Integral hoops and integral filters

1 Introduction

Much of human reasoning and decision making is based on an environment of imprecision, uncertainty, incompleteness of information, partiality of truth and partiality of possibility in short, on an environment of imperfect information. Hence how to represent and simulate human reasoning become a crucial problem in information science field. For this reason, various logical algebras have been proposed as the semantical systems of non classical logic systems, for example, MV-algebras, BL-algebras, MTL-algebras, NM-algebras, residuated latticese and hoops. Among these logical algebras, hoops [7] are very basic and important algebraic structures. In the last few years, the theory of hoops has been enriched with deep structure theorems [1–3, 13]. Many of these results have a strong impact with fuzzy logics. In particular, from the structure theorem of finite basic hoops, one obtains an elegant short proof of the completeness theorem for propositional basic logic, which introduced by H a ´ jek [10]. It was proved that a hoop has the divisibility condition and it is a meet-semilattice, so a bounded Rℓ-monoid can be viewed as a bounded hoop together with the join-semilattice property. Blok introduced the notion of cancellative hoop in [3], and proved that every cancellative hoop is isomorphism to an abelian ℓ-group. She also proved cancellative hoops form a subvariety of the variety of Wajsberg hoops. That is to say, hoops are very important and basic residuated structures in which the community of many-valued logicians got interested in the last years as they are building blocks for several interesting structures being the algebraic semantics for relevant many-valued logics such as BL-algebras, MV-algebras and so on. In a words, hoops contains all algebraic structures that induce by continuous t-norms and their residue. Therefore, hoops are more general algebraic structure and play an important role in studying fuzzy logics and the related algebraic structures.

The filter theory of the hoops plays an important role in studying these algebras and the completeness of the correspondence logic system. Recently, the filters on hoops have been widely studied and some important results have been obtained [3, 11]. In particular, Blok introduced the idea of filters in hoops and investigate some important properties of it [3]. After then, the concepts of prime, maximal and primary filters were defined in hoops in [12]. In [11], Kondo was the first to systematically study filter theory in hoops, in which the relations between kinds of filters were obtained and and some of their characterizations were presented. In the last few years, in order to study the consequence operators and MP rules in Basic fuzzy logic, Borzooei [4] introduced a new type filter in BL-algebras, called an integral filter, and obtained some important result about this filters. After then, Borzooei [5] introduced n-fold integral filters in BL-algebras and discuss the relations between other n-fold filters and n-fold integral filters in BL-algebras. As well known, from a logic point of view, various filters in hoops have natural interpretation as various sets of provable formulas in the correspondence fuzzy logic, so is integral filters. Also, in hoops, filters and deductive systems are equivalent and hence filters offer an algebraic foundation for inference in fuzzy logic.

As we know, cancellative hoops form a subvariety of the variety of Wajsberg hoops, and hence it is a subvariety of the variety of MV-algebras. It is a natural problem that is there a class of hoops, which has similar properties with cancellative hoops but is not a subclass of MV-algebras? In order to study the problem we will introduce a notion of integral hoops, which is a generalization of cancellative hoops and is not MV-algebras. This is one of motivations of this manuscript.

In this paper, we introduced integral hoops and integral filters. We consider fundamental properties of integral hoops and integral filters and give some characterizations of them. Also, we prove that integral hoops are prefect and local hoops. Especially, we proved that every cancellative hoop is an integral hoop but the converse is not true. Finally, we discuss the relation between integral filters and some types of filters (obstinate, primary, prefect filters) of hoops and prove a filter F is an integral filter if and only if L/F is an integral hoop.

2 Preliminaries

In this section, we summarize some definitions and results about hoops which will be used in the following sections.

Definition 2.1. [7] An algebra (L, ⊙, →, 1) of type (2,2,0) is called a hoop if it satisfies the following conditions: for all x, y, z ∈ L,

(L, ⊙, 1) is a commutative monoid,

In what follows, by L we denote the universe of a hoop (L, ⊙, →, 1). For any x ∈ L, we define x⁰ = 1 and x ⁿ = x^n-1 ⊙ x for any natural number n.

On a hoop (L, ⊙, →, 1), we define x ≤ y iff x → y = 1 for all x, y ∈ L. It is easy to check that ≤ is a partial order relation on L and for all x ∈ L, x ≤ 1. Moreover, an algebra (L, ⊙, →, 0, 1) is a bounded hoop if (L, ⊙, →, 1) is a hoop and there exists an element 0 ∈ L such that 0 ≤ x for all x ∈ L. In a bounded hoop (L, ⊙, →, 0, 1), we define the negation ¬ : ¬ x = x → 0 for all x ∈ L. If x ⊙ x = x, that is, x² = x for all x ∈ L, then the hoop L is said to be idempotent. It is easy to check that an idempotent hoop is equivalent to an Brouwerian semilattice [7].

Proposition 2.2. [2, 13] In a hoop (L, ⊙, →, 1), the following properties hold: for any x, y, z ∈ L,

x ∧ y = x ⊙ (x → y),

Proposition 2.3. [9, 13] Let (L, ⊙, →, 0, 1) be a bounded hoop, then the following properties hold: for any x, y, z ∈ L,

¬0 =1, ¬1 =0,

Definition 2.4. [11] A filter of a hoop L is a nonempty subset F of L such that for any x, y ∈ L, we have:

if x, y ∈ F, then x ⊙ y ∈ F,

Let (L, ⊙, →, 1) be a hoop. A non-empty set F of L is a filter if and only if it satisfies x, x → y ∈ F implies y ∈ F, for any x, y ∈ L. A filter of L is called a proper filter if F ≠ L. A proper filter of L is called a prime filter of L, if x ∨ y ∈ F implies x ∈ F or y ∈ F, for any x, y ∈ L. A proper filter F of L is called a maximal filter of L, if it is not properly contained in any other proper filters of L [11, 12].

Definition 2.5. [11, 12] A nonempty subset F of L is called:

An implicative filter of L, if 1 ∈ F and x → (y → z) ∈ F and x → y ∈ F imply that x → z ∈ F, for any x, y, z ∈ L.

Theorem 2.6. [11, 12] Let L be a hoop and F be a filter of L. Then the following hold:

F is a fantastic filter of L if and only if ((x → 0) →0) → x ∈ F, for any x ∈ L,

Definition 2.7. [9] Let L₁ and L₂ be two hoops. A function f : L₁ ⟶ L₂ is called a homomorphism of hoops if and only if:

f (1) =1,

If f is bijective, then the homomorphism f is called hoop isomorphism. In this case we write L₁ ≅ L₂.

Theorem 2.8. [11] Let L be a hoop and F be a filter of L. Then the binary relation ≡ _F which is definedby

x ≡ _F y if and only if x → y ∈ F and y → x ∈ F

is a congruence relation on L. Define ·, ⇀, ⊔, ⊓ on L/F, the set of all congruence classes of L as follows: [x] · [y] = [x ⊙ y], [x] ⇀ [y] = [x → y],

[x] ⊔ [y] = [x ∨ y], [x] ⊓ [y] = [x ∧ y].

Then (L/F, ·, ⇀, ⊔, ⊓, [0], [1]) is a hoop which is called quotient hoop with respect to F.

Theorem 2.9. [12] The order of x ∈ L, in symbols ord(x), is the smallest n ∈ N such that x ⁿ = 0. If no such n exists, then ord(x) =∞.

Theorem 2.10. [12] L is called a prefect hoop if ord(x)< ∞, then ord(¬ x) = ∞, and if ord(x) =∞, then ord(¬ x) < ∞, for any x ∈ L.

Theorem 2.11. [12] L is called a local hoop if ord(x)< ∞ or ord(¬ x) < ∞, for any x ∈ L.

Theorem 2.12. [11] Let L be bounded hoop and F be a filter of L. Then the following conditions are equivalent:

F is a primary filter of L,

Theorem 2.13. [11] Let L be bounded hoop and F be a filter of L. Then the following conditions are equivalent:

F is a prefect filter of L,

Theorem 2.14. [12] A filter F of hoop L is maximal and implicative if and only if F is an obstinate filter.

Theorem 2.15. [12] Let L be a bounded hoop. Then the following conditions are equivalent:

L is a local hoop,

Integral hoops

In this section, we introduce a new class of hoops, called integral hoops and give some characterizations of them. Also, we discuss the relation between integral hoops and other hoops, for example, local and prefect hoops.

Definition 3.1. A bounded hoop L called an integral hoop if x ⊙ y = 0, then x = 0 or y = 0, for any x, y ∈ L.

Example 3.2. Let L = {0, a, b, c, 1} with 0 ≤ a ≤ b, c ≤ 1. Define ⊙ and → as follows: OPEN IN VIEWER

Then (L, ⊙, →, 0, 1) is a bounded hoop. One can easily to check that if x ⊙ y = 0, then x = 0 or y = 0, for any x, y ∈ L and hence L is an integralhoop.

The following example shows that any bounded hoop may not be an integral hoop.

Example 3.3. Let L = [0, 1] be the real interval. If for x, y ∈ L, we define x ⊙ y =max{0, x + y - 1} and x → y =min{1, 1 - x + y}, then (L, ⊙, →, 0, 1) is a bounded hoop, which is not an integral hoop. Since if x = 1/2, y = 1/2, x ⊙ y = 0, butx = y ≠ 0.

Example 3.4. Let L = {0, a, b, 1} with 0 ≤ a ≤ b ≤ 1. Consider the operation ⊙ and → given by the following tables: OPEN IN VIEWER

Then (L, ⊙, →, 0, 1) is a bounded hoop, which is not an integral hoop since a ⊙ b = 0 for a, b ≠ 0.

Theorem 3.5. Let F be a filter of L and L/F be an integral hoop. Then F is a primary filter of L.

Proof. Let ¬ (x ⊙ y) ∈ F, for any x, y ∈ L. Then (x ⊙ y) →0 ∈ F. Since 0 → (x ⊙ y) =1 ∈ F, then [x ⊙ y] = [0] and so [x] · [y] = [x ⊙ y] = [0]. Since L/F be an integral hoop, then [x] = [0] or [y] = [0]. Hence, ¬x = x → 0 ∈ F or ¬y = y → 0 ∈ F and so F is a primary filter of L.

The following example shows that the converse of Theorem 3.5 is not true in general.

Example 3.6. Considering the hoop L in the example 3.4, it is easy to check that F = {b, 1} is a primary filter, but L/F is not an integral hoop. Since a → 0 = b ∈ F and 0 → a = 1 ∈ F, then [a] = [0] and so [b] · [b] = [b ⊙ b] = [a] = [0]. But [b] ≠0. Therefore, L/F is not an integral hoop.

Theorem 3.7. Let L be a integral hoop. Then

L is a local hoop,

Proof. (1) Since {1} is a filter of L, then by Theorem 2.8, L/{1} is well defined. Let f : L ⟶ L/{1} be canonical epimorphism. Now, we will prove that f is a isomorphism, if f (x) = [x] = [y] = f (y) for any x, y ∈ L. Then x → y, y → x ∈ {1} and so x ≤ y and y ≤ x. Hence x = y and so f is isomorphism. Now, since L ≅ L/{1} and L is an integral hoop, then L/{1} is an integral hoop and by Theorem 3.5, {1} is a primary filter of L. Hence, by Theorem 2.12, we have L/{1} is a local hoop and so L is a localhoop.

(2) Let 0 ≠ x ∈ L ∖ M (L). Then there exists minimal element n ∈ N ∪ {0} such that x ⁿ = 0 and x ^m ≠ 0 for any m ≤ n. Now, since x^n-1 ⊙ x = 0 and L is an integral hoop, then x^n-1 = 0 or x = 0, which is impossible. Hence M (L) ⊆ L ∖ {0} ⊆ M (L) and so M (L) = L ∖ {0}.

(3) Since L is an integral hoop, then by (1), L is a local hoop. Hence, by (2) and Theorem 2.15, for all 0 ≠ x ∈ L, ord(x)=∞. Moerover, if x = 0, then ord(¬ x) = ∞. Therefore, L is a perfecthoop.

Theorem 3.8. If F is an obstinate and implicative filter of L. Then L/F is an integral hoops and so F is a primary filter.

Proof. Let F be an obstinate and implicative filter of L and [x] · [y] = [0], for any [x], [y] ∈ L/F. Then by Proposition 2.2 (7) and (8), x → (y → 0) = (x ⊙ y) →0 ∈ F and y → (x → 0) = x → (y → 0) = (x ⊙ y) →0 ∈ F. If x ∈ F, then y → 0 ∈ F and so [y] = [0]. Similarly, if y ∈ F, then x → 0 ∈ F and so [x] = [0]. Now, let x, y ∈ F. Then x ⊙ y ∈ F and so 0 ∈ F, that is impossible. If x, y ∉ F, since F is an obstinate, then x → y ∈ F and y → x ∈ F. Since F is an implicative filter, then x → 0 ∈ F and y → 0 ∈ F and so [x] = [0] and [y] = [0]. Hence L/F is an integral hoops and so by Theorem 3.5, F is a primary filter.

Theorem 3.9. Let L be a bounded hoop and F be a primary filter of L. If [x] · [y] = [0], for any [x], [y] ∈ L/F, then [x] or [y] is nilpotent.

Proof. Let [x] · [y] = [0], for any [x], [y] ∈ L/F. Then ¬ (x ⊙ y) ∈ F. Since F is a primary filter, then ¬ (x ⁿ ) ∈ F or ¬ (y ⁿ ) ∈ F, for some n ∈ N ∪ {0}. Therefore, [x] ⁿ = [0] or [y] ⁿ = [0] and so [x] or [y] is nilpotent.

Theorem 3.10. Let L be an idempotent bounded hoop and F be a filter of L. Then the following conditions are equivalent:

F is a primary filter of L,

Proof. (1) ⇒ (2) Let F be a primary filter of L and [x] · [y] = [0], for any [x], [y] ∈ L/F. Then by Theorem 3.9, there exist n, m ∈ N, such that [x] ⁿ = [0] or [y] ^m = [0] and so [x ⁿ ] = [0] or [y ^m ] = [0]. Since L is an idempotent bounded hoop, then x ⁿ = x and y ^m = y and so [x] = [0] or [y] = [0]. Hence L/F is an integral hoop.

(2) ⇒ (1) If L/F is an integral hoop, then by Theorem 3.5, F is a primary filter of L.

Theorem 3.11. Let L be an idemopent bounded hoop and F be a filter of L. Then the following conditions are equivalent:

L is an integral hoop,

Proof. (1) ⇒ (2) Let L be an integral hoop. Then by Theorem 3.7 (1), L is a local hoop.

(2) ⇒ (1) Let L be a local hoop. Then by L ≅ L/{1}, L/{1} is a local hoop and by Theorem 2.12, {1} is a primary filter. Hence, by Theorem 3.10, L/{1} is an integral hoop and so L is an integralhoop.

In [3], Blok introduced a hoop called cancellative hoop if it satisfies x ⊙ y = x ⊙ z imply y = z for any x, y, z ∈ L.

Theorem 3.12. Every cancellative hoop is an integral hoop.

Proof. Assume that x ⊙ y = 0 for any x, y ∈ L. If x ≠ 0, then x ⊙ y = 0 = x ⊙ 0 and hence x ⊙ y = x ⊙ 0, since L is a cancellative hoop, so y = 0. By the similar way, we can proof that if y ≠ 0, then x = 0. Therefore, L is an integralhoop.

The following example shows that the converse of Theorem 3.12 is not true ingeneral.

Example 3.13. Considering the hoop L in the example 3.2, it is easy to check that L is an integral hoop, but L is not a cancellative hoop. Since a ⊙ b = a = a ⊙ c, but b ≠ c.

The following example shows that integral hoop is not an MV-algebra.

Example 3.14. Considering the hoop L in the example 3.2, it is easy to check that L is an integral hoop, but L is not an MV-algebra. Since ¬¬ a = (a → 0) →0 = 0 →0 = 1 ≠ a.

The following example shows that MV-algebra is not an integral hoop.

Example 3.15. Let L = {0, a, b, c, d, 1}, where 0 ≤ c, d ≤ a ≤ 1, 0 ≤ c ≤ b ≤ 1. Define ⊙ and → as follows: OPEN IN VIEWER

Then (L, ⊙, →, 0, 1) is an MV-algebra, which is not an integral hoop, since a ⊙ c = 0, but a ≠ 0 and c ≠ 0.

Remark: From [3], we know that cancellative hoops form a subvariety of the variety of Wajsberg hoops, and a Wajsberg hoop is equivalent to an MV-algebra. We also proved that every cancellative hoop is an integral hoop. That is to say, we find a new class of hoops, which is not equivalent to class of MV-algebras. The relation between cancellative hoop, MV-algebra and integral hoop as follows:

OPEN IN VIEWER

4 Integral filters in hoops

In this section, we introduce a new class of filters in hoops, called integral filters and give some characterizations of them. Also, we discuss the relation between integral filters and other filters in hoops, for example, primary, obstinate and prefect filters.

Definition 4.1. A proper filter F of a bounded hoop L is called an integral filter, if ¬ (x ⊙ y) ∈ F implies ¬x ∈ F or ¬y ∈ F, for all x, y ∈ L.

Example 4.2. Let L = {0, a, b, c, d, 1}, where 0 ≤ a, b ≤ c ≤ 1, 0 ≤ a, b ≤ d ≤ 1. Define ⊙ and → as follows: OPEN IN VIEWER

Then (L, ⊙, →, 0, 1) is a bounded hoop and F = {1, a, c} is an integral filter of L. Since if ¬ (x ⊙ y) ∈ F, then ¬ (x ⊙ y) = a or c or 1, and so x ⊙ y = b or d or 0. If x ⊙ y = b, then x = b or c and y = d. If x = b, then ¬b = c ∈ F and if x = c, ¬c = b ∉ F, but ¬d = a ∈ F. By the similar way for x ⊙ y = d or 0, we conclude that F is an integral filter.

Theorem 4.3. Every integral filter is a primary filter of L.

Proof. Let n = 1 in Definition 2.5 (4). Then the proof is clear.

The following example shows that the converse of Theorem 4.3 is not true in general.

Example 4.4. Considering the bounded hoop L in the Example 4.2. One can easy to check that F = {1, d} is a primary filter while it is not an integral filter since ¬ (c ⊙ c) = ¬ a = d ∈ F and¬c = b ∉ F.

Theorem 4.5. Let F, G be two filters of L such that F ⊆ G. If F is an integral filter of L, then G is also an integral filter of L.

Proof. Let ¬ (x ⊙ y) ∈ G, for any x, y ∈ L. Then by Proposition 2.3 (2) and (5), ¬ ((x ⊙ y) ⊙ ¬ (x ⊙ y)) = ¬0 = 1 ∈ F. Since F is an integral filter, then we get ¬ (x ⊙ y) ∈ F or ¬¬ (x ⊙ y) ∈ F. If ¬ (x ⊙ y) ∈ F, then ¬x ∈ F ⊆ G or ¬y ∈ F ⊆ G and so G is an integral filter. If ¬¬ (x ⊙ y) ∈ F ⊆ G, then by ¬ (x ⊙ y) ∈ G and by Proposition 2.3 (5), we have ¬¬ (x ⊙ y) ⊙ ¬ (x ⊙ y) =0 ∈ G. Therefore, G = L and so G is an integral filter.

Theorem 4.6. Let L be a bounded hoop and F be a proper filter of L. Then the following conditions are equivalent:

F is an integral filter of L,

Proof. (1) ⇒ (2) let F be an integral filter and [x] · [y] = [0], for any [x], [y] ∈ L/F. Then [x] · [y] ⇀ [0] = [x ⊙ y] ⇀ [0] = [(x ⊙ y) →0] ∈ L/F, that is ¬ (x ⊙ y) ∈ F, and so ¬x ∈ F or ¬y ∈ F. Hence, [x] = [0] or [y] = [0].

(2) ⇒ (1) Let ¬ (x ⊙ y) ∈ F, for any x, y ∈ L. Then [x ⊙ y] = [x] · [y] = [0]. Since L/F is an integral hoop, then [x] = [0] or [y] = [0]. Therefore, ¬x ∈ F or ¬y ∈ F.

Theorem 4.7. Let L be a bounded hoop. Then the following conditions are equivalent:

{1} is an integral filter of L,

Proof. (1) ⇔ (2) By Theorem 4.5, the proof is clear.

(1) ⇒ (3) Since L ≅ L/{1} and {1} is an integral filter, then by Theorem 4.6, L/{1} is an integral hoop and so L is an integral hoop.

(3) ⇒ (1) By Theorem 4.6, the proof is clear.

Theorem 4.8. Let L be a bounded hoop. Then the following conditions are equivalent:

L is an integral hoop,

Proof. (1) ⇒ (2) Let L be an integral hoop. Since 0 → 0 =1 and so 0 ∉ L₀. Hence L₀ ⊆ L \ {0}. Now, if x ∈ L \ {0}, from Proposition 2.3(5), we have x ⊙ ¬ x = 0 and so ¬ (x ⊙ ¬ x) =1. By Theorem 4.7, we have {1} is an integral filter, then ¬x = 1 or ¬¬ x = 1. If ¬x = 1, then x ⊙ ¬ x = x ⊙ 1 = x and so x = 0, that is a contradiction. Hence, ¬¬ x = 1, by Proposition 2.3(3), we have ¬x = ¬¬ ¬ x = 0 and hence x ∈ L₀. Therefore, L₀ = L \ {0}.

(2) ⇒ (1) Let x ⊙ y = 0, for any x, y ∈ L. Then x ⊙ y ∉ L₀ = L \ {0}. Since L₀ is a proper filter, then x ∉ L₀ or y ∉ L₀. Therefore, x = 0 or y = 0 and so L is an integral hoop.

Theorem 4.9. Let L be a bounded hoop and F be an integral filter of L. Then F is a perfect filter of L.

Proof. Let F be an integral filter of L, then by Theorem 4.6, we have L/F is an integral hoop and by Theorem 3.7(3), L/F is a prefect hoop. Hence by Theorem 2.13, we have F is a prefect filter of L.

Theorem 4.10. Let L be a bounded hoop and F be a proper obstinate filter of L. Then F is an integral filter of L.

Proof. Let F be a proper obstinate filter, ¬ (x ⊙ y) ∈ F but ¬x ∉ F and ¬y ∉ F, for any x, y ∈ L. Since F is an obstinate filter and 0 ∉ F, then (¬ x) →0 = ¬¬ x ∈ F. Now, by Theorem 2.6(2), F is a fantastic filter and by Theorem 2.6(1), ¬¬ x → x ∈ F, and so x ∈ F. By the similar way, we can proof that y ∈ F. Hence, x ⊙ y ∈ F and so 0 ∈ F, which is a contradiction. Therefore, ¬x ∈ F or ¬y ∈ F and so F is an integral filter.

The following example shows that the converse of Theorem 4.10 is not correct in general.

Example 4.11. Let L = {0, a, b, c, 1}, where 0 ≤ c ≤ a, b ≤ 1. Define ⊙ and → as follows: OPEN IN VIEWER

Then (L, ⊙, →, 0, 1) is a bounded hoop and it is clear that F = {b, 1} is an integral filter but is not an obstinate filter. Since, a ∉ F and 0 ∉ F, but a → 0 =0 ∉ F.

From the above example, one can see that every integral filter may not be an obstinate filter. However, if the conditions are strengthened, we can obtain that every integral filter and fantastic filter is an obstinate filter.

Theorem 4.12. Let L be a bounded hoop, then the following condition are equivaent:

F is an integral filter and fantastic filter of L,

Proof. (1) ⇒ (2) Let x, y ∉ F, for any x, y ∈ L. We will show that x → y ∈ F and y → x ∈ F. Since x ∉ F, then ¬x ∈ F. Since if ¬x ∉ F, by Proposition 2.3(5), ¬ (x ⊙ ¬ x) =1 ∈ F and since F is an integral filter, then ¬¬ x ∈ F. Now, since F is a fantastic filter, then by Theorem 2.6(1), ¬¬ x → x ∈ F and so x ∈ F, which is a contradiction. Now, by Proposition 2.2 (6), ¬x ≤ x → y and since F is a filter and ¬x ∈ F, we get that x → y ∈ F. By the similar way we can prove that, y → x ∈ F. Therefore, F is an obstinate filter.

(2) ⇒ (1) By Theorem 2.6 and Theorem 4.10, we can get the proof.

5 Conclusions

In this paper, firstly motivated by the problem “is there a class of hoops, which has similar properties with cancellative hoops but is not a subclass of MV-algebras”, we introduce and study integral hoops, which is a generalization of cancellative hoops and is not MV-algebras. Moreover motivated by the previous research of integral filters in BL-algebras, we extended the concept of integral filters to a more generally algebraic structure, i.e. hoops. We have also presented several different characterizations and many important properties of integral filters in hoops. We also discuss the relations between integral hoops and other types of hoops and prove that an integral hoop is a perfect and local hoop. Moreover, we discuss the relations between integral filter and some types of filters in hoops. The work of this manuscript will provide some new field for study of hoops. In our future work, we will consider relations between integral hoops and abelian ℓ-groups.