Characterizations of hoops based on stabilizers

1 Introduction

Various kinds of fuzzy logical algebras as the semantical systems of fuzzy logic systems have been extensively introduced and studied, for example, MV-algebras, BL-algebras, MTL-algebras, residauted lattices and hoops. Among these logical algebras, hoops are the most significant because most of the others are particular cases of hoops such as BL-algebras and MV-algebras. In fact, hoops contain some algebras induced by continuous t-norm and its residua [1]. In the last few years, the theory of hoops has been enriched with deep structure theorems [1, 3]. 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. It is proved that a hoop is a meet-semilattice ordered residuated, integral, divisible and commutative monoid.

The notion of stabilizer is helpful for studying structures and properties in algebraic systems. From a logical point of view, stabilizer can be used in studying the consequence connectives in the correspondence logic system [7]. Since stabilizer was successful in several distinct tasks in various branches of mathematics, it has been extended to various logical algebras, for example, Haveshki was first discussed the relations between stabilizers and filters in BL-algebras [9]; Borzooei showed that the (semi) normal filters and fantastic filters are equal in BL-algebras via stabilizers [4], they also used the right stabilizers produce a basis for a topology on hoops and showed that the generated topology by this basis is Baire, connected, locally connected and separable [5]; Borumand Saeid discussed the relation between stabilizers and some other ideals in MV-algebras and also proved that the lattices of ideals of MV-algebras forms a pseudocomplement lattice via stabilizers [7]; Motamed introduced the notion of RS-BL-algebras via stabilizer and proposed some open problems related to stabilizers [11]; Turunen proved that RS-BL-algebras are equivalent to MV-algebras [15]; the first author [17] characterized some subclasses of MTL-algebras in terms of stabilizers and gave answers to some open problems in [11]. This paper is the continuation of the paper [5], one of our aims is to obtain some characterizations of subclasses of hoops by means of stabilizers. In particular, we will obtain the following main results: (1) every bounded hoop is an MV-algebra if and only if its left and right stabilizers are equivalent (see Theorem 4.1). Indeed, this results essentially generalize the following important results in [15]: every BL-algebra is an MV-algebra if and only if its left and right stabilizers are equivalent; (2) every bounded hoop is a simple Gödel hoop if and only if L _a  = 1 for a ≠ 1 (see Theorem 4.4); (3) every finite bounded stabilizer hoop is equivalent to an MV-algebra (see Theorem 4.10). On the other hand, we further determine the relationship between filters and stabilizers and obtain some improvement results (see Theorems 4.11-4.13).

This paper is structured in four sections. In order to make the paper as self-contained as possible, we recapitulate in Section 2 the definition of hoops, and review their basic properties. In Section 3, we further study left and right stabilizers and give some characterizations of them. In Section 4, we characterize some special classes of hoops in terms of left and right stabilizers and clarify the relationship between filters and stabilizers in hoops.

2 Preliminaries

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

Definition 2.1. [1, 3] An algebraic structure (L, ⊙ , → , 1) of type (2, 2, 0) is called a hoop, if for any x, y, z ∈ L, it satisfies the following conditions: (i) (L, ⊙ , 1) is a commutative monoid, (ii) x → x = 1, (iii) (x ⊙ y) → z = x → (y → z), (iv) x ⊙ (x → y) = y ⊙ (y → x).

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, we define x → y = 1 if and only if x ≤ y 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.

Proposition 2.2. [3, 18] The following properties hold in any hoop L, (i) (L, ∧ , 1) is a ∧-semilattice with upper bounded 1, (ii) x ⊙ y ≤ x ∧ y, (iii) x → (y ∧ z) = (x → y) ∧ (x → z), (iv) x ≤ y → x, for all x, y, z ∈ L.

A hoop L is bounded, if there exists an element 0 ∈ L such that 0 ≤ x for all x ∈ L. In a bounded hoop L, we define the negation ¬ : ¬ x = x → 0 for all x ∈ L. If ¬ (¬ x) = x for all x ∈ L, then L is said to have a double negation property(DNP, for short). A hoop is called a semihoop, if the condition divisibility x ∧ y = x ⊙ (x → y) for any x, y ∈ L, does not hold. 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) =∞ (See [13, 18]).

Proposition 2.3. [3] In any bounded hoop L, the following properties hold: (i) ¬x = ¬¬ ¬ x, (ii) x ⊙ ¬ x = 0, (iii) if ¬¬ x = x, then x → y = ¬ y → ¬ x, (iv) ¬¬ x = x if and only if (x → y) → y = (y → x) → x, for all x, y, z ∈ L.

A nonempty subset F of L is called a filter of L, if it satisfies: (i) x, y ∈ F implies x ⊙ y ∈ F; (ii) x ∈ F, y ∈ L and x ≤ y imply y ∈ F. A filter F of L is called a proper filter if F ≠ L. If X is a nonempty subset of L, then we denote the filter generated by X by 〈X〉. Clearly, we have 〈 X 〉 = { x ∈ L ∣ x 1 ⊙ x 2 ⊙ ⋯ ⊙ x n ≤ x , for some x i ∈ X } = { x ∈ L ∣ x 1 → ( x 2 → ( ⋯ ( x n → x ) ⋯ ) ) = 1 , x i ∈ X } . A filter F of L is called a sub-summand of L, if there exists a filter G of L such that F ∩ G = {1} and L = 〈F ∪ G〉 (See [10, 16]).

Definition 2.4. [3, 18] Let L be a hoop. Then L is called a: (i) Gödel hoop, if x ⊙ x = x, for any x ∈ L, (ii) Wajsberg hoop, if (x → y) → y = (y → x) → x, for any x, y ∈ L, (iii) simple hoop, if it has exactly two filters: {1} and L, (iv) local hoop, if ord (x)< ∞ or ord (¬ x) < ∞, for any x ∈ L.

Definition 2.5. [10] Let L be a hoop and F be a filter of L. Then F is called a (an):

(i) implicative filter, if x → (y → z) ∈ F and x → y ∈ F, then x → z ∈ F, for any x, y, z ∈ L, (ii) positive implicative filter, if x → ((y → z) → y) ∈ F and x ∈ F, then y ∈ F, for any x, y, z ∈ L, (iii) fantastic filter, if z → (y → x) ∈ F and z ∈ F, then ((x → y) → y) → x ∈ F, for any x, y, z ∈ L, (iv) integral filter, if ¬ (x ⊙ y) ∈ F, then ¬x ∈ F or ¬y ∈ F, for any x, y ∈ L.

Theorem 2.6. [10] Let L be a hoop. Then the following conditions are equivalent: (i) {1} is an implicative filter (positive implicative filter, fantastic filter), (ii) every filter of L is an implicative filter (positive implicative filter, fantastic filter), (iii) L is a Gödel hoop (Boolean algebra, Wajsberg hoop).

3 Some properties of stabilizers in hoops

In the section, we further study left and right stabilizers in hoops and discuss the relation between them.

Definition 3.1. [5] Let L be a hoop and X be a nonempty subset of L. The left and right stabilizers of X are defined as follows, respectively, X l = { a ∈ L | a → x = x , for all x ∈ X } , X r = { a ∈ L | x → a = a , for all x ∈ X } . The set X _s  = X _l  ∩ X _r is called the stabilizer of X. For convenience, the stabilizer, left stabilizer and right stabilizer of X = {x} are denoted by S _x , L _x and R _x , respectively.

The following example shows that X _l  ≠ X _r , in general.

Then (L, ⊙ , → , 0, 1) is a bounded hoop. If we put X = {b}, then X _r  = {0, a, 1} ≠ X _l  = {1} and hence X _s  = {1}.

The following theorems, Theorems 3.3 and 3.4, we give a characterization of left and right stabilizer in hoops.

Theorem 3.3. Let L be a hoop, a ∈ L and ∅ ≠ X ⊆ L. Then the following assertions are equivalent: (i) a ∈ X _l , (ii) a ⁿ  → x = x, for any n ∈ ℕ and x ∈ X.

Proof. (i) ⇒ (ii) Let a ∈ X _l . Then a → x = x, for any x ∈ X. So by Definition 2.1(iii), for any n ∈ ℕ and x ∈ X, we have x = a → x = a → ( a → x ) = a → ( a → ( a → x ) ) = ( a ⊙ a ⊙ a ) → x = ⋯ = ( a ⊙ a ⊙ ⋯ ⊙ a ) → x = a n → x .

(ii) ⇒ (i) Let n = 1 in (ii). Then a → x = x, and so a ∈ X _l . □

Theorem 3.4. Let L be a hoop, a ∈ L and ∅ ≠ X ⊆ L. Then the following assertions are equivalent: (i) a ∈ X _r , (ii) x ⁿ  → a = a, for any n ∈ ℕ and x ∈ X.

Proof. The proof is similar to the proof of Theorem 3.3. □

The following proposition provides some useful properties of stabilizers in a bounded hoop.

Proposition 3.5. Let L be a bounded hoop and X, Y be two nonempty subsets of L. Then the following properties hold: (i) X _r  = ∩ _x∈XR _x , X _l  = ∩ _x∈XL _x and X _s  = ∩ _x∈XS _x , (ii) if X ⊆ Y, then Y _r  ⊆ X _r , Y _l  ⊆ X _l , and Y _s  ⊆ X _s , (iii) 〈X〉 _r  = X _r , (iv) X = {1} if and only if X _r  = X _l  = X _s  = L, (v) L _r  = L _l  = L _s  = {1}, (vi) R₀ = {1} and so S₀ = {1}, (vii) if a, b ∈ X _r , then a ∧ b, a → b ∈ X _r , (viii) X _l is a filter of L, (ix) 〈X〉 ∩ X _r  = {1} = 〈X〉 ∩ X _s .

Proof. The proofs of (i) , (ii) , (iv) , (v) , (vi), (viii) and (ix) are easy.

(iii) Since X ⊆ 〈X〉, by (ii), we have 〈X〉 _r  ⊆ X _r . On the other hand, suppose a ∈ X _r . Then x → a = a, for all x ∈ X. For any y ∈ 〈X〉, there exist y₁, y₂ ⋯ y _n  ∈ X such that y₁ ⊙ y₂ ⊙ . . . ⊙ y _n  ≤ y. Thus y → a ≤ ( y 1 ⊙ y 2 ⊙ . . . ⊙ y n ) → a . Since a ∈ X _r , for any y _i  ∈ X, y _i  → a = a, and so y → a ≤ y 1 → ( y 2 → ( . . . y n → a ) . . . ) = a . On the other hand, a ≤ y → a, for all y ∈ 〈X〉. Hence, y → a = a, and so a ∈ 〈X〉 _r , that is, 〈X〉 _r  = X _r .

(vii) For all a, b ∈ X _r and x ∈ X, by Proposition 2.2(iii), we have x → ( a ∧ b ) = ( x → a ) ∧ ( x → b ) = a ∧ b , that is, a ∧ b ∈ X _r . Now, we show that for any a, b ∈ X _r , a → b ∈ X _r . Since a, b ∈ X _r , for any x ∈ X, x → a = a and x → b = b. Then a → b = ( x → a ) → ( x → b ) = x → ( ( x → a ) → b ) = x → ( a → b ) . □

In the following example, we show that Proposition 3.5(iii), may not be true for left stabilizer of a bounded hoop, in general.

Example 3.6. Let L = {0, a, b, c, d, 1} be a set, where 0 ≤ c, d ≤ a, b ≤ 1. Consider the operations ⊙ and → on L as the following tables:

Then (L, ⊙ , → , 0, 1) is a bounded hoop. If X = {c}, then X _l  = {a, 1} and 〈X〉 _l  = {1}. Hence X _l  ≠ 〈X〉 _l .

The following example shows that Proposition 3.5(viii) may not be true for right stabilizer of a bounded hoop, in general.

Example 3.7. Consider the bounded hoop L in Example 3.2. Let X = {b}, then X _r  = {0, a, 1} is not a filter of L.

Let L be a hoop and for any x, y ∈ L, x ∨ y = ( ( x → y ) → y ) ∧ ( ( y → x ) → x ) . If ∨ is the join operation on L, then hoop L is called a ∨-hoop such that (L, ∨ , ∧) is a distributive lattice. If L is a ∨-hoop, then we have (x ∧ y) → z = (x → z) ∨ (y → z), for any x, y, z ∈ L(See [8]). Also, a sub-hoop of a hoop L, is a subset S of L which is closed under two operations ⊙ and →. It means that, for any x, y ∈ S, x ⊙ y ∈ S and x → y ∈ S.

Proposition 3.8. Let L be a ∨-hoop and ∅ ≠ X ⊆ L. Then X _l is a sub-hoop of L.

Proof. Let a, b ∈ X _l . Then, for any x ∈ X, a → x = b → x = x. Thus, we have ( a ⊙ b ) → x = a → ( b → x ) = a → x = x , and ( a → b ) → x = ( a → b ) → ( a → x ) = ( a ⊙ ( a → b ) ) → x = ( a ∧ b ) → x = ( a → x ) ∨ ( b → x ) = x ∨ x = x . Hence a ⊙ b ∈ X _l and a → b ∈∈ X _l . Therefore X _l is a sub-hoop of L. □

Example 3.9. Consider the bounded hoop L in Example 3.6. By routine calculations, it is easy to see that L is not a ∨-hoop. Now, let X = {a}, then X _l  = {b, 1}. Since b ⊙ b = c ∉ X _l . Hence, X _l is not a sub-hoop of L.

In the following example, we show that the set X _r  ∪ {0} is not a sub-hoop of a bounded hoop, in general.

Then (L, ⊙ , → , 0, 1) is a bounded hoop. Let X = {b}, then X _r  ∪ {0} = {0, a, 1} is not a sub-hoop of L since a → 0 = b ∉ X _r  ∪ {0}.

Remark 3.11 It is well known that bounded Gödel hoops are exactly the {∧ , → , 0, 1}-subreducts of Gödel algebras, and so the set X _r  ∪ {0} is not a subalgebra of a Gödel algebra, since it is not closed under residuated implication.

The following example shows that the condition “〈X〉” in the Proposition 3.5(ix), is necessary.

Example 3.12. Consider the bounded hoop L in Example 3.2. If X = {b}, then X _r  = {0, a, 1}, and hence ∅ = X ∩ X _r  = X ∩ X _s  ≠ {1}.

The following theorem gives a stronger version of Proposition 3.5(ix).

Theorem 3.13. Let L be a hoop and F, G be two filters of L. Then the following assertions are equivalent: (i) F ∩ G = {1}, (ii) F ⊆ G _s , (iii) F ⊆ G _r , (iv) F ⊆ G _l .

Proof.(i) ⇒ (ii) If x ∈ F, then (x → y) → y ∈ F ∩ G = {1}, for any y ∈ G. Hence x → y = y for any y ∈ G, that is, x ∈ G _l . Similarity, we can obtain that (y → x) → x ∈ F ∩ G = {1}, that is, y → x = x, for all y ∈ G. Then x ∈ G _r , and hence x ∈ G _s . Therefore, F ⊆ G _s . (ii) ⇒ (i) If x ∈ F ∩ G, then 1 = x → x = x. Thus, F ∩ G = {1}. (i) ⇒ (iii) If F ∩ G = {1}, then by (ii), we have F ⊆ G _s . Hence F ⊆ G _r . (iii) ⇒ (i) Let x ∈ F ∩ G. Since F ⊆ G _r , we get y → x = x, for any y ∈ G. Thus 1 = x → x = x. Hence F ∩ G = {1}. (i) ⇔ (iv) The proof is similar to the proof of (i) ⇔ (iii). □

The following example shows that the condition F and G are two filters of L in the above theorem is necessary.

Then (L, ⊙ , → , 0, 1) is a bounded hoop. Let F = {a, c, 1} and G = {b, 1}. It is clear that F is a filter of L and G is not a filter of L because b ⊙ b = 0 ∉ G. Also, F ∩ G = {1} but G ⊈ F _s  = {1}.

The following theorem shows that stabilizer X _s is equivalent to co-annihilator ⊥ X = { a ∈ L | a ∨ x = 1 , for all x ∈ X } , which was introduced in [14].

Theorem 3.15. Let L be a bounded Wajsberg hoop and X be a nonempty subset of L. Then (i) ^⊥X = X _r  ∩ X _l  = X _s , (ii) X _l  = X _r .

Proof. (i) Let a ∈ L such that a ∨ x = 1, for all x ∈ X. Since L is a bounded Wajsberg hoop, we have (x → y) → y = (y → x) → x, and so x ∨ y = (x → y) → y. Then 1 = x ∨ a = ( x → a ) → a = ( a → x ) → x . Suppose that (x → a) → a = 1, then by Proposition 2.2(iv), a ≤ x → a, and so a → (x → a) =1. Thus x → a = a, for all x ∈ X. Hence a ∈ X _r . By the similar way, we can see that a → x = x, for all x ∈ X, and hence a ∈ X _l . Therefore, a ∈ X _r  ∩ X _l . Conversely, if a ∈ X _r  ∩ X _l , then x → a = a and a → x = x, for all x ∈ X. Thus (x → a) → a = 1 and (a → x) → x = 1, for all x ∈ X and so a ∨ x = ( ( x → a ) → a ) ∧ ( ( a → x ) → x ) = 1 , that is, a ∈ ^⊥X. Therefore,^⊥X = X _r  ∩ X _l  = X _s .

(ii) We note that a bounded Wajsberg hoop satisfies ( x → y ) → y = ( y → x ) → x , for all x, y ∈ L. Now, let a ∈ X _l , then a → x = x, for all x ∈ X and so (a → x) → x = 1. On one hand, since L is a Wajsberg hoop, so (x → a) → a = 1. On the other hand, from Proposition 2.2(iv), we have a ≤ x → a. Thus x → a = a, and so a ∈ X _r . Hence X _l  ⊆ X _r . By the similar way, we have X _r  ⊆ X _l . Therefore, X _l  = X _r . □

4 Characterizations of subclasses of hoops based on stabilizers

In this section, we will characterize some special classes of hoops, for example, Wajsberg hoops, Gödel hoops and local hoops, in terms of stabilizers. Also, we further determine the relationship between stabilizers and filters in hoops and obtain some improvement results of [5, Theorem 3.9].

Theorem 4.1. Let L be a bounded Wajsberg hoop and ∅ ≠ X ⊆ L and x, y ∈ L. Then the following assertions hold: (i) X _r  = X _l  = X _s , (ii) X _l  = 〈X〉 _l , (iii) x → y = y if and only if y → x = x, (iv) X _r is a filter of L, (v) S _x  = R _x  = L _x , (vi) L₀ = S₀ = ^⊥ {0}, (vii) if x → y, y → x ∈ L₀, then x = y.

Proof. (i) It follows from Theorem 3.15.

(ii) It follows from (i) and Proposition 3.5(iii).

(iii) Let x, y ∈ L such that x → y = y. Then x ∈ L _y and by (ii) we have x ∈ 〈y〉 _l . Since (y → x) → x ∈ 〈y〉, we have x ∈ L_(y→x)→x and 1 = ( y → x ) → ( x → x ) = x → [ ( y → x ) → x ] = ( y → x ) → x and finally y → x = x. Similarly, we can prove that y → x = x implies x → y = y for any x, y ∈ L.

(iv) From (iii), we have R x = { a ∈ L | x → a = a } = { a ∈ L | a → x = x } = L x for any x ∈ L. From Proposition 3.5(viii), we know that L _x is a filter of L, then R _x is a filter of L, for any x ∈ L.

(v) By (iv), for any x ∈ L, R _x is a filter of L. Let a ∈ R _x . Since a ≤ (a → x) → x, we obtain (a → x) → x ∈ R _x . Also, x ≤ (a → x) → x, for x ∈ 〈x〉, and so (a → x) → x ∈ R _x  ∩ 〈x〉 = {1}, by Proposition 3.5(ix). Therefore, a → x = x, for all x ∈ X, that is, a ∈ R _x  ∩ L _x . Hence a ∈ S _x , which implies R _x  ⊆ S _x . On the other hand, we have S _x  ⊆ R _x , thus S _x  = R _x  = L _x .

(vi) Taking X = {0} in (v).

(vii) Let x → y, y → x ∈ L₀. Since L₀ = ^⊥ {0} = {1}, we have x → y = y → x = 1, and so x = y.

□

Corollary 4.2. It is well known that hoops are a class of residuated structure with the divisibility, in this case, the double negation property is equivalent to the (x → y) → y = (y → x) → x. Using this result, we can prove that every bounded hoop with (DNP) is a bounded Wajsberg hoop, which is equivalent to MV-algebra, if and only if X _l  = X _r for any nonempty subset X of L from Theorem 4.1. As Rℓ-monoinds and BL-algebras are subclasses of hoops, we have that (i) every Rℓ-monoind is an MV-algebra if and only if its left stabilizers are equivalent to right stabilizers, (ii) every BL-algebra is an MV-algebra if and only if its left stabilizers are equivalent to right stabilizers.

In the following example, we show that there exists a bounded semihoop such that X _l  = X _r , for any nonempty subset X of L, that is not an MV-algebra.

Routine calculation shows that L with these operations is a bounded semihoop. And it is easy to see that X _l  = X _r , for any nonempty subset of L such as X. But L is not an MV-algebra.

Theorem 4.4. Let L be a Gödel hoop. Then the following assertions are equivalent: (i) L is simple, (ii) L _a  = 1, for any a ≠ 1.

Proof.(i) ⇒ (ii) If L is simple, then for some a ≠ 1 such that L _a  ≠ 1, that is, there exists x ≠ 1 such that x ∈ L _a . So, we have x → a = a. By the simplicity of L, we get <x > = L and hence for some n ≥ 1, x ⁿ  → a = 1 as a ∈ L. On the other hand, since L is a Gödel hoop, it follows from Definition 2.1(iii), that 1 = x n → a = x → ( x n - 1 → a ) = ( x ⊙ x ) n - 1 → a = x n - 1 → a = ⋯ = x → a = a . This shows a = 1, which is a contradiction. Therefore L _a  = 1 for any a ≠ 1. (ii) ⇒ (i) Suppose L _a  = 1 for all a ≠ 1 in L. To prove that L is simple, it suffice to show that <y > = L for any y ≠ 1. Now if x ∈ L, x ≠ 1 such that y ⁿ  → x ≠ 1, then by hypothesis y → (y ⁿ  → x) ≠ y ⁿ  → x. That is yⁿ⁺¹ → x ≠ y ⁿ  → x which contradict the fact that L is a Gödel hoop. This indicates y ⁿ  → x = 1 and hence x∈ < y >. Therefore <y > = L for any y ≠ 1. □

Theorem 4.5. Let L be a bounded Gödel hoop. Then the following assertions are equivalent: (i) L is a local hoop, (ii) L is an integral hoop, (iii) L₀ = L ∖ 0.

Proof. (i) ⇔ (ii) It follows from [16, Theorem 3.11].

(ii) ⇒ (iii) Let L be an integral hoop. Since 0 → 0 =1, we get 0 ∉ L₀. Hence L₀ ⊆ L ∖ {0}. Now, if x ∈ L ∖ {0}, then by Proposition 2.3(i), we have x ⊙ ¬ x = 0 and so ¬ (x ⊙ ¬ x) =1. By Theorem 2.5, {1} is an integral filter, then ¬x = 1 or ¬¬ x = 1. If ¬x = 1, then x ⊙ ¬ x = x ⊙ 1 = x, and so x = 0, which is a contradiction. Hence, ¬¬ x = 1, by Proposition 2.2(i), we have ¬x = ¬¬ ¬ x = 0 and hence x ∈ L₀. Therefore, L₀ = L ∖ {0}. (iii) ⇒ (ii) Let x ⊙ y = 0, for any x, y ∈ L. Then (x ⊙ y) ∉ L₀ = L ∖ {0}. Since L₀ is a proper filter of L, we get that x ∉ L₀ or y ∉ L₀. Therefore, x = 0 or y = 0, and so L is an integral hoop. □

Definition 4.6. A hoop L is called a stabilizer hoop, if R _a is a sub-summand of L, for any a ∈ L.

Example 4.7. Let L be the hoop as in Example 3.6. It is easy to see that L is a stabilizer hoop.

Proposition 4.8. Every bounded stabilizer hoop is a bounded Wajsberg hoop.

Proof. If L is a bounded stabilizer hoop and R _a is a sub-summand of L, then R _a is a filter of L. Thus, L is a bounded Wajsberg hoop. □

In the following example we show that there exists an MV-algebra that is not a bounded stabilizer hoop.

Then (L, ⊙ , → , 0, 1) is an MV-algebra. Since R _b  = {a, 1}, and a ⊙ a = 0, it is clear that R _b is not a filter of L, and so L is not a bounded stabilizer hoop.

The following theorem shows that every finite bounded stabilizer hoop is equivalent to MV-algebra.

Theorem 4.10. Let L be a finite bounded hoop. Then the following assertions are equivalent: (i) L is a bounded stabilizer hoop, (ii) L is a bounded Wajsberg hoop, (iii) 〈a〉 is a sub-summand of L, for any a ∈ L, (iv) R _a is a sub-summand of L, for any a ∈ L.

Proof. (i) ⇒ (ii) It follows from Proposition 4.8.

(ii) ⇒ (iii) It is easy to see that a → x ≤ a 2 → x ≤ ⋯ ≤ a m → x ≤ ⋯ , for any x, a ∈ L and m ∈ ℕ . Since L is a bounded finite hoop, we obtain that there exists n ∈ ℕ such that a n → x = a n + 1 → x = a → ( a n → x ) . Thus a ⁿ  → x ∈ R _a . Moreover, we have x ≥ a ⁿ  ⊙ (a ⁿ  → x). So x ∈ 〈〈a〉 ∪ R _a 〉. Hence L = 〈〈a〉 ∪ R _a 〉. Also, by Proposition 3.5(ix), we have 〈a〉 ∩ R _a  = {1}. Thus, 〈a〉 is a sub-summand of L. (iii) ⇒ (iv) If 〈a〉 is a sub-summand of L, then there exists a filter F such that L = 〈〈a〉 ∪ F〉 and 〈a〉 ∩ F = {1}. If 〈a〉 ∩ F = {1}, then it is easy to check that F ⊆ 〈a〉 _r , and hence F ⊆ R _a . On the other hand, if x ∈ R _a , then x ∈ L = 〈〈a〉 ∪ F〉, thus x ≥ a ⁿ  ⊙ y, for some n ∈ ℕ , y ∈ F. So y ≤ a ⁿ  → x and hence a ⁿ  → x ∈ F. Since x ∈ R _a , a → x = x, and so a ⁿ  → x = x. Hence x ∈ F, that is, F = R _a . Therefore R _a is a sub-summand of L, for any a ∈ L. (iv) ⇒ (i) By Definition 4.6, the proof is clear. □

In the following, we discuss the relations between the stabilizers and types of filters in hoops. This results essentially improve the results of [11, 3.9].

Theorem 4.11. Let L be a hoop and X be a nonempty subset of L. Then the following assertions are equivalent: (i) L is a Gödel hoop, (ii) for all nonempty subset X of L, X _l is an implicative filter of L.

Proof. (i) ⇒ (ii) It follows from [5, Theorem 3.9(i)]. (ii) ⇒ (i) If X _l is an implicative filter of L, for any nonempty subset X of L, then by Proposition 3.5(v), we obtain that {1} is an implicative filter of L, thus by Theorem 2.6, L is a Gödel hoop. □

Theorem 4.12. Let L be a hoop and X be a nonempty subset of L. Then the following assertions are equivalent: (i) L is a Wajsberg hoop, (ii) for all nonempty subset X of L, X _l is a fantastic filter of L,

(iii) for all nonempty subset X of L, X _r is a fantastic filter of L.

Proof. It follows from [5, Theorem 3.9(ii)], Theorems 2.6 and 3.15. □

Theorem 4.13. Let L be a hoop and X be a nonempty subset of L. Then the following assertions are equivalent: (i) L is a Boolean algebra, (ii) for all nonempty subset X of L, X _l is a positive implicative filter of L, (iii) for all nonempty subset X of L, X _r is a positive implicative filter of L.

Proof. It follows from [5, Theorem 3.9(iii)], Theorems 2.6 and 3.15. □

5 Conclusions

The aim of this paper is to develop the stabilizer of hoops. In the paper, some useful properties of particular stabilizers are discussed. And, we characterize some special class of hoops, for example, Wajsberg hoops, local hoops, Gödel hoops, via these stabilizers. Finally, we discuss the relationship between stabilizers and filters. These results will provide a more general algebraic foundation for consequence operators in fuzzy logic based on continuous t-norms. In our future work, we will consider some applications of stabilizers in fuzzy logic and its corresponding algebraic system.

Compliance with ethical standards:

Funding: This study for the first author, was funded by a grant of National Natural Science Foundation of China (61976244,11961016), the Natural Science Basic Research Plan in Shaanxi Province of China (2020JQ-762) and the Natural Science Foundation of Education Committee of Shannxi Province (20JK0626).

Conflict of interest:

Authors declare that they have no conflict of interest.

Ethical approval:

This article does not contain any studies with animals performed by any of the authors.