Stabilizers in MTL-algebras

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 kinds of fuzzy logical algebras as the semantical systems of fuzzy logic systems have been extensively introduced and studied, for example, MV-algebras [3], BL-algebras [7], MTL-algebras [4], NM-algebras [4]. Among these logical algebras, MTL-algebras are the most significant because the others are all particular cases of MTL-algebras. As a more general residuated structure based continuous t-norm and its residua, an MTL-algebra is a BL-algebra without the divisibility. In fact, MTL-algebras contain all algebras induced by left-continuous t-norm and its residua. Therefore, MTL-algebras play an important role in studying fuzzy logics and their related structures.

The notion of stabilizers 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, which has a very close relationship with fuzzy set and rough set [12, 17–19]. Since stabilizer was successful in several distinct tasks in various branches of mathematics [14], it has been extended to various logical algebras, for example, Haveshki was first introduced the stabilizers in BL-algebras and investigate some basic properties of them. Also, they discussed the relations between stabilizers and filters in BL-algebras in [9]. Inspired by this, Borzooei introduced some new types of stabilizers and determined the relations among stabilizers in BL-algebras, they also show that the (semi) normal filters and fantastic filters are equal in BL-algebras via stabilizers in [1]. Besides, Saeid introduced two kinds of stabilizers and discussed the relation between stabilizers and some other ideals in MV-algebras, they also prove that the lattices of ideals of MV-algebras forms a pseudocomplement lattice via stabilizers in [5]. Recently, Motamed has introduced the notion of right stabilizers in BL-algebras and two class of BL-algebras, called RS-BL-algebras and semi RS-BL-algebras, and has discussed the relations between them and (semi)local BL-algebras, they also proposed some open problems related to stabilizers in [11], for example, “Let X be a nonempty subset of a BL-algebra L. Is X _r ∪ {0} a subalgebra of L?” and “If L is a RS-BL-algebra and F is an any filter of L, then (F _r ) _r = F?”. After then, Turunen proved that RS-BL-algebras are equivalent to MV-algebras in [16].

In this paper, we will study stabilizers on MTL-algebras. One of our aims is to give answers to serval open problems related to stabilizers in BL-algebras in [11]. On the other hand, the main focus of existing research about stabilizers on MV-algebras, BL-algebras, etc. All the above-mentioned algebraic structures satisfy the divisibility condition x ∧ y = x ⊙ (x → y). In this case, the conjunction ⊙ on the unit interval corresponds to a continuous t-norm. However, there are few research about the stabilizer on residuated structures without the divisibility condition so far. Therefore, it is meaningful to study stabilizers in MTL-algebras for providing a solid algebraic foundation for consequence operations in MTL logic. This is the motivation for us to investigate stabilizer theory on MTL-algebras.

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 MTL-algebras, and review their basic properties. In Section 3, we introduce implicative stabilizers and characterize some special classes of MTL-algebras in terms of these stabilizers. In Section 4, we introduce multiplicative stabilizers and investigate related properties of them. Using multiplicative stabilizers, we give some characterizations of Gödel algebras and linearly order Gödel algebras. Finally, we discuss the relations between implicative stabilizers and multiplicative stabilizers.

2 Preliminaries

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

Definition 2.1. [4] An algebraic structure (L, ∧, ∨, ⊙, →, 0, 1) of type (2, 2, 2, 2, 0, 0) is called an MTL-algebra if it satisfies the following conditions:

(L, ∧, ∨, 0, 1) is a bounded lattice,

In what follows, by L we denote the universe of an MTL-algebra (L, ∧, ∨, ⊙, →, 0, 1). For any x ∈ L and a natural number n, we define ¬x = x → 0, ¬¬ x = ¬ (¬ x), x⁰ = 1 and x ⁿ = x^n-1 ⊙ x for all n ≥ 1.

Proposition 2.2. [13] In any MTL-algebra L, the following properties hold:

x ≤ y if and only if x → y = 1,

Definition 2.3. [13] Let L be an MTL-algebra. Then L is called:

a BL-algebra if x ∧ y = x ⊙ (x → y) for any x, y ∈ L,

A nonempty subset F of L is called a filter of L if it satisfies: (1) x, y ∈ F implies x ⊙ y ∈ F; (2) x ∈ F, y ∈ L and x ≤ y implies y ∈ F. We denote by F [L] the set of all filers of L. A filter F of L is called a proper filter if F ≠ L. A proper filter F of L is called a prime filter if for each x, y ∈ L, x ∨ y ∈ F, implies x ∈ F or y ∈ F. 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 ≥ x₁ ⊙ x₂ ⊙ ⋯ ⊙ x _n , x _i ∈ X} = {x ∈ L|x₁ → (x₂ → (⋯ (x _n → x) ⋯)) =1, x _i ∈ X}, see [2, 8].

If (L, ∧, ∨, →, ⊙, 0, 1) is an MTL-algebra, we denote by G (L) the set of all idempotent elements of (L, ⊙, 1). The set G (L) is the universe of a Gödel subalgebra of L, which is called the Gödel center of L [10].

Proposition 2.4. [10] Let L be an MTL-algebra. For every x, y ∈ L and e ∈ G (L), we have:

e ⊙ e = e,

A nonempty subset I of an MTL-algebra (L, ∧, ∨, →, ⊙, 0, 1) is called a lattice ideal of L if it satisfies: (i) for all x, y ∈ I, x ∨ y ∈ I; (ii) for all x, y ∈ L, if x ∈ I and y ≤ x, then y ∈ I. That is, a lattice ideal of an MTL-algebra L is the notion of ideal in the underlying lattice (L, ∧, ∨). A lattice ideal I of L is called to be prime if it satisfies for all x, y ∈ L, x ∧ y ∈ I implies x ∈ I or y ∈ I. For any nonempty subset H of L, the smallest lattice ideal containing H is called the lattice ideal generated by H. The lattice ideal generated by H will be denoted by (H]. In particular, if H = {t}, we write (t] for ({t}], (t] is called a principal lattice ideal of L. It is easy to check that (t] = ↓ t = {x ∈ L|x ≤ t} [6].

3 Implicative stabilizers in MTL-algebras

In the section, we investigate left and right implicative stabilizers and discuss the relation between them. Then, we give some characterizations of IMTL-algebras, integral MTL-algebras and MV-algebras via implicative stabilizers.

Definition 3.1. Let L be an MTL-algebra and X be a nonempty subset of L. The left and right implicative stabilizer 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 implicative stabilizer of X. For convenience, the implicative stabilizer, left implicative stabilizer and right implicative stabilizer of X = {x} are denoted by S _x , L _x and R _x , respectively.

The following example shows that X _r ≠ X _l and X _r is not a filter of an MTL-algebra, in general.

Then (L, ∧, ∨, ⊙, →, 0, 1) is an MTL-algebra. If we put X = {b}, then X _l = {1} ≠ X _r = {a, 1} and hence X _s = {1}, in this case, X _r is not a filter of L.

Example 3.3. Let S 3 = { 0 , 1 2 , 1 } . For any x, y ∈ L, we define x ∧ y =min{x, y},x ∨ y =max{x, y}, ¬x = 1 - x, x → y = 1 if x ≤ y; otherwise x → y = ¬ x ∨ y, and x ⊙ y = 0 if x ≤ ¬ y; otherwise x ∧ y. Then (S₃, ∧, ∨, ⊙, →, 0, 1) is an MTL-algebra. Consider the MTL-algebra L = S₃ × S₃ is a product of S₃ and S₃. Then R_(1,0) = {(0, 0), ( 1 2 , 0), (1, 0), (1, 1)}, L_(1,0) and L_{1,0} = {(0, 1), ( 1 2 , 1), (1, 1), (1, 0)}. Thus, S_{1,0} = {(1, 1)}.

The following proposition provides some useful properties of implicative stabilizers in an MTL-algebra.

Proposition 3.4. Let L be an MTL-algebra and X, Y be two nonempty subsets of L. Then the following properties hold:

X _r = ∩ _x∈XR _x , X _l = ∩ _x∈XL _x and X _s = ∩ _x∈XS _x ,

Proof. The proofs of (1), (2), (4), (5), (6), (8) and (9) are easy.

(3) Since X ⊆ 〈X〉, by (2), we have 〈X〉 _r ⊆ X _r . On the other hand, suppose that a ∈ X _r , and so 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)) =1, hence y → a ≤ y₁ → (y₂ → ⋯ y _n ) → a. Thus, a = y₁ → a = y₁ → (y₂ → a) ⋯ y₁ → (y₂ → y₃ → ⋯ (y _n → a))) ≤ y → a. On the other hand, we have y → a ≤ a for all y ∈ 〈X〉, and so a ∈ 〈X〉 _r , that is, 〈X〉 _r = X _r .

(7) For all a, b ∈ X _r , from Proposition 2.2(3), we have x → (a ∧ b) = (x → a) ∧ (x → b) = a ∧ b, that is, a ∧ b ∈ X _r . Now, we will show that for any a, b ∈ X _r , a → b ∈ X _r . Since a → b = (x → a) → (x → b) = x → ((x → a) → b) = x → (a → b) and hence a → b ∈ X _r . Also, from Proposition 2.2(8), one can prove that a ∨ b ∈ X _r , for any a, b ∈ X _r .

The following example shows X _l ≠ 〈X〉 _l , for any nonempty subset X of L, not holds, in general.

Then (L, ∧, ∨, ⊙, →, 0, 1) is an MTL-algebra. If X = {b}, then X _l = {a, 1} and 〈X〉 _l = {1}. Hence X _l ≠ 〈X〉 _l .

The following theorem shows X _l = 〈X〉 _l , for any nonempty subset X, under certain conditions in an MTL-algebra.

Theorem 3.6. Let L be an MTL-algebra. Then the following conditions are equivalent:

X _l = 〈X〉 _l , for any nonempty subset X of L,

Proof. (1) ⇒ (2) Let a, b ∈ L be such that a → b = b, that is, a ∈ L _b . Then we have a ∈ 〈b〉 _l by (1). Since ((b → a) → a ∈ 〈b〉, we have a ∈ L_(b→a)→a and 1 = (b → a) → (a → a) = a → [(b → a) → a] = (b → a) → a, and finally b → a = a. Similarly, we can prove that b → a = a implies a → b = a for any a, b ∈ L.

(2) ⇒ (3) From (2), we have R _x = {a ∈ L|x → a = a} = {a ∈ L|a → x = x} = L _x for any x ∈ L. From Proposition 3.4(8), we know that L _x is a filter of L and hence R _x is a filter of L, for any x ∈ L.

(3) ⇒ (4) By hypothesis, for any x ∈ L, R _x is a filter of L. Let a ∈ R _x . Since a ≤ (a → x) → x and hence (a → x) → x ∈ R _x . Also, x ≤ (a → x) → x, for x ∈ 〈x〉, and so (a → x) → x ∈ R _x ∩ 〈x〉 = {1}, by Proposition 3.4(9). Therefore, a → x = x, for all x ∈ X, that is, a ∈ R _x ∩ L _x = 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 .

(4) ⇔ (5) It follows from Proposition 3.4(1).

(5) ⇒ (1) It follows from (5) and Proposition 3.4(3).

From Definition 2.3(3), we note that a Gödel algebra is an MTL-algebra satisfies x ⊙ y = x ∧ y, for all x, y ∈ L. Applying Proposition 3.4(7), one can obtain that the set X _r is closed under the operations ∧, ∨, →. Now, we have the following natural questions:

For any nonempty subset X of a Gödel algebra L, is X _r ∪ {0} a subalgebra of L?

For the first question, we have a negative answer as the following example shows:

Then (L, ∧, ∨, ⊙, →, 0, 1) is a Gödel algebra. Let X = {b}, then X _r ∪ {0} = {0, a, 1} is not a subalgebra of L since a → 0 = d ∉ X _r ∪ {0}.

Remark 3.8. It was described that in [11] that if L is a Gödel algebra and X is a nonempty subset of L, then X _r ∪ {0} is a subalgebra of L. From Example 3.7, we know that the above statement is not true. Moreover, Example 3.7 also gives a negative answer to the open problem in [11] that “Let X be a nonempty subset of a BL-algebra L. Is X _r ∪ {0} a subalgebra of L?" since a Gödel algebra is a subclass of aBL-algebra.

As to the second question, we have the following proposition.

Proposition 3.9. Let L be a Gödel algebra and X a nonempty subset of L. Then (L₀) _r is a subalgebra of L.

Proof. For any x ∈ L₀, we have x → 1 =1 and x → 0 =0 and so 0, 1 ∈ (L₀) _r . By Proposition 3.4(7), we have (L₀) _r is closed under the operations ∧, ∨, →. Therefore, (L₀) _r is a subalgebra of L.

From the Proposition 3.4(8), it is natural to ask that whether there exists a nonempty subset X such that X _l = F for given filter F in an MTL-algebra L. For this question, we give the positive answer under some suitable conditions by Theorem 3.6 if F = 〈X〉 _l . Furthermore, we have the first open problem:

For any filter F of a MTL-algebra L, whether there exists a nonempty subset X such that X _l = F?

The following theorem shows that implicative stabilizer X _s is equivalent to ^⊥X = {a ∈ L|a ∨ x = 1, for all x ∈ X}, which was introduced in [15].

Theorem 3.10 Let L be an MTL-algebra and X a nonempty subset of L. Then ^⊥X = X _r ∩ X _l = X _s .

Proof. Let a ∈ L such that a ∨ x = 1, for all x ∈ X. By Proposition 2.2(8), we have 1 = x ∨ a = ((x → a) → a) ∧ ((a → x) → x), hence (x → a) → a = 1 and (a → x) → x = 1. Suppose that (x → a) → a = 1, then by Proposition 2.2(9), a ≤ a → x, and so a → (x → a) =1. Thus, x → a = a, for all x ∈ X. Therefore, a ∈ X _r . In 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 we have 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 hence a ∨ x = ((x → a) → a) ∧ ((a → x) → x) =1, that is, a ∈ ^⊥X. Therefore, ^⊥X = X _r ∩ X _l = X _s .

In the following theorems, we give some characterizations of IMTL-algebras and integral MTL-algebras via implicative stabilizers.

Theorem 3.11 Let L be an MTL-algebra. Then the following assertions are equivalent:

L is an IMTL-algebra,

Proof. (1) ⇒ (2) Let L be an IMTL-algebra and a ∈ L₀. Then a → 0 =0 and so a = ¬¬ a = (a → 0) →0 = 0 →0 = 1. Thus, L₀ = {1}. Furthermore, from Proposition 3.4(6) and Theorem 3.10, we have L₀ = S₀ = ^⊥ {0}.

(2) ⇒ (1) If L₀ = S₀ = ^⊥ {0}, then L₀ = {1}. Now, for any x ∈ L, we have x → ¬¬ x = x → ((x → 0) →0) = (x → 0) → (x → 0) =1, and so ¬ (¬¬ x → x) =0, then (¬¬ x → x) →0 = 0. Thus, ¬¬ x → x ∈ L₀ = {1}. Therefore, ¬¬ x → x = 1 and so ¬¬ x = x for any x ∈ L. Thus, L is an IMTL-algebra.

(2) ⇒ (3) Let x → y, y → x ∈ L₀. Since L₀ = ^⊥ {0} = {1}, then x → y = y → x = 1 and so x = y.

(3) ⇒ (2) Let x ∈ L₀. Since 1 ∈ L₀, we have x → 1 =1 ∈ L₀ and 1 → x = x ∈ L₀, then by (3), we obtain that x = 1. Hence L₀ = {1}. Therefore, L₀ = S₀ = ^⊥ {0}.

Theorem 3.12. Let L be an MTL-algebra. Then the following assertions are equivalent:

L is an integral MTL-algebra,

Proof. (1) ⇒ (2) Let L be an integral MTL-algebra. Since 0 → 0 =1 and so 0 ∉ L₀. Hence L₀ ⊆ L ∖ {0}. Now, if x ∈ L ∖ {0}, from Proposition 2.2(2), we have x ⊙ ¬ x = 0 and so ¬ (x ⊙ ¬ x) =1. By Theorem 2.5, we have that {1} is an integral filter, then ¬x = 1 or ¬¬ x = 1. If ¬x = 1, the x ⊙ ¬ x = x ⊙ 1 = x and so x = 0, that is a contradiction. Hence, ¬¬ x = 1, by Proposition 2.2(10), 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 MTL-algebra.

Theorem 3.11(3.12) suggests a method of how to check an MTL-algebra is an IMTL-algebra (integral MTL-algebra). In what follows, we give some examples to show the applications of Theorem 3.11(3.12).

Then (L, ∧, ∨, ⊙, →, 0, 1) is an MTL-algebra. One can check that L₀ = S₀ = {1} = ^⊥ {0}, by Theorem 3.11, we know that (L, ∧, ∨, ⊙, →, 0, 1) is an IMTL-algebra.

Then (L, ∧, ∨, ⊙, →, 0, 1) is an MTL-algebra. One can check that L₀ = {c, a, b, 1} = L ∖ {0}, by Theorem 3.12, we know that (L, ∧, ∨, ⊙, →, 0, 1) is an integral MTL-algebra.

The following theorem shows that the left and right implicative stabilizers are equivalent in any MV-algebra.

Theorem 3.15. Let L be an MTL-algebra. If L is an MV-algebra, then X _l = X _r = X _s = ^⊥X for any nonempty subset X of L.

Proof. We note that an MV-algebra satisfies (x → y) → y = (y → x) → x for all x, y ∈ L. Now, we will prove X _l = X _r . Let a ∈ X _l , then a → x = x for all x ∈ X and hence (a → x) → x = 1. On one hand, since L is an MV-algebra and so (x → a) → a = 1. On the other hand, from Proposition 2.2(9), we have a ≤ x → a. Combining them, one can obtain that x → a = a. Thus, a ∈ X _l . Hence X _l ⊆ X _r . In the similar way, we have X _r ⊆ X _l . Therefore, X _l = X _r . From Theorem 3.10, we can see that X _l = X _r = X _s = ^⊥X.

From the above theorems, we give some characterizations of MV-algebra via implicativestabilizers.

Theorem 3.16. Let L be a BL-algebra. Then the following assertions are equivalent:

L is an MV-algebra,

Proof. (1) ⇒ (2) It follows from Theorem 3.15.

(2) ⇒ (3) Taking X = {0} in (2).

(3) ⇒ (1) It follows from Theorem 3.11.

(4) ⇒ (5), (5) ⇒ (6), (6) ⇒ (7), (7) ⇒ (4) follow from Theorem 3.6.

Proposition 3.17. Let L be a BL-algebra and F be any filter of L. If F = (〈F〉 _r ) _r , then L is an MV-algebra.

Proof. By Proposition 3.4(3) and hypothesis, we have 〈a〉 = (〈a〉 _r ) _r = (R _a ) _r . For any x ∈ R _a , since a ∈ 〈a〉 = (R _a ) _r , we obtain y → a = a for any y ∈ R _a . Thus x → a = a, that is, x ∈ L _a . Therefore, R _a ⊆ L _a . On the other hand, for any x ∈ L _a , we have x → a = a, thus a ∈ R _x and hence a ∈ L _x , which implies that a → x = x, that is x ∈ R _a . Based on the above, we get that L _a = R _a , for all a ∈ L. From Theorem 3.16 (7) ⇒ (1), we get that L is anMV-algebra.

The following example shows that the converse of the above proposition is not true, in general.

One can easily check that (L, ∧, ∨, ⊙, →, 0, 1) is an MV-algebra and F = {a, 1} is a filter of L. However, (F _r ) _r = {a, b, 1} ≠ F.

Remark 3.19. It was proved that in [16] that RS-BL-algebras, which were defined in [11], are equivalent to MV-algebras. Therefore, Example 3.18 gives a negative answer to the another open problem in [11] that “ Is the converse of the Proposition 3.17 true?".

As we known, BL-algebras are a class of residuated structure based on continuous t-norm and its residua, then it satisfies the condition of divisibility x ∧ y = x ⊙ (x → y), in this case, the condition ¬¬ x = x is equivalent to the (x → y) → y = (y → x) → x. Using this important result, one can prove that a BL-algebra L is an MV-algebra if and only if X _l = X _r for any nonempty subset X of L from Theorem 3.16. Compared to BL-algebras, MTL-algebras are a more general residuated structure based on left-continuous t-norm and its residua, which not satisfies the condition of divisibility. Based on the above consideration, we have the second open problem:

Let L be an MTL-algebra and X _l = X _r for any nonempty subset X of L. Is L an MV-algebra?

4 Multiplicative stabilizers in MTL-algebras

In this section, we investigate related properties of multiplicative stabilizers and discuss the relations between implicative stabilizers and multiplicative stabilizers. Also, we prove that the left and right multiplicative stabilizers form two MTL-algebras. Finally, using multiplicative stabilizers, we give some characterizations of Gödel algebras and linearly order Gödel algebras.

Definition 4.1. Let L be an MTL-algebra and X be a nonempty subset of L. The left and right multiplicative stabilizer of X are defined as follow, 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 multiplicative stabilizer of X. For convenience, the multiplicative stabilizer, left multiplicative stabilizer and right multiplicative stabilizer of X = {x} are denoted by S x ∗ , L x ∗ and R x ∗ , respectively.

Example 4.2. Considering the MTL-algebra in Example 3.2. If X = {a, b}, then X l ∗ = { 1 , b } , X r ∗ = { 0 , b } and hence X s ∗ = { b } . In this case, X r ∗ is not a filter of L.

Proposition 4.3. Let L be an MTL-algebra and X, Y be two nonempty subsets of L. Then the following assertions hold:

X r ∗ = ∩ x ∈ X R x ∗ , X l ∗ = ∩ x ∈ X L x ∗ and X s ∗ = ∩ x ∈ X S x ∗ ,

Proof. The proofs of (1),(2),(4),(5),(6) and (7) are similar to that of Proposition 3.4.

(3) Since X ⊆ 〈X〉, by (2), we have 〈 X 〉 r ∗ ⊆ X r ∗ . On the other hand, suppose that a ∈ X r ∗ , so a ⊙ x = a, for all x ∈ X. For any y ∈ 〈X〉, there exist y₁, y₂, ⋯ y _n ∈ X such that y₁ ⊙ y₂ ⋯ ⊙ y _n ≤ y and hence a ⊙ y ≥ a ⊙ y₁ ⊙ y₂ ⋯ ⊙ y _n = a. Moreover, we have a ⊙ y ≤ a, for all y ∈ 〈X〉, and so a ∈ 〈 X 〉 r ∗ . Therefore, 〈 X 〉 r ∗ = X r ∗ .

(8) For all a , b ∈ X r ∗ , we have a = x ⊙ a and b = x ⊙ b. It follows that a ⊙ b = (x ⊙ a) ⊙ (x ⊙ b) ≤ x ⊙ (a ⊙ b) ≤ a ⊙ b, which implies that a ⊙ b = x ⊙ (a ⊙ b). Thus, a ⊙ b ∈ X r ∗ . On the other hand, we have a ∨ b = (a ⊙ x) ∨ (b ⊙ x) = x ⊙ (a ∨ b) for all a , b ∈ X r ∗ . Hence we can obtain x ⊙ (a ∨ b) = a ∨ b, that is, a ∨ b ∈ X r ∗ . One can also check that X l ∗ is closed under the operations ⊙ and ∨.

The proofs of (9),(10),(11) are easy and hence we omit them.

The following example shows that X l ∗ = 〈 X 〉 l ∗ not holds, in general.

Then (L, ∧, ∨, ⊙, →, 0, 1) is an MTL-algebra. If X = {b}, then X l ∗ = { b , 1 } and 〈 X 〉 l ∗ = { 1 } and hence X l ∗ ≠ 〈 X 〉 l ∗ .

From Proposition 4.3(8), we know that X r ∗ is closed under ∨, in general, the following example shows that X r ∗ is not a lattice ideal of L.

Then (L, ∧, ∨, ⊙, →, 0, 1) is an MTL-algebra, which does not satisfy the divisibility condition x ∧ y = x ⊙ (x → y), for all x, y, z ∈ L. Let X = {a, c}, one can check that X r ∗ = { 0 , a , c } is not a lattice ideal of L, since c ∈ X r ∗ and b ≤ c, but b ∉ X r ∗ .

Then following proposition shows that if L is a BL-algebra, we can obtain that X r ∗ is a lattice ideal of L.

Proposition 4.6. Let L be a BL-algebra. Then X r ∗ is a lattice ideal of L.

Proof. From Proposition 4.3(8), we conclude that X r ∗ is closed under ∨, that is, a ∨ b ∈ X r ∗ , for all a , b ∈ X r ∗ . Moreover, for all a ∈ X r ∗ and b ∈ L, let b ≤ a, we will show b ∈ X r ∗ . It follows from b ≤ a that b = b ∧ a. Since x ⊙ b = x ⊙ (a ∧ b) = x ⊙ (a ⊙ (a → b)) = (x ⊙ a) ⊙ (a → b) = a ⊙ (a → b) = a ∧ b = b. Thus, b ∈ X r ∗ . Therefore, X r ∗ is a lattice ideal of L.

The following theorems show that the R x ∗ and L x ∗ have the same structure as an MTL-algebra under which suitable conditions, which reveals the essence of the stabilizer in MTL-algebras.

Theorem 4.7 Let L be an MTL-algebra and x be an boolean element of L. Then ( L x ∗ , ⊙ , ∧ , ∨ , → , x , 1 ) is an MTL-algebra.

Proof. First, we show that ( L x ∗ , ∧ , ∨ , x , 1 ) is a bounded lattice with x as the smallest element and 1 as the greatest element. From Proposition 4.3(8), we have that L x ∗ is closed under ∨. On the other hand, we have x ⊙ (a ∧ b) = x ⊙ (a ∧ b) = x ∧ (a ∧ b) = (x ∧ a) ∧ (x ∧ b) = (x ⊙ a) ∧ (x ⊙ b) = x ∧ x = x, that is, a ∧ b ∈ L x ∗ . Therefore ( L x ∗ , ∧ , ∨ ) is a lattice. For all a ∈ L x ∗ , one can easily check that a ∧ x = a ⊙ x = x and a ∨ 1 =1. Thus, x is the smallest element and 1 is the greatest element in L x ∗ , respectively.

Next, we prove that ( L x ∗ , ⊙ , 1 ) is a commutative monoid with 1 as neutral element. From Proposition 4.3(8), we have L x ∗ is closed under ⊙. It follows that ( L x ∗ , ⊙ ) is a commutative semigroup. For all a ∈ L x ∗ , we can obtain that a ⊙ 1 = a, that is, 1 is a unital element.

From Proposition 2.4(2), for all a , b ∈ L x ∗ , we have x ⊙ (a → b) = x ⊙ [(x ⊙ a) → (x ⊙ b)] = x ⊙ (x → x) = x, that is, a → b ∈ L x ∗ .

Therefore, we obtain that ( L x ∗ , ⊙ , ∧ , ∨ , → , x , 1 ) is an MTL-algebra.

Theorem 4.8 Let L be an MTL-algebra and x be an idempotent element of L. Then ( R x ∗ , ⊙ , ∧ , ∨ , ⇝ , 0, x) is an MTL-algebra, where a ⇝ b = x ⊙ (a → b), for all a, b ∈ L.

Proof. First, we show that ( R x ∗ , ∧ , ∨ , 0 , x ) is a bounded lattice with 0 as the smallest element and x as the greatest element. From Proposition 4.3(8), we have that R x ∗ is closed under ∨. On the other hand, we have x ⊙ (a ∧ b) = x ∧ (a ∧ b) = (x ∧ a) ∧ (x ∧ b) = (x ⊙ a) ∧ (x ⊙ b) = a ∧ b, that is, a ∧ b ∈ R x ∗ . Therefore ( R x ∗ , ∧ , ∨ ) is a lattice. For all a ∈ R x ∗ , one can easily check that a ∨ x = (x ⊙ a) ∨ x = x and a ∧ 0 =0. Thus, 0 is the smallest element and x is the greatest element in R x ∗ , respectively.

Next, we prove that ( R x ∗ , ⊙ , x ) is a commutative monoid with x as neutral element. From Proposition 4.3(8), we have R x ∗ is closed under ⊙. It follows that ( R x ∗ , ⊙ ) is a commutative semigroup. For all a ∈ R x ∗ , we can obtain that a ⊙ x = a, that is, x is a unital element.

For all a , b ∈ R x ∗ , we define a ⇝ b = x ⊙ (a → b). Now, we will show that a ⊙ b ≤ c if and only if b ≤ a ⇝ c for all a , b , c ∈ R x ∗ . One can easily check that x ⊙ a ≤ b if and only if x ⊙ a ≤ x ⊙ b. Hence we have a ⊙ b ≤ c if and only if b ≤ a → c if and only if x ⊙ b ≤ a → c if and only if x ⊙ b ≤ x ⊙ (a → c) if and only if x ⊙ b ≤ a ⇝ b if and only if b ≤ a ⇝ c for all a , b , c ∈ R x ∗ .

For all a , b ∈ R x ∗ , we have (a ⇝ b) ∨ (b ⇝ a) = (x ⊙ (a → b)) ∨ (x ⊙ (b → a)) = x ⊙ ((a → b) ∨ (b → a)) = x ⊙ 1 = x.

Therefore, we obtain that ( R x ∗ , ⊙ , ∧ , ∨ , ⇝ , 0 , x ) is an MTL-algebra.

From Theorems 4.7 and 4.8, we obtain that ( R x ∗ , ⊙ , ∧ , ∨ , ⇝ , 0 , x ) and ( L x ∗ , ⊙ , ∧ , ∨ , → , x , 1 ) form two MTL-algebras, respectively. Now, we have the third open problem:

Are ( R x ∗ , ⊙ , ∧ , ∨ , ⇝ , 0 , x ) and ( L x ∗ , ⊙ , ∧ , ∨ , → , x , 1 ) isomorphic?

characterizations of Gödel algebras and linearly ordered Gödel algebras via multiplicative stabilizers.

Theorem 4.9. Let L be an MTL-algebra. Then the following assertions are equivalent:

L is a Gödel algebra,

Proof. (1) ⇒ (2) Assume that L is a Gödel algebra, we have that x ⊙ x = x for all x ∈ L. Thus, x ∈ L x ∗ . Moreover, we have that L x ∗ is a filter of L by combining Proposition 4.3(7). That is, for all x ∈ L, if x ≤ a, we can obtain a ∈ L x ∗ , which implies that [ x ) ⊆ L x ∗ . Next, we will show that L x ∗ ⊆ [ x ) . For all a ∈ L x ∗ , we have x ∧ a = x ⊙ a = x, which implies that x ≤ a, that is, a ∈ [x). It follows that L x ∗ = [ x ) . Therefore, we obtain L x ∗ = [ x ) . Moreover, one can easily check that if x is an idempotent element of L, then 〈x〉 = [x).

(2) ⇒ (1) Suppose that L x ∗ = [ x ) for all x ∈ L. Since x ∈ [ x ) = L x ∗ , that is, x ⊙ x = x for all x ∈ L. Next, we will prove that x ⊙ y = x ∧ y = x ⊙ (x → y) for all x, y ∈ L. For all x, y ∈ L, we have x ⊙ (x → y) ≤ (x ⊙ x) → (x ⊙ y), that is, x ⊙ (x → y) ≤ x → (x ⊙ y). Then we have x → y ≤ x → (x → (x ⊙ y)) = (x ⊙ x) → (x ⊙ y) = x → (x ⊙ y). It follows that x ⊙ (x → y) ≤ x ⊙ y. On the other hand, from y ≤ x → y, we obtain x ⊙ y = x ⊙ (x → y). Thus, we get x ⊙ y = x ⊙ (x → y). Finally, we will show x ⊙ y = x ∧ y. For all x, y ∈ L, we have x ⊙ y ≤ x ∧ y by Proposition 2.2(2). Now, for all u ∈ L, if u ≤ x and u ≤ y, we can obtain u ⊙ u ≤ x ⊙ y. Hence we have u ≤ x ⊙ y. It follows that x ⊙ y = x ∧ y = x ⊙ (x → y). Using Definition 2.3(3), we have that L is a Gödel algebra.

(1) ⇒ (3) Assume that L is a Gödel algebra, we have that x ⊙ x = x for all x ∈ L. It follows that x ⊙ x = x for all x ∈ L. Thus, x ∈ R x ∗ . Moreover, we have that R x ∗ is a lattice ideal of L by combining Proposition 4.6. That is, for all x ∈ L, if a ≤ x, we can obtain a ∈ R x ∗ , which implies that ( x ] ⊆ R x ∗ . Next, we will show that R x ∗ ⊆ ( x ] . For all a ∈ R x ∗ , we have x ∧ a = x ⊙ a = a, which implies that a ≤ x, that is, a ∈ (x]. It follows that R x ∗ = ( x ] . Therefore, we obtain R x ∗ = ( x ] .

(3) ⇒ (1) The proof is similar to (2) ⇒ (1).

Theorem 4.10. Let L be an MTL-algebra. Then the following assertions are equivalent:

L is a linearly ordered Gödel algebra,

Proof. (1) ⇒ (2) Assume that L is a linearly ordered Gödel algebra, using Theorem 4.9 (1) ⇒ (2), we have L x ∗ = [ x ) is a filter. Moreover, if x ∨ y ∈ L x ∗ . Since L is a linearly ordered Gödel algebra, then x ≤ y or y ≤ x. Assume x ≤ y, then x = x ⊙ (x ∨ y) = x ⊙ y. It follows that y ∈ L x ∗ . This shows that L x ∗ is a prime filter.

(2) ⇒ (1) First, from Theorem 4.9 (2) ⇒ (1), we obtain that L is a Gödel algebra. Now, we will prove that L is a linearly ordered Gödel algebra. For x, y ∈ L, consider L x ∨ y ∗ , which is induced by x ∨ y. Then L x ∨ y ∗ is a prime filter by hypothesis. Note that x ∨ y ∈ L x ∨ y ∗ . Hence x ∈ L x ∨ y ∗ or y ∈ L x ∨ y ∗ . Assume that x ∈ L x ∨ y ∗ , then x ∨ y = x ⊙ (x ∨ y) = x ∧ (x ∨ y) = x. So y ≤ x. This means that L is a linearly ordered Gödel algebra.

(1) ⇒ (3) Assume that L is a linearly ordered Gödel algebra, using Theorem 4.9 (1) ⇒ (3), we have R x ∗ = ( x ] is a lattice ideal. Moreover, if x ∧ y ∈ R x ∗ . Since L is a linearly ordered Gödel algebra, then x ≤ y or y ≤ x. Assume x ≤ y, then x = x ∧ y = x ⊙ (x ∧ y) = x ⊙ x. It follows that x ∈ R x ∗ . This shows that R x ∗ is a prime lattice ideal.

(3) ⇒ (1) First, from Theorem 4.9 (3) ⇒ (1), we obtain that L is a Gödel algebra. Now, we will prove that L is a linearly ordered Gödel algebra. For x, y ∈ L, consider the R x ∧ y ∗ , which induced by x ∧ y. Then R x ∧ y ∗ is a lattice prime ideal by hypothesis. Note that x ∧ y ∈ R x ∧ y ∗ . Hence x ∈ R x ∧ y ∗ or y ∈ R x ∧ y ∗ . Assume that x ∈ R x ∧ y ∗ , then x = x ⊙ (x ∧ y) = x ∧ (x ∧ y). So x ≤ y. This means that L is a linearly ordered Gödel algebra.

In the following, we discuss the relation between implicative stabilizers and multiplicative stabilizers. In particular, we shows the sets R _x (L _x ) and R x ∗ are order isomorphic.

Theorem 4.11 Let L be an MTL-algebra and x be an idempotent element of L. Then sets R _x and R x ∗ are order isomorphic.

Proof. For all x ∈ L, let g : R x ∗ → R x be defined by g (a) = x → a for all a ∈ R x ∗ . Clearly, g is a map from R x ∗ to R _x , that is, g is well defined.

For all a , b ∈ R x ∗ , that is, a = x ⊙ a and b = x ⊙ b, if g (a) = g (b), then x → a = x → b. From x ⊙ a ≤ a, we have a ≤ x → a, then a ≤ x → b. It follows that x ⊙ a ≤ b, which means a ≤ b. Similarly, we can prove b ≤ a. Then a = b. Consequently, we obtain that g is injective.

Theorem 4.12. Let L be an MV-algebra and x be an idempotent element of L. Then sets L _x and R x ∗ are order isomorphic.

Proof. It follows from Theorems 3.15 and 4.11.

5 Conclusions

The aim of this paper is to develop the stabilizer theory of MTL-algebras. In the paper, some useful properties of particular stabilizers are discussed. And, we characterize some special class of MTL-algebras, for example, IMTL-algebras, integral MTL-algebra, Gödel algebras and MV-algebras, via these stabilizers. Finally, we discuss the relation between these stabilizers and obtain that the right implicative stabilizers are isomorphic to the right multiplicative stabilizers. There are still three open problems:

For any filter F of an MTL-algebra L, whether there exists a nonempty subset X such that X _l = F?

In our future work, we will consider these problems and some applications of stabilizers in fuzzy logic and its corresponding algebraic system.