Some types of ideals in residuated lattices

Abstract

It is clear to characterize logic algebras, ideals and filters play important role, therefore in this research, we survey the structure of various ideals and find some types of ideals such as positive implicative ideals, strong ideals and MV-ideals in residuated lattices. We also introduce the concept of quasi ideals and show that any ideal is a quasi ideal, but the converse does not hold in general. We clarify the relations between these ideals in residuated lattices, strong residuated lattices and BL-algebras. For instance, we prove that in strong residuated lattices the concept of quasi ideals and ideals are the same and in BL-algebras the concepts of quasi ideals, ideals, strong ideals and MV-ideals coincide, whereas they are different in residuated lattices and MTL-algebras. We detect the relation between these ideals with MV-algebras, strong algebras and Boolean algebras.

Keywords

Residuated lattice ideal quasi ideal positive implicative ideal strong ideal MV-ideal

1 Introduction

It is well-known that certain information process is based on classical logic (Boolean logic) and artificial intelligence deal with uncertain information that this process is based of non-classical logic. Many-valued logic, as the extension and development of classical logic, has always been a crucial direction in non-classical logic. To formalize the fuzzy logic induced by continuous t-norms on the real unit interval [0,1], in 1998, Hájek introduced a general many-valued logic, called Basic Logic that BL-algebras are the algebraic structure of this logic [8]. As is known to all result that a t-norm has a residuum if and only if the t-norm is left-continuous, so this shows that Basic Logic is not the most general t-norm- based logic. In fact, a logic weaker than Basic Logic, called Monoidal t-norm-based logic (MTL for short), was defined by Esteva and Godo in [7] and proved to be the logic of left-continuous t-norms and their residua. Thus, the MTL is indeed the logic of left-continuous t-norms, and MTL-algebras are the algebraic counterpart of this logic. Residuated lattices are very basic and important algebraic structure because the other logical algebras such as BL-algebras and MTL-algebras are all particular cases of residuated lattices. They have been investigated by Krull, Ward and Dilworth, Balbes and Dwinger and Pavelka (see [1 , 18]). Y. Liu et al. in [10] introduced and characterized the concept of an ideal on residuated lattices, they also study fuzzy ideals and congruence relation induced by an ideal on a residuated lattice. In fact by Theorem 2.9, Definitions 2.8 and 2.10, the concept of ideals and LI-ideals that studied on lattice implication algebras and MTL-algebras coincides, respectively (See [11, 19]). In area of logic algebras, many types of filters in residuated lattices or in special classes of residuated lattices (BL-algebras, MTL-algebras, etc.) have been studied. Many researchers (see [4, 9]) by the structures of some logical algebras such as MV-algebras, Boolean algebras and Gődel algebras introduced some types of filters such as MV-filters (fantastic filters), positive implicative filters (Boolean filters) and implicative filters (Hyting filters or G-filters). Similarly to characterize logical algebras ideals are important and we study some types of ideals in residuated lattices and special classes of them. In this paper, by the concept of ideals and structure of some classes of residuated lattices, we introduce the concepts of positive implicative ideals, quasi ideals, strong ideals and MV-ideals. We characterize the relationships between these ideals and survey these concepts of ideals in BL-algebras and strong algebras and show that in strong residuated lattices, the concept of quasi ideals and ideals coincides and in BL-algebras, the concepts of quasi ideals, ideals, strong ideals and MV-ideals coincide. Finlay, we show the relation between these ideals by a figure.

2 Preliminaries

Definition 2.1. [2 , 8]

A resituated lattice is an algebra (L, ∨ , ∧ , ⊙ , → , 0, 1) of type (2,2,2,2,0,0) such that

(L, ∨ , ∧ , 0, 1) is a bounded lattice with the greatest element 1 and the smallest element 0,

(L, ⊙ , 1) is a commutative monoid,

(⊙ , →) is an adjoint couple on L.

A resituated lattice L is called a strong residuated lattice or Glivenko residuated lattice if (x ^** → x) ^** = 1 for all x, y ∈ L.

A resituated lattice L is called an MTL-algebra, if it satisfies the pre-linearity equation: (x → y) ∨ (y → x) =1 for all x, y ∈ L.

An MTL-algebra L is called an IMTL-algebra, if (x → 0) →0 = x, for all x ∈ L.

An MTL-algebra L is called a BL-algebra if x ∧ y = x ⊙ (x → y), for all x, y ∈ L.

A BL-algebra L is said to be an MV-algebra if (x → y) → y = (y → x) → x or x ^** = 0 that x ^* = x → 0.

Proposition 2.2. [2 , 6–8] label 1.2 The following properties are hold for any resituated lattice ((R1)-(R11)), MTL-algebra ((R1)-(M8)) and BL-algebra ((R1)-(B1)) for all x, y, z ∈ L:

x ≤ y ⇔ x → y = 1.

1 → x = x, x → 1 =1, x → x = 1, 0 → x = 1, x → (y → x) =1.

x ≤ y → z ⇔ y ≤ x → z.

x → (y → z) = (x ⊙ y) → z = y → (x → z).

x ≤ y implies z → x ≤ z → y and y → z ≤ x → z.

z → y ≤ (x → z) → (x → y), z → y ≤ (y → x) → (z → x).

(x → y) ⊙ (y → z) ≤ x → z.

x ^* = x ^***, x ≤ x ^**.

x ^* ∧ y ^* = (x ∨ y) ^*.

x ∨ x ^* = 1 implies x ∧ x ^* = 0.

(x → y ^**) ^** = x → y ^**.

x ⊙ y ≤ x ∧ y.

x ≤ y implies x ⊙ z ≤ y ⊙ z.

y → z ≤ x ∨ y → x ∨ z.

x ^* ∨ y ^* = (x ∧ y) ^*.

(x ∨ y) → z = (x → z) ∧ (y → z).

x ∨ y = ((x → y) → y) ∧ ((y → x) → x).

x → (y ∨ z) = (x → y) ∨ (x → z).

x ∧ (y ∨ z) = (x ∧ y) ∨ (x ∧ z).

(x ∧ y) ^** = (x ^** ∧ y ^**),(x ∨ y) ^** = (x ^** ∨ y ^**), (x ⊙ y) ^** = (x ^** ⊙ y ^**).

In a residuated lattice, the binary operation ⊕ is defined by x ⊕ y = x ^* → y for any x, y ∈ L.

Note. Because each MTL-algebra is a residuated lattice, we can extend the definitions and theorems that follow similar axioms from MTL-algebras to residuated lattices. Since a bounded BCK-algebras that mentioned by R. Cignoli and A. Torrens (2004) (see [6]) is a residuated lattice, we use Lemma 2.12 and property (R11) for residuated lattices. Also Theorem 2.9, Definitions 2.8 and 2.10 show that, the concept of LI-ideals and ideals coincides in any residuated lattices.

Definition 2.3. [3 , 17] Let F be a nonempty subset of residuated lattice L. Then

F is called a filter (deductive system) if 1 ∈ F and x ∈ F, x → y ∈ F imply y ∈ F.

let F be a filter of L. Then F is called a strong filter of L if for all x ∈ L, (x ^** → x) ^** ∈ F.

Theorem 2.4. [3] Both IMTL-algebras and BL-algebras are strong MTL-algebras, but the converse does not hold in general.

Definition 2.5. [5, 15] The residuated lattice L is called commutative if (x → y) → y = (y → x) → x for all x, y ∈ L.

Lemma 2.6. [5, 15] Let L be a commutative residuated lattice. Then for all x, y ∈ L:

x ^** = x.

(x → y) → y = (y → x) → x = x ∨ y.

Proposition 2.7. [5, 15] Let L be a residuated lattice. Then the following statements are equivalent for all x, y ∈ L:

(L, ⊕ , * , 0) is a MV-algebra.

(x → y) → y = (y → x) → x.

((x → y) → y) → x = y → x.

Definition 2.8. [10] Let L be a residuated lattice and ∅ ≠ I ⊆ L. Then I is said to be an ideal of L, if I satisfies:

for any x, y ∈ L, if x ≤ y and y ∈ I, then x ∈ I,

for any x, y ∈ I, x ⊕ y ∈ I.

Theorem 2.9. [10] Let L be a residuated lattice. I is an ideal of L if and only if I satisfies the following conditions:

0 ∈ I,

for any x, y ∈ L, if (x ^* → y ^*) ^* ∈ I and x ∈ I, then y ∈ I.

Definition 2.10. [19] Let I be a nonempty subset of residuated lattice L and 0 ∈ I. Then for all x, y ∈ L: I is called an LI-ideal if (x ^* → y ^*) ^* ∈ I and x ∈ I imply y ∈ I.

Proposition 2.11. [20] Let L be a residuated lattice. Then the following statements are equivalent:

L is a Boolean algebra.

(x → y) → x = x, for every x, y ∈ L.

x ^* → x = x, for every x ∈ L.

Lemma 2.12. [6] Let L be a residuated lattice. Then the following statements are equivalent for all x, y ∈ L:

(x ^** → x) ^** = 1.

(x → y) ^** = x → y ^**.

3 Quasi ideals and strong ideals in residuated lattices

Definition 3.1 A non-empty subset I of residuated lattice L is called quasi ideal if for any x, y ∈ L:

0 ∈ I and

(x → y) ^* ∈ I and y ∈ I imply x ∈ I.

Lemma 3.2. Let I be an ideal of residuated lattice L. Then I is a quasi ideal.

Proof. Suppose that I is an ideal, x ∈ I and (y → x) ^* ∈ I. Since x ^* → y ^* ≥ y → x, (x ^* → y ^*) ^* ≤ (y → x) ^*, hence by Definition 2.8 (I1), we have (x ^* → y ^*) ^* ∈ I, and it follows that y ∈ I. □

The following example shows that the converse of Lemma 3.2, does not hold in general.

Example 3.3. Let L = {0, a, b, c, 1}. Then we define the operations “→” and “⊙” as follows: $\begin{matrix} \begin{matrix} \to & 0 & a & b & c & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ a & c & 1 & 1 & 1 & 1 \\ b & c & c & 1 & 1 & 1 \\ c & b & b & b & 1 & 1 \\ 1 & 0 & a & b & c & 1 \end{matrix} & \begin{matrix} ⊙ & 0 & a & b & c & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & 0 & a \\ b & 0 & 0 & 0 & 0 & b \\ c & 0 & 0 & 0 & c & c \\ 1 & 0 & a & b & c & 1 \end{matrix} \end{matrix}$

Suppose that ∧ and ∨ are defined on L by min and max, respectively. Then (L, ∧ , ∨ , ⊙ , 0, 1) is a residuated lattice. It is easy to show that {0, a} is a quasi ideal, but {0, a} is not an ideal, since (a ^* → b ^*) ^* = (c → c) ^* = 0 ∈ {0, a}, but b ∉ {0, a}.

Lemma 3.4. Let I be a quasi ideal of residuated lattice L, x ≤ y and y ∈ I. Then x ∈ I.

Proof. Suppose that I is a quasi ideal, x ≤ y and y ∈ I. Then (x → y) ^* = 1^* = 0 ∈ I, hence x ∈ I. □

Definition 3.5. An ideal I of residuated lattice L is called a strong ideal if (x ^** → x) ^* ∈ I, for all x ∈ L.

The following example shows that strong ideals exist.

Example 3.6. Let L = {0, a, b, c, 1}. Then we define the operations “→” and “⊙” as follows: $\begin{matrix} \begin{matrix} \to & 0 & a & b & c & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ a & c & 1 & 1 & 1 & 1 \\ b & b & b & 1 & 1 & 1 \\ c & a & a & b & 1 & 1 \\ 1 & 0 & a & b & c & 1 \end{matrix} & \begin{matrix} ⊙ & 0 & a & b & c & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & 0 & a \\ b & 0 & 0 & 0 & b & b \\ c & 0 & 0 & b & c & c \\ 1 & 0 & a & b & c & 1 \end{matrix} \end{matrix}$

Suppose that ∧ and ∨ are defined on L by min and max, respectively. Then (L, ∧ , ∨ , ⊙ , 0, 1) is a residuated lattice. It is easy to show that I = {0, a} and I = {0} are strong ideals, but in Example 3.3, {0, a} is not a strong ideal, since (a ^** → a) ^* = (b → a) ^* = c ^* = b ∉ {0, a}.

Lemma 3.7. Let I and J be ideals of residuated lattice L and I ⊆ J. If I is a strong ideal, so is J.

Proof. By the definition of a strong ideal, the proof is clear. □

Theorem 3.8. Let L be a residuated lattice. Then the following statements are equivalent:

{0} is a strong ideal.

every ideal is a strong ideal.

L is a strong residuated lattice.

Proof. The proof holds by Lemmas 2.12 and 3.7. □

Theorem 3.9. In Strong residuated lattice L, the concept of ideals and quasi ideals coincides.

Proof. Suppose that L is a strong residuated lattice. Then by Theorem 3.2, we have any ideal is a quasi ideal, so it is sufficient to prove that if I is a quasi ideal, x ∈ I and (x ^* → y ^*) ^* ∈ I, then y ∈ I. To solve this problem we present two ways, first way: $\begin{matrix} (y \to x)^{*} \to (x^{*} \to y^{*})^{*} \\ = ((y \to x)^{*} \to (x^{*} \to y^{*})^{*})^{* *} \\ = ((x^{*} \to y^{*}) \to (y \to x)^{* *})^{* *} \\ = ((y \to x^{* *}) \to (y \to x)^{* *})^{* *} \\ \geq ((y \to x^{* *}) \to (y \to x))^{* *} \\ \geq (x^{* *} \to x)^{* *} = 1 \end{matrix}$

Therefore ((y → x) ^* → (x ^* → y ^*) ^*) ^* = ((x ^** → x) ^**) ^* = 0 ∈ I, and it follows that (y → x) ^* ∈ I. Now, since I is a quasi ideal and x ∈ I, we have y ∈ I.

Second way: by Lemma 2.12, we have (x ^* → y ^*) ^** = (y → x ^**) ^** = (y → x) ^**, and it follows that (y → x) ^* = (x ^* → y ^*) ^* ∈ I, hence by the definition of a quasi ideal, we have y ∈ I. □

Corollary 3.10. Let L be a residuated lattice. Then the following statements are equivalent:

{0} is a strong ideal.

every quasi ideal is a strong ideal.

L is a strong residuated lattice.

Proof. The proof holds by Theorems 3.8 and 3.9. □

Theorem 3.11. Let I be a nonempty set of residuated lattice L. Then F is a strong filter if and only if F ^* is a strong ideal.

Proof. (⇒) Let F be a strong filter. Since for x ∈ L we have (x ^** → x) ^** ∈ F ⇔ (x ^** → x) ^* ∈ F ^*. Therefore by Definitions 2.3 and 3.7, the proof is clear. □

4 Positive implicative ideals in residuated lattices

Definition 4.1. An nonempty subset I of aresiduated lattice L is said to be a positive implicative ideal of L if it satisfies:

(z ^* → ((x ^* → y ^*) → x ^*) ^**) ^* ∈ I and z ∈ I implies x ∈ I for all x, y, z ∈ L.

Theorem 4.2. Any positive implicative ideal of residuated lattice L is an ideal.

Proof. Suppose that I is a positive implicative ideal of L, (x ^* → y ^*) ^* ∈ I and x ∈ I. Then by Definition 4.1, we have $\begin{matrix} (x^{*} \to ((y^{*} \to 0^{*}) \to y^{*})^{* *})^{*} \\ = (x^{*} \to (1 \to y^{*})^{* *})^{*} \\ = (x^{*} \to y^{* * *})^{*} \\ = (x^{*} \to y^{*})^{*} \in I \end{matrix}$

Since I is a positive implicative ideal, hence y ∈ I. □

The following example shows that the converse of Theorem 4.2, does not hold in general.

Example 4.3. Let L = {0, a, b, 1}. Then we define the operations “→” and “⊙” as follows: $\begin{matrix} \begin{matrix} \to & 0 & a & b & 1 \\ 0 & 1 & 1 & 1 & 1 \\ a & b & 1 & 1 & 1 \\ b & a & a & 1 & 1 \\ 1 & 0 & a & b & 1 \end{matrix} & \begin{matrix} ⊙ & 0 & a & b & 1 \\ 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & a \\ b & 0 & 0 & b & b \\ 1 & 0 & a & b & 1 \end{matrix} \end{matrix}$

Suppose that ∧ and ∨ are defined on L by min and max, respectively. Then (L, ∧ , ∨ , ⊙ , 0, 1) is a residuated lattice. It is easy to show that {0, a} is a positive implicative ideal, but {0} is not a positive implicative ideal, since ((a ^* → 1^*) → a ^*) ^* = (a ^** → a ^*) ^* = (a → b) ^* = 0 ∈ {0}, but a ∉ {0}.

Lemma 4.4. Let I be an ideal of residuated lattice L. Then the following statements are equivalent for all x, y ∈ L:

I is a positive implicative ideal.

((x ^* → y ^*) → x ^*) ^* ∈ I implies x ∈ I.

(x → x ^*) ^* ∈ I implies x ∈ I.

Proof. (i) ⇒ (ii) Let I be a positive implicative ideal and ((x ^* → y ^*) → x ^*) ^* ∈ I. Since 0 ∈ I and $\begin{matrix} (0^{*} \to ((x^{*} \to y^{*}) \to x^{*})^{* *})^{*} \\ = (x^{*} \to y^{*}) \to x^{*})^{* * *} \\ = (x^{*} \to y^{*}) \to x^{*})^{*} \in I \end{matrix}$

Therefore by Definition 4.1, we have x ∈ I.

(ii) ⇒ (i) The proof is straightforward.

(ii) ⇒ (iii) Let (x → x ^*) ^* ∈ I. Then ((x ^* → 1^*) → x ^*) ^* = (x ^** → x ^*) ^* = (x → x ^*) ^* ∈ I, so by (ii) we have x ∈ I.

(iii) ⇒ (ii) Let ((x ^* → y ^*) → x ^*) ^* ∈ I. Then x ^* → 0 ≤ x ^* → y ^*, so (x ^* → y ^*) → x ^* ≤ (x ^* → 0) → x ^* = (x ^** → x ^*). Consequently (x ^** → x ^*) ^* ≤ ((x ^* → y ^*) → x ^*) ^*, hence by Definition 2.8 (I1), we have (x → x ^*) ^* = (x ^** → x ^*) ^* ∈ I, and it follows that x ∈ I. □

Theorem 4.5. Let I be an ideal of residuated lattice L. Then the following statements are equivalent for all x, y, z ∈ L:

I is a positive implicative ideal.

(x ^* → (x ^* → y ^*)) ^* ∈ I implies (x ^* → y ^*) ^* ∈ I.

(z ^* → (x ^* → y ^*)) ^* ∈ I implies (z ^* → x ^*) → (z ^* → y ^*)) ^* ∈ I.

Proof. (i) ⇒ (ii) Let I be a positive implicative ideal of L and (y ^* → (y ^* → x ^*)) ^* ∈ I. Then $\begin{matrix} ((y^{*} \to x^{*})^{* *} \to x^{*}) \to (y^{*} \to x^{*})^{* *} \\ \geq ((y^{*} \to x^{*})^{* *} \to x^{*}) \to (y^{*} \to x^{*}) \\ \geq y^{*} \to (y^{*} \to x^{*})^{* *} \\ \geq y^{*} \to (y^{*} \to x^{*}) \end{matrix}$

Therefore $\begin{matrix} (((y^{*} \to x^{*})^{* *} \to x^{*}) \to (y^{*} \to x^{*})^{* *})^{*} \\ \leq (y^{*} \to (y^{*} \to x^{*}))^{*} \in I . \end{matrix}$

Now, by Definition 2.8 (I1), we have (((y ^* → x ^*) ^** → x ^*) → (y ^* → x ^*) ^**) ^* ∈ I, hence by the definition of a positive implicative ideal (y ^* → x ^*) ^* ∈ I.

(ii) ⇒ (iii). Let I be an ideal of L and (z ^* → (x ^* → y ^*)) ^* ∈ I. Since x ^* ≤ z ^* → x ^*, hence (z ^* → x ^*) → y ^* ≥ x ^* → y ^* and we have $\begin{matrix} z^{*} \to (z^{*} \to ((z^{*} \to x^{*}) \to y^{*})) \\ \geq z^{*} \to (z^{*} \to (x^{*} \to y^{*})) \\ \geq z^{*} \to (x^{*} \to y^{*}) \end{matrix}$

So (z ^* → (z ^* → ((z ^* → x ^*) → y ^*))) ^* ≤ (z ^* → (x ^* → y ^*)) ^*.

Since I is an ideal and (z ^* → (x ^* → y ^*)) ^* ∈ I, we have (z ^* → (z ^* → ((z ^* → x ^*) → y ^*))) ^* ∈ I. Therefore by (i) we have (z ^* → x ^*) → (z ^* → x ^*)) ^* = (z ^* → ((z ^* → x ^*) → y ^*)) ^* ∈ I.

(iii) ⇒ (ii) It is sufficient in (iii), we put z = x.

(ii) ⇒ (i) Let (x → x ^*) ^* ∈ I. Since $\begin{matrix} (x^{* *} \to (x^{* *} \to (x^{* *} \to x^{*})^{*}))^{*} \\ = (x^{* *} \to ((x^{* *} \to x^{*}) \to x^{*})^{*} \\ = ((x^{* *} \to x^{*}) \to (x^{* *} \to x^{*}))^{*} \\ = 1^{*} = 0 \in I \end{matrix}$ Therefore by (ii) we have $\begin{matrix} (x^{* *} \to (x^{* *} \to x^{*})^{*})^{*} \in I \end{matrix}$

On the other hand since $\begin{matrix} ((x \to x^{*})^{* *} \to x^{*})^{*} & = & (x^{* *} \to (x \to x^{*})^{*})^{*} \\ = & (x^{* *} \to (x^{* *} \to x^{*})^{*})^{*} \end{matrix}$

Hence ((x → x ^*) ^** → x ^*) ^* ∈ I and by the definition of an ideal we have x ∈ I. Therefore by Lemma 4.4, I is a positive implicative ideal. □

Lemma 4.6. Let L be a residuated lattice. Then for all x, y ∈ L, x → y ^* = (x → y ^*) ^**.

Proof. Suppose that L is a residuated lattice and x, y ∈ L. Then by Lemma 2.2 (R11), we have $\begin{matrix} (x \to y^{*})^{* *} & = & (x \to (y^{*})^{* *})^{* *} = x^{* *} \to (y^{*})^{* * *} \\ = & x^{* *} \to y^{* * *} = x^{* *} \to y^{*} \\ = & y \to x^{* * *} = y \to x^{*} \\ = & x \to y^{*} □ \end{matrix}$

Theorem 4.7. Let I and J be ideals of residuated lattice L, I ⊆ J. If I is a positive implicative ideal, so is J.

Proof. Let I be a positive implicative ideal, J be an ideal, I ⊆ J and (x ^* → (x ^* → y ^*)) ^* ∈ J. If we put U = x ^* → (x ^* → y ^*), then by Lemma 4.6, we have $\begin{matrix} x^{*} \to (x^{*} \to (U \to y^{*})^{* *}) \\ = x^{*} \to (x^{*} \to (U \to y^{*})) \\ = U \to (x^{*} \to (x^{*} \to y^{*})) \\ = 1 \end{matrix}$

Thus (x ^* → (x ^* → (U → y ^*) ^**)) ^* = 0 ∈ I. Now, by Lemma 4.5 (ii), we have (x ^* → (U → y ^*) ^**) ^* ∈ I, hence by Lemma 4.6, we have (U → (x ^* → y ^*)) ^* ∈ I. On the other hand since $\begin{matrix} U^{* *} \to (x^{*} \to y^{*})^{* *} & \geq & U \to (x^{*} \to y^{*}) \end{matrix}$

Therefore ((U ^** → (x ^* → y ^*) ^**) ^* ≤ (U → (x ^* → y ^*)) ^*. Since I is an ideal, so ((U ^** → (x ^* → y ^*) ^**) ^* ∈ I ⊆ J. Now, since U ^* ∈ J and J is an ideal, we have (x ^* → y ^*) ^* ∈ J and this completes the proof. □

Theorem 4.8. Let I be an ideal of residuated lattice L. Then I is a positive implicative ideal of L if and only if for all a ∈ L, $A_{a}^{I} = {x \in L : (a^{*} \to x^{*})^{*} \in I}$ is an ideal of L.

Proof. Suppose that for all a ∈ L, $A_{a}^{I}$ is an ideal of L and (x ^* → (x ^* → y ^*)) ^* ∈ I. Now, by Theorem 4.5, it is sufficient to show that (x ^* → y ^*) ^* ∈ I. Since by Lemma 4.6, (x ^* → (x ^* → y ^*) ^**) ^* = (x ^* → (x ^* → y ^*)) ^* ∈ I, $(x^{*} \to y^{*})^{*} \in A_{x}^{I}$ . On the hand since $x \in A_{x}^{I}$ and $A_{x}^{I}$ is an ideal, hence $y \in A_{x}^{I}$ , consequently (x ^* → y ^*) ^* ∈ I.

Conversely, if I is a positive implicative ideal and $(x^{*} \to y^{*})^{*} \in A_{x}^{I}$ , $y \in A_{x}^{I}$ , then by the definition of $A_{x}^{I}$ and Lemma 4.6, (x ^* → (x ^* → y ^*)) ^* = (x ^* → (x ^* → y ^*) ^**) ^* ∈ I. Now, since I is a positive implicative ideal, by Theorem 4.5, (x ^* → y ^*) ^* ∈ I and it implies $y \in A_{x}^{I}$ . Therefore $A_{x}^{I}$ is an ideal. □

Corollary 4.9. Let I be a positive implicative ideal of residuated L. Then for a ∈ L, $A_{a}^{I}$ is the least ideal of L containing I and a.

Proof. Suppose that I is a positive implicative ideal and x ∈ I. Then by Theorem 4.8, for a ∈ L, $A_{a}^{I}$ is an ideal of L also it is clear that $a \in A_{a}^{I}$ . On the other hand, since (x ^* → (a ^* → x ^*) ^**) ^* = (x ^* → (a ^* → x ^*)) ^* = (a ^* → (x ^* → x ^*)) ^* = 1^* = 0 ∈ I and I is also ideal, so (a ^* → x ^*) ^* ∈ I, that implies $x \in A_{a}^{I}$ , hence $I \subset A_{a}^{I}$ . If D is an ideal containing I and a, then for $x \in A_{a}^{I}$ , we have (a ^* → x ^*) ^* ∈ I, hence (a ^* → x ^*) ^* ∈ D. Since D is an ideal and a ∈ D, x ∈ D, and it follows that $A_{a}^{I} \subseteq D$ . □

Theorem 4.10. Let L be a commutative residuated lattice. Then the following statements are equivalent:

{0} is a positive implicative ideal.

every ideal is a positive implicative ideal.

for all a ∈ L, $A (a) = A_{a}^{{0}}$ is a positive implicative ideal.

L is a Boolean algebra.

Proof. (i) ⇔ (ii) By Theorem 4.7, obviously holds.

(ii) ⇒ (iii) Since {0} is an ideal, by hypothesis, {0} is a positive implicative ideal. Hence by Theorem 4.8, $A (a) = A_{a}^{{0}}$ is an ideal, and by (ii), we have A (a) is a positive implicative ideal.

(iii) ⇒ (iv) Since L is a commutative residuated lattice for x, y ∈ L, there exist s, t ∈ L such that x = s ^*, y = t ^*. Now, we put u = ((x → y) → x) ^*. Since ((s ^* → t ^*) → s ^*) ^* = ((x → y) → x) ^* = u ∈ A (u) and A (u) is a positive implicative ideal, by Lemma 4.4 (ii) we have s ∈ A (u). Therefore (u ^* → s ^*) ^* ∈ {0}, hence [((x → y) → x) → x] ^* = [((x → y) → x) ^** → x] ^* = 0, consequently ((x → y) → x) → x = 1, so (x → y) → x ≤ x. It is clear that x ≤ (x → y) → x. Hence (x → y) → x = x and by Proposition 2.11, L is a Boolean algebra.

(iv) ⇒ (i) Suppose that (x → x ^*) ^* ∈ {0}. Then by (iv), we have x ^* → x = (x → 0) → x = x. Hence by Proposition 2.11, (x → x ^*) ^* = ((x ^* → x) → x ^*) ^* = (x ^*) ^* = x ^** ∈ {0}, this implies x = 0 ∈ {0}. Therefore by Lemma 4.4, {0} is a positive implicative ideal. □

5 MV-ideals in residuated lattices

Definition 5.1. An ideal I of residuated lattice L is called MV-ideal if (x ^* → y ^*) ^* ∈ I imply (((y ^* → x ^*) → x ^*) → y ^*) ^* ∈ I.

Theorem 5.2. Let L be a BL-algebra. Then the concepts of quasi ideals, ideals, strong ideals and MV-ideals coincide.

Proof. Suppose that L be a BL-algebra. Then by Theorems 2.4, 3.8 and 3.9, the concepts of quasi ideals, ideals and strong ideals coincide, so it is sufficient to prove that any ideal is a MV-ideal. Suppose that I is an ideal and (x ^* → y ^*) ^* ∈ I. Since $\begin{matrix} (((y^{*} \to x^{*}) \to x^{*}) \to y^{*})^{*} \\ = (((x \to y^{* *}) \to x^{*}) \to y^{*})^{*} \\ = (((x \to y^{* *}) ⊙ x)^{*} \to y^{*})^{*} \\ = ((x \land y^{* *})^{*} \to y^{*})^{*} \\ = (x^{*} \lor y^{*}) \to y^{*})^{*} \\ = ((x^{*} \to y^{*}) \land (y^{*} \to y^{*}))^{*} \\ = (x^{*} \to y^{*})^{*} \end{matrix}$ Therefore (((y ^* → x ^*) → x ^*) → y ^*) ^* ∈ I and I is a MV-ideal. □

Theorem 5.3. Let L be a strong residuated lattice. Then every MV-ideal is a strong ideal.

Proof. Let I be a MV-ideal of strong residuated lattice L. Since any MV-ideal is an ideal, hence by Theorem 3.8, I is a strong ideal. □

The following example shows that MV-ideals exist and the converse of Theorem 5.3, does not hold in general.

Example 5.4. Let L = {0, a, b, c, 1}. Then we define the operations “→” and “⊙” as follows: $\begin{matrix} \begin{matrix} \to & 0 & a & b & c & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ a & b & 1 & 1 & 1 & 1 \\ b & a & a & 1 & 1 & 1 \\ c & 0 & a & b & 1 & 1 \\ 1 & 0 & a & b & c & 1 \end{matrix} & \begin{matrix} ⊙ & 0 & a & b & c & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & a & a \\ b & 0 & 0 & b & b & b \\ c & 0 & a & b & c & c \\ 1 & 0 & a & b & c & 1 \end{matrix} \end{matrix}$

Suppose that ∧ and ∨ are defined on L by min and max, respectively. Then (L, ∧ , ∨ , ⊙ , 0, 1) is a strong residuated lattice. It is easy to show that {0, a} is a MV-ideal and {0} is a strong ideal, but {0} is not a MV-ideal, since (b ^* → a ^*) ^* = (a → b) ^* = 1^* = 0 ∈ {0}, but (((a ^* → b ^*) → b ^*) → a ^*) ^* = (((b → a) → a) → b) ^* = ((a → a) → b) ^* = b ^* = a ∉ {0}.

Theorem 5.5. Any positive implicative ideal of residuated lattice L is a MV-ideal.

Proof. Let I be a positive implicative ideal and (x ^* → y ^*) ^* ∈ I. If we put u = ((y ^* → x ^*) → x ^*), then we have $\begin{matrix} (u \to y^{*})^{*} \to (u \to y^{*})^{* *} \\ = (u \to y^{*})^{*} \to (u \to y^{*}) \\ = u \to ((u \to y^{*})^{*} \to y^{*}) \\ = u \to (y \to (u \to y^{*})^{* *}) \\ = u \to (y \to (u \to y^{*})) \\ = u \to (u \to (y \to y^{*})) \end{matrix}$

Therefore $\begin{matrix} ((u \to y^{*})^{*} \to (u \to y^{*})^{* *})^{*} \to (x^{*} \to y^{*})^{*} \\ = (u \to (u \to (y \to y^{*})))^{*} \to (x^{*} \to y^{*})^{*} \\ = (x^{*} \to y^{*}) \to (u \to (u \to (y \to y^{*})))^{* *} \\ = (x^{*} \to y^{*}) \to (u \to (u \to (y \to y^{*}))) \\ = u \to (u \to ((x^{*} \to y^{*}) \to (y \to y^{*}))) \\ \geq u \to (u \to (y \to x^{*})) \\ = u \to (((y^{*} \to x^{*}) \to x^{*}) \to (y \to x^{*})) \\ \geq u \to (y \to (y^{*} \to x^{*})) \\ = u \to ((y ⊙ y^{*}) \to x^{*}) \\ = u \to (0 \to x^{*}) \\ = 1 \end{matrix}$

So ((u → y ^*) ^* → (u → y ^*) ^**) ^* ≤ (x ^* → y ^*) ^*. Since (x ^* → y ^*) ^* ∈ I, we have ((u → y ^*) ^* → (u → y ^*) ^**) ^* ∈ I, hence by Lemma 4.4, we have (u → y ^*) ^* = (((y ^* → x ^*) → x ^*) → y ^*) ^* ∈ I, this completes the proof. □

The following example shows that the converse of Theorem 5.5, does not hold in general.

Example 5.6. Let L = {0, a, b, 1}. Then we define the operations “→” and “⊙” as follows: $\begin{matrix} \begin{matrix} \to & 0 & a & b & 1 \\ 0 & 1 & 1 & 1 & 1 \\ a & b & 1 & 1 & 1 \\ b & b & b & 1 & 1 \\ 1 & 0 & a & b & 1 \end{matrix} & \begin{matrix} ⊙ & 0 & a & b & 1 \\ 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & a \\ b & 0 & 0 & 0 & b \\ 1 & 0 & a & b & 1 \end{matrix} \end{matrix}$

Suppose that ∧ and ∨ are defined on L by min and max, respectively. Then (L, ∧ , ∨ , ⊙ , 0, 1) is a residuated lattice. It is easy to show that {0} is a MV-ideal, but {0} is not a positive implicative ideal, since ((a ^* → 1^*) → a ^*) ^* = (a ^** → a ^*) ^* = (b → b) ^* = 0 ∈ {0}, but a ∉ {0}.

Lemma 5.7. Let I and J be ideals of residuated lattice L and I ⊆ J. If I is a MV-ideal, so is J.

Proof. Let I be a MV-ideal, J be an ideal, I ⊆ J and (x ^* → y ^*) ^* ∈ J. Since $\begin{matrix} (x^{*} \to ((x^{*} \to y^{*}) \to y^{*}))^{*} \\ = ((x^{*} \to y^{*}) \to (x^{*} \to y^{*}))^{*} = 1^{*} = 0 \in I . \end{matrix}$

Hence by the definition of a MV-ideal we have $(((((x^{*} \to y^{*}) \to y^{*}) \to x^{*}) \to x^{*}) \to y^{*})^{*} \in J$

On the other hand since $\begin{matrix} (((y^{*} \to x^{*}) \to x^{*}) \to y^{*})^{*} \to \\ (((((x^{*} \to y^{*}) \to y^{*}) \to x^{*}) \to x^{*}) \to y^{*})^{*} \\ \geq (((((x^{*} \to y^{*}) \to y^{*}) \to x^{*}) \to x^{*}) \to y^{*}) \\ \to (((y^{*} \to x^{*}) \to x^{*}) \to y^{*}) \\ \geq ((y^{*} \to x^{*}) \to x^{*}) \to ((((x^{*} \to y^{*}) \to y^{*}) \\ \to x^{*}) \to x^{*}) \\ \geq (((x^{*} \to y^{*}) \to y^{*}) \to x^{*}) \to (y^{*} \to x^{*}) \\ \geq y^{*} \to ((x^{*} \to y^{*}) \to y^{*}) = 1 \end{matrix}$

Therefore by Definition 2.8 (I1), we have $(((y^{*} \to x^{*}) \to x^{*}) \to y^{*})^{*} \in J$ and this completes the proof. □

Theorem 5.8. Let L be a residuated lattice. Then the following statements are equivalent for all x, y ∈ L:

{0} is a MV-ideal.

every ideal is a MV-ideal.

(x ^* → y ^*) → y ^* = (y ^* → x ^*) → x ^*.

((x ^* → y ^*) → y ^*) → x ^* = y ^* → x ^*.

Proof. (i) ⇔ (ii) By Lemma 5.7, it is clear.

(ii) ⇒ (iii) By suppose we have {0} is a MV-ideal, if u = ((x ^* → y ^*) → y ^*) ^*, then by Lemma 4.6, we have $\begin{matrix} (x^{*} \to u^{*})^{*} & = & (x^{*} \to ((x^{*} \to y^{*}) \to y^{*})^{* *})^{*} \\ = & (x^{*} \to ((x^{*} \to y^{*}) \to y^{*}))^{*} \\ = & ((x^{*} \to y^{*}) \to (x^{*} \to y^{*}))^{*} \\ = & (1)^{*} = 0 \in {0} \end{matrix}$

Therefore by the definition of a MV-ideal, we have (((u ^* → x ^*) → x ^*) → u ^*) ^* ∈ {0} (a). On the other hand since y ^* ≤ u ^*, u ^* → x ^* ≤ y ^* → x ^*, it follows that (y ^* → x ^*) → x ^* ≤ (u ^* → x ^*) → x ^*, hence ((u ^* → x ^*) → x ^*) → u ^* ≤ ((y ^* → x ^*) → x ^*) → u ^*, and by (a) we have (((y ^* → x ^*) → x ^*) → u ^*) ^* ≤ (((u ^* → x ^*) → x ^*) → u ^*) ^* ∈ {0}. Consequently (((y ^* → x ^*) → x ^*) → u ^*) ^* ∈ {0}, and it follows that (((y ^* → x ^*) → x ^*) → u ^*) ^** = (((y ^* → x ^*) → x ^*) → u ^*) =1. Therefore ((y ^* → x ^*) → x ^*) ≤ u ^* = ((x ^* → y ^*) → y ^*) ^** = ((x ^* → y ^*) → y ^*). Similarly (y ^* → x ^*) → x ^* ≥ (x ^* → y ^*) → y ^*, so we have (y ^* → x ^*) → x ^* = (x ^* → y ^*) → y ^*.

(iii) ⇒ (iv) ((x ^* → y ^*) → y ^*) → x ^* = ((y ^* → x ^*) → x ^*) → x ^* = y ^* → x ^*.

(iv) ⇒ (ii) Suppose that I is an ideal and (x ^* → y ^*) ^* ∈ I. Then by (iv) we have (((y ^* → x ^*) → x ^*) → y ^*) ^* = (x ^* → y ^*) ^* ∈ I, and this completes the proof. □

Corollary 5.9. Let L be a commutative residuated lattice. Then the following statements are equivalent for all x, y ∈ L:

{0} is a MV-ideal.

every ideal is a MV-ideal.

(x ^* → y ^*) → y ^* = (y ^* → x ^*) → x ^*.

((x ^* → y ^*) → y ^*) → x ^* = y ^* → x ^*.

L is a MV-algebra.

Proof. The proof holds by Theorem 5.8 and Proposition 2.7. □

In the following diagram we can see the relation between some kinds of ideals in residuated lattices.

6 Conclusion

It is well known that the use of filters and ideals play an important role in characterizing the structure of a logical system. Many researcher studied various filters in logical structures such as BL-algebras, MTL-algebras and residuated lattices, but because of some limitation various ideals are not defined and consequently by the structure of sub-logical algebras such as MV-algebras and strong algebras, we introduce some types of ideals in residuated lattices and detect the relations between these ideals in residuated lattices.

Footnotes

Acknowledgments

The author would like to thank the referees and editor for their valuable suggestions and comments.

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