition 4.7, ((b ⁿ  → a) ⇝ a) → b = u ∈ F. Therefore, F ∈ F F n ( H ) . □

Theorem 5.14. Let H be bounded and F ∈ F F n ( H ) ∩ PI F n ( H ) . Then F ∈ I F n ( H ) .

Proof. Let (a ⁿ  → 0) ⇝ a ∈ F, for a ∈ H. Since F ∈ F F n ( H ) , by (NF2), ((a ⁿ  → (a ⁿ  → 0)) ⇝ (a ⁿ  → 0)) → a ∈ F. By Proposition 2.5(v), ((a²ⁿ → 0) ⇝ (a ⁿ  → 0)) → a ∈ F. Since F ∈ PI F n ( H ) , by Proposition 3.8(i), a ⁿ  → a²ⁿ ∈ F. By Proposition 2.5(iii), a ⁿ  → a²ⁿ ≤ (a²ⁿ → 0) ⇝ (a ⁿ  → 0), and so by (F2), (a²ⁿ → 0) ⇝ (a ⁿ  → 0) ∈ F. Since ((a²ⁿ → 0) ⇝ (a ⁿ  → 0)) → a ∈ F, we get a ∈ F. Then by Proposition 4.7, F ∈ I F n ( H ) . □

6 Conclusions and future works

In this paper, the notions of n-fold (implicative, positive implicative, and fantastic) filters in pseudo-hoops are introduced and investigated some properties of them. Several characterizations of these filters are derived, and the relation among these kinds of filters are studied. In the other word, the notion of filter, (positive) implicative and fantastic filters on pseudo hoop are generalized that help us to make different kinds of logical algebras by the quotient of pseudo hoop. Moreover, in future, other kinds of filters such as primary, integral and (semi) maximal filters, on pseudo hoop will be introduecd and by using them we can study pseudo-hoops in different fields such as defining a topological structure on them (Zariski topology), making an algebraic structure on the set of all these kinds of filter, studing softness(roghness) theory on pseudo hoop, and etc.