On nodal and conodal ideals in residuated lattices

Abstract

In this article, we put forward the concepts of nodal and conodal ideals in a residuated lattice and study some properties. We state some examples and theorems. We investigate the inverse image of a nodal (conodal) ideal under a homomorphism. In addition, we pay attention to the relationships with the other types of ideals and special sets in varieties of residuated lattices. At the same time, we give a characterization of nodal ideals in terms of congruences and we show that if L is an MTL-algebra and I is a non-principal nodal ideal, then L/I is a chain. We propose a characterization for Boolean residuated lattices (L is a Boolean residuated lattice if and only if L is an involution semi-G-agebra) and we discuss briefly the applications of our results in varieties of residuated lattices. Finally, we introduce the concept of a fuzzy (nodal) ideal of a residuated lattice, and give some related results. After that we define the concept of fuzzy ideal of a residuated lattice with respect to a t-conorm briefly, S-fuzzy ideals and we prove Representation Theorem in residuated lattices.

Keywords

Residuated lattice obstinate ideal implicative ideal nodal ideal conodal ideal

1. Introduction

Residuation is a fundamental concept of ordered structures and categories. The origin of residuated lattices is in Mathematical Logic without contraction. The general definition of a residuated lattice was given by Galatos et al. [7]. They first developed the structural theory of this kind of algebra about residuated lattices.

Ideal theory plays an important rule in studying residuated lattices. From logical point of view, various ideals correspond to various sets of provable formula. In residuated lattices, the filters and ideals are not dual, in consequence, the ideals have a proper meaning in residuated lattices, from a purely algebraic point of view. In 2010, Y. Zhu et al. [21] presented interesting aspects of the theory of filters in residuated lattices. In 2013, C. Lele et al. [14] studied the relationship between ideals and filters in BL-algebras. In 2012, A. Borumand Saeid et al. [1, 8] studied the concept of obstinate filters and co-annihilators in residuated lattices. Soon after, in 2014, F. Forouzesh et al. [5] published a study on obstinate ideals in MV-algebras, they investigate some relationships between obstinate ideals and other ideals of an MV-algebra. In 2015, D. Buşneag et al. [4] investigated the variety of Stonean residuated lattices from the view of ideal theory and proved some theorems specific to distributive residuated lattices. In 2016, Yu Xi Zou et al. [23] studied ideals and annihilator ideals in BL-algebras. In 2017, L.C. Holdon [11] introduced a new class of residuated lattices called De Morgan residuated lattices (L is called De Morgan residuated lattice if the De Morgan law (x ∧ y) ^* = x^* ∨ y^*, for all x, y ∈ L hold) and investigated it from the view of ideal theory, the author proved that various notions of ideals presented in literature such as right ideals, left ideals [14, 23] and implicative ideals [4] coincide in residuated lattices. In 2018, L.C. Holdon [12] studied a new topology based on upsets in residuated lattices, in a dual sense, ideals are down-sets in residuated lattices, hence a new topology based on down-sets can be generated.

J.C. Varlet (1973) [19] presented the first study of nodal filters in semilattices. Recently, Raza Tayebi Khorami and Arsham Borumand Saeid [17] studied nodal elements and nodal filters in BL-algebras, and Arsham Borumand Saeid et al. [2] extend the research of nodal elements and nodal filters in residuated lattices. Soon after, in 2016, F. Forouzesh [6] studied nodal and conodal ideals in MV-algebras. However, in residuated lattices, the notions of nodal filters and nodal ideals are not dual. We can conclude that there is a great deal of interest on this subject, that because of its applications in fuzzy sets theory.

Since the notion of ideals in residuated lattices has been defined and investigated in papers [4 , 23], we think it is a new direction to study the ideals by the concept of nodal (conodal) ideals, which will enrich and develop the theory of ideals in residuated lattices. We mention that, in residuated lattices, the notions of nodal filters and nodal ideals are not dual.

The paper is organized as follows: In Section 2, we review some basic definitions and results about residuated lattices. In Section 3, we introduce the concept of nodal ideal in residuated lattices. We note that the lattice nodal ideals and nodal ideals are not always the same. We state some examples and characterization theorems. We pay attention to the relationships between the nodal (conodal) ideals with other ideals, likeness Boolean ideals, prime ideals, maximal ideals, primary ideals, implicative ideals and ⊙-prime ideals. We conclude that the class of nodal (conodal) ideals is a new class of ideals in residuated lattices. We state the characterization theorems for nodal (conodal) ideals in Theorems 4 and 11. In Theorem 13, we show that for a MTL-algebra L, if I is a nodal ideal, then its quotient algebra L/I is a chain. Using nodal ideals, we define an equivalence relation denoted by R (the elements x and y of L are connected, in symbol (x, y) ∈ R) if there is no nodal ideal which separates them). In Theorem 9, we present some interesting properties of R-classes. In Theorem 12, we present a characterization for nodal ideals in term of congruences. In Section 4, we discuss briefly the applications of our results in varieties of residuated lattices. Finally, we propose a new characterization for Boolean residuated lattices and we present some applications. In conclusion, our results generalize some theorems regarding nodal (conodal) ideals from MV-algebras [6] in the general case of residuated lattices. This study has applications in fuzzy sets theory and, consequently, in decision-making theory and graph theory. In Section 5, we define the notion of fuzzy (nodal) ideal of a residuated lattice, we show that fuzzy ideals and fuzzy filters are not dual notions in residuated lattices. We pay attention to characterization theorems for fuzzy (nodal) filters and fuzzy (nodal) ideals, after that, we introduce the concept of a fuzzy (nodal) ideal of a residuated lattice, with respect to a t-conorm briefly, S-fuzzy ideals, and we prove the Representation Theorem in residuated lattices.

2. Preliminaries

In this section, we present some preliminaries including the basic definitions, some results on ideals in residuated lattices, some examples of residuated lattices, rules of calculus and theorems that are needed in the sequel. All results with proofs are original.

Definition 1. [7] A bounded residuated lattice is an algebra (L, ∨ , ∧ , ⊙ , → , 0, 1) of type (2, 2, 2, 2, 0, 0) such that

(Lr₁) (L, ∨ , ∧ , 0, 1) is a bounded lattice;

(Lr₂) (L, ⊙ , 1) is a commutative monoid;

(Lr₃) ⊙ and → form an adjoint pair, i.e., a ⊙ x ≤ b if and only if x ≤ a → b .

We call them simply residuated lattices. Usually, (Lr₃) is called the residuation property. For examples of residuated lattices see Galatos et al.(2007), Hájek (1998), D. Buşneag et al.(2013), Freytes (2004), Kowalski and Ono (2002), Iorgulescu (2008), Piciu (2007) and Turunen (1999). If L is totally ordered, then L is called a chain.

In what follows by L we denote the universe of a residuated lattice (unless otherwise specified). For x ∈ L and n ≥ 0, we define x^* = x → 0, x^** = (x^*) ^*, x⁰ = 1 and xⁿ = x^n-1 ⊙ x, for n ≥ 1 .

We refer to Hájek (1998), D. Buşneag et al.(2013), Piciu (2007)-Ward and Dilworth (1939) for detailed proofs of these well-known results:

Lemma 1. For every x, y, z ∈ L, we have:

(c₁) x → x = 1, x → 1 =1, 1→ x = x ;

(c₂) x ≤ y iff x→ y = 1 ;

(c₃) if x ≤ y, then z ⊙ x ≤ z ⊙ y, z → x ≤ z → y, y→ z ≤ x → z ;

(c₄) if x ≤ y, then y^* ≤ x^*, x^**≤ y^** ;

(c₅) x ⊙ x^* = 0, x ≤ x^**, x^*** = x^*, 0^* = 1, 1^* = 0 ;

(c₆) x ⊙ y = 0 if and only if x≤ y^* ;

(c₇) x⊙ (y ∨ z) = (x ⊙ y) ∨ (x ⊙ z) ;

(c₈) x∨ (y ⊙ z) ≥ (x ∨ y) ⊙ (x ∨ z) ;

(c₉) x→ (y ∧ z) = (x → y) ∧ (x → z) ;

(c₁₀) x⊙ (x → y) ≤ x ∧ y ;

(c₁₁) (x∨ y) ^* = x^* ∧ y^* ;

(c₁₂) x^* ⊙ y^* ≤ (x ⊙ y) ^*, x^**⊙ y^** ≤ (x ⊙ y) ^** ;

(c₁₃) (x→ y^**) ^** = x → y^** ;

(c₁₄) x → (y → z) = y → (x → z) = (x ⊙ y) → z, (x ⊙ y) ^* = x → y^* = y → x^* .

We consider the identities:

(i₁) x ∧ y = x ⊙ (x → y) (divisibility) ;

(i₂) (x^* ∧ y^*) ^* = [x^* ⊙ (x^* → y^*)] ^* (semi - divisibility) ;

(i₃) (x → y) ∨ (y → x) =1 (prelinearity) ;

(i₄) x^* ∨ x^** = 1 (Stoneproperty) ;

(i₅) x² = x (idenpotence) ;

(i₆) x = x^** (involution) ;

(i₇) (x²) ^* = x^* ;

(i₈) (x∧ y) ^* = x^* ∨ y^* ;

(i₉) x ∨ x^* = 1 .

Then the residuated lattice L is called:

Divisible if L verifies (i₁);

Semi-divisible if L verifies (i₂);

MTL-algebra if L verifies (i₃);

BL-algebra if L verifies (i₁) and (i₃);

Stonean if L verifies (i₄) ;

G-algebra(Heyting algebra) if L verifies (i₅) ;

Involution if L verifies (i₆) ;

semi-G-algebra if L verifies (i₇) ;

De Morgan if L verifies (i₈) ;

Boolean algebra if L verifies (i₉) .

Definition 2. [9] A filter is a non-empty subset F of L such that

(F₁) if x ≤ y and x ∈ F, then y∈ F ;

(F₂) if x, y ∈ F, then x ⊙ y ∈ F .

We denote by $F (L)$ the set of all filters of L .

Definition 3. [2, 17] A filter F of L is called a nodal filter of L, if F is a node of the lattice $(F (L), \subseteq)$ of all filters of L .

Theorem 1. [2] Let F be a filter of L . The following assertions are equivalent:

Let G be an filter of L . Then we have either F ⊆ G or G⊆ F ;

F is a node of the lattice $(F (L), \subseteq);$

Let x ∈ F and y ∉ F . Then x ≩ y .

For every x, y ∈ L, we define: $x \oplus y = (x^{*} ⊙ y^{*})^{*} \overset{(c_{14})}{=} x^{*} \to y^{* *} .$ (1)

The operator x ⊕ y will be called strong addition.

Lemma 2. For every x, y, z, t ∈ L, we have:

(c₁₅) x⊕ 0 = x^**, x ⊕ 1 =1, x ⊕ x^* = 1 ;

(c₁₆) x⊕ y = y ⊕ x, x, y ≤ x ⊕ y ;

(c₁₇) x⊕ (y ⊕ z) = (x ⊕ y) ⊕ z ;

(c₁₈) if x ≤ y, then x⊕ z ≤ y ⊕ z ;

(c₁₉) if x ≤ y, z ≤ t, then x ⊕ z ≤ y ⊕ t .

Remark 1. By Lemma 2, we conclude that the operator ⊕ is commutative, associative and compatible with the order relation.

Definition 4. [4, 11] An ideal is a nonempty subset I of L such that

(I₁) if x ≤ y and y ∈ I, then x∈ I ;

(I₂) if x, y ∈ I, then x ⊕ y ∈ I .

We denote by $I (L)$ the set of all ideals of L .

Remark 2. [4, 11] It is known that every ideal is a lattice ideal in the lattice (L, ∧ , ∨ , 0, 1) , but the converse is not true. Moreover, the intersection of two ideals becomes an ideal, too.

For an MV-algebra (A, ⊕ , * , 0) , we recall (Definition 1.2 [5]): an ideal of A is a nonempty subset I of A such that

(MV₁) if x ≤ y and y ∈ I, then x∈ I ;

(MV₂) if x, y ∈ I, then x ⊕ y ∈ I .

The operation ⊙ is defined as follow: x ⊙ y = (x^* ⊕ y^*) ^*, for all x, y ∈ A .

Remark 3. We note that, in MV-algebras, the notions of ideals defined in Definition 1.2 [5] and Definition 4 coincide. Indeed, if A is an MV-algebra and x, y ∈ L, if we denote x^* = a and y^* = b, it follows that x = a^* and y = b^* . We obtain successively x ⊙ y = (x^* ⊕ y^*) ^*, a^* ⊙ b^* = (a^** ⊕ b^**) ^*, a^* ⊙ b^* = (a ⊕ b) ^*, (a^* ⊙ b^*) ^* = (a ⊕ b) ^**, (a^* ⊙ b^*) ^* = a ⊕ b . Therefore, the operator ⊕ used in both definitions coincide. However, in the general case of residuated lattices, the operator ⊕ (used in both definitions) has different meanings.

Proposition 1. [11] If I is an ideal of L, then x ∈ I iff x^** ∈ I iff nx ∈ I, for all n ≥ 1 .

For x ∈ L and n ≥ 0, we define 0 · x = 0 and n · x = [(n - 1) · x] ⊕ x for n ≥ 1 . For simplicity, we denote nx : = n · x .

Proposition 2. [4] Let S ⊆ L be a nonempty subset, a ∈ L and $I \in I (L) .$ Then:

(S] = {x∈ L : x ≤ s₁ ⊕ . . . ⊕ s_n, forsome n ≥ 1 and s₁, . . . , s_n ∈ S} ;

(a] = {x∈ L : x ≤ na forsome n ≥ 1} ;

I (a) = {x∈ L : x ≤ i ⊕ na forsome i ∈ I and n ≥ 1} ;

$(I (L), \subseteq)$ is a complete lattice, where for $I_{1}, I_{2} \in I (L),$ I₁ ∨ I₂ = (I₁ ∪ I₂] =

{x ∈ L : x ≤ i₁ ⊕ i₂ with i₁ ∈ I₁ and i₂ ∈ I₂} .

Proposition 3. For x ∈ L, we have:

(x] = (x^**] = (nx] , for all n≥ 1 ;

if L is a semi-G-algebra, then nx = x^**, for all n≥ 1 ;

if L is a semi-G-algebra and x, y are incomparable element of L, then x ∉ (y] and y ∉ (x] .

Proof.

For x ∈ L, (x] is an ideal of L . By Proposition 1, it follows that x ∈ (x] iff x^** ∈ (x] iff nx ∈ (x] , for all n ≥ 1 .

Let L be a semi-G-algebra (that is, (x²) ^* = x^*, for all x ∈ G). Hence x ⊕ x = (x^* ⊙ x^*) ^* = [(x^*) ²] ^* = x^** . By the associativity of operator ⊕ and (c₅) (x^*** = x^*, for all x ∈ L), it follows that nx = x^**, for all n ≥ 1 .

It follows from (i) and (ii) .□

Let X be a nonempty subset of L . The set of complemented elements (with respect to X) is defined by D (X) : = {x ∈ L : x^* ∈ X} .

Proposition 4. Let I be an ideal of L . Then x ∈ I iff x^* ∈ D (I) iff x^** ∈ I iff nx ∈ I, for all n ≥ 1 .

Proof. Let I be an ideal of L and x ∈ I . We show that x ∈ I iff x^* ∈ D (I) . By Proposition 1, it follows that x^** ∈ I . Hence x^* ∈ D (I) . Conversely, let x^* ∈ D (I) . Since (x^**) ^* = x^* ∈ D (I) , it follows that x^** ∈ I . By Proposition 1, we conclude that x^** ∈ I iff x ∈ I . Hence x ∈ I iff x^* ∈ D (I) iff x^** ∈ I. Finally, by Proposition 1, we have that x ∈ I iff x^** ∈ I iff nx ∈ I, for all n ≥ 1 . We conclude that x ∈ I iff x^* ∈ D (I) iff x^** ∈ I iff nx ∈ I, for all n ≥ 1 . □

In what follows, we recall various types of ideals from literature (see [4–6 , 23]).

Definition 5.

A proper ideal $I \in I (L)$ is called prime if and only if, for all a, b ∈ L, if a ∧ b ∈ I, then a ∈ I or b ∈ I . We denote by Spec_Id (L) the set of all prime ideals of L ;

An ideal $M \in I (L)$ is called maximal if M is not strictly contained in a proper ideal of L . We denote by Max_Id (L) the set of all maximal ideals of L ;

An ideal I of L is called a Boolean ideal if x ∧ x^* ∈ I, for all x∈ L ;

P is a primary ideal of L if it is a proper ideal such that for every x, y ∈ L such that x ⊙ y ∈ P, there exists an integer n ≥ 1 such that xⁿ ∈ P or yⁿ∈ P ;

An ideal I of L is called an implicative ideal if for any x, y, z ∈ L such that z ⊙ (y^* ⊙ x^*) ∈ I and y ⊙ x^* ∈ I, then z⊙ x^* ∈ I ;

An ideal I of L is called an ⊙-prime ideal, if for all x, y ∈ L, x ⊙ y ∈ I implies x ∈ I or y∈ I ;

An ideal I of L is called an obstinate ideal, if for all x, y ∈ L \ I, x^* ⊙ y ∈ I and x ⊙ y^* ∈ I .

The ⊙-prime ideals were introduced in residuated lattices [11] in order to establish the relationship between ideals and filters.

3. Nodal ideals of a residuated lattice

In this section, we define the concept of nodal (conodal) ideals in residuated lattices and we study their properties in details. We recall that an element x of a poset X is called a node of X, if it is comparable with every element of X .

Definition 6. An element a of L is called a node of L, if it is comparable with every element of L . We denote by Nod (L) the set of all nodes of L .

In Example 1, we have that Nod (L) = {0, m, 1} . If an element x ∈ L is not a node of L, it will be called conode of L . We denote by CoNod (L) the set of all conodes of L . In Example 1, the set of all conodes of L is CoNod (L) = {a, b, c, d, e, f} . In [2, 17], it proved that nodes are different elements than dense, nilpotent, boolean and regular elements. Also, in [16, 18] some interesting relationships have been shown between dense, nilpotent, boolean and regular elements. In what follows, we present some results on nodes in residuated lattices.

Remark 4.

By (c₂) , we note that an element a ∈ L is a node if and only if a → x = 1 or x→ a = 1 ;

If a ∈ L is a node, then a^* ⊙ x = 0 or a ⊙ x^* = 0, for all x ∈ L . Indeed, if a ∈ L is a node, then a ≤ x or x ≤ a, for all x ∈ L . By (c₃) , it follows that a ⊙ x^* ≤ x ⊙ x^* = 0 or a^* ⊙ x ≤ a^* ⊙ a = 0 . Hence a ⊙ x^* = 0 or a^* ⊙ x = 0 .

Example 1. In this example, we show that the converse of Remark 4(ii) does not hold.

Let L = {0, a, b, c, d, e, f, m, 1} with the Hasse diagram:

Then ([13], page 252) L is a residuated lattice with respect to the following operations:

If we consider the element c fixed, then c^* ⊙ x = 0 or c ⊙ x^* = 0, for all x ∈ L, but c is not a node of L .

Proposition 5. If L is an involution residuated lattice, then a ∈ L is a node if and only if a^* ⊙ x = 0 or a ⊙ x^* = 0, for all x ∈ L .

Proof. It follows from Remark 4(ii) . For the converse, let x ∈ L be an arbitrary element such that a^* ⊙ x = 0 or a ⊙ x^* = 0 . By residuation property (Lr₃) , it follows that a^* ⊙ x = 0 or a ⊙ x^* = 0 if and only if a^* ⊙ x ≤ 0 or a ⊙ x^* ≤ 0 if and only if a ≤ x^** = x or x ≤ a^** = a if and only if a ≤ x or x ≤ a . Hence a is a node of L . □

Example 2. In this example, we present the set of all nodes (conodes) of L .

We consider L = {0, n, a, b, c, d, e, f, m, 1} with the Hasse diagram:

Then ([13], page 242) L is a residuated lattice with respect to the following operations:

It is easy to ascertain that Nod (L) = {0, n, m, 1} and CoNod (L) = {a, b, c, d, e, f} . Moreover, L is an involution residuated lattice and the condition from Proposition 5 hold in L .

We recall [2, 17] that a filter F is called nodal filter of L if it is a node of the lattice $(F (L), \subseteq)$ of all filters of L .

Remark 5. We note that nodal ideals and nodal filters are not dual notions in residuated lattices. Indeed, in Example 2, the elements 0 and n are nodes of L, I₁ = {0} is a nodal ideal, but L \ I₁ = {n, a, b, c, d, e, f, m, 1} is not a nodal filter of L (because a ⊙ b = 0 ∉ L \ I₁).

We recall (Definition 1.6 [5]) that a nonzero element m of a poset P with 0 is a molecule if whenever 0 < x, y < m, then {x, y} has a nonzero lower bound. Thus m ∈ L is a molecule if and only if whenever x, y ∈ L satisfy 0 < x, y < m . Hence x ∧ y > 0 . We denote by Mol (L) the set of all molecules of L . A residuated lattice is called totaly ordered (or simply, a chain) if every two elements are comparable.

Proposition 6. If L is a totally ordered residuated lattice, then all nodes of L \ {0} are molecules.

Proof. Let x, y ∈ L be such that 0 < x, y < m . Since L is totaly ordered, we conclude that all elements of L are nodes. Since x, y are nodes, it follows that x ∧ y = x > 0 or x ∧ y = y > 0 . Hence m is a molecule. □

Example 3. In the residuated lattice L from Example 2, Nod (L) = {0, n, m, 1} and Nod (L) ∩ Mol (L) = {n, m, 1} , but L is not a totally ordered residuated lattice. Moreover, we can see that f is a molecule, but it is not a node of L .

Example 4. In this example, we present a node which is not molecule. If we consider the residuated lattice L from Example 1, it is easy to ascertain that the set of all nodes of L is Nod (L) = {0, m, 1} . Since 0 < a, f < m and a ∧ f = 0, it follows that m is not a molecule. Hence m is a node of L, but it is not a molecule.

We recall [9] that an element e ∈ L is boolean element if and only if e ∨ e^* = 1 . We denote by B (L) the set of all boolean elements of L .

Theorem 2. In L, Nod (L) ∩ B (L) = {0, 1} .

Proof. Clearly, {0, 1} ⊆ Nod (L) ∩ B (L) . Assume that there is an element 0 ≠ a ∈ Nod (L) ∩ B (L) . We will show that a = 1 . Since a ∈ B (L) , it follows that there exists its complement b ∈ L such that a ∧ b = 0 and a ∨ b = 1 . Since a ∈ Nod (L) , it follows that a ≤ b or b ≤ a . Hence a ∧ b = a, a ∨ b = b or a ∧ b = b, a ∨ b = a . Therefore, a = 1 . □

Remark 6. We note that every node a of L is a distributive element of L (that is, a ∧ (x ∨ y) = (a ∧ x) ∨ (a ∧ y) , for all x, y ∈ L). By Theorem 2, we conclude that examples of distributive elements different than nodes are the boolean elements.

Definition 7. An ideal I of L is called nodal ideal of L, if I is a node of the lattice $(I (L), \subseteq) .$

Clearly, {0} and L are nodal ideals.

Remark 7. Clearly, every nodal ideal is a lattice nodal ideal, but the converse does not always hold. Indeed, in Example 1, we have I = {0, a, b, c, d, e, f, m} is a lattice nodal ideal, but it is not a nodal ideal because m ⊕ m = 1 . Hence the notion of nodal ideal has a proper meaning in a residuated lattice.

Example 5. In this example, we present a residuated lattice L with two ideals which are not nodal ideals. We consider L = {0, a, b, n, c, d, 1} with the Hasse diagram:

Then ([13], page 191) L is a distributive residuated lattice with respect to the following operations:

It easy to ascertain that I₁ = {0, a} , I₂ = {0, b} and I₃ = {0} are all proper ideals of L . Since I₁ and I₂ are incomparable, we conclude that they are not nodal ideals of L . Clearly, n is a node of L . Moreover, the lattice ideal generated by element n is {0, a, b, n} , but the ideal generated by n is (n] = L . Hence the lattice ideal and the ideal generated by n are different. In conclusion, every ideal is a lattice ideal, but the converse may not always hold.

Proposition 7. Let I be an ideal of L . If for all x ∈ I and for all y ∉ I, the relation x < y is satisfied, then I is a nodal ideal of L .

Proof. Assume that there is an ideal J incomparable with I . Then there are elements x, y ∈ L such that x ∈ I \ J, y ∈ J \ I and xnleqy, that is a contradiction with the hypothesis. Hence every ideal J of L is comparable with I . Therefore, I is a nodal ideal of L.□

Theorem 3. Let I be a nodal ideal of L . For every x ∈ I and e ∉ I, if e ∈ B (L) , then the relation x < e is satisfied.

Proof. Since I is a nodal ideal of L, so for all x, e ∈ L such that x ∈ I and e ∉ I, we have [x) ⊆ I and I ⊆ [e) . Hence, [x) ⊆ I ⊆ [e) , so x ∈ [e) , that is x ≤ ne, for some n ≥ 1 . Since e ∈ B (L) , it follows that 2e = e ⊕ e = (e^* ⊙ e^*) ^* = [(e^*) ²] ^* = e^** = e . By mathematical induction with respect to n, it follows that ne = e, for all n ≥ 1 and e ∈ B (L) . Since e ∈ B (L) , we conclude that x < ne = e . □

Example 6. The converse of Theorem 3 may not always hold. Indeed, if we consider the residuated lattice L from Example 2, I = {0, n} is a nodal ideal of L and n ∈ I . We have that m ∉ I and n < m, but m ∉ B (L) (because m ∨ m^* = m ∨ n = m ≠ 1).

Proposition 8. If x is a node of L, then the principal ideal (x] is a nodal ideal.

Proof. Let x ∈ L be a node and I an ideal of L . If x ∈ I, then (x] ⊆ I . Now, let x ∉ I . If I ⊈ (x] , then there is y ∈ I such that y ∉ (x] , then for all n ≥ 1, ynleqnx . Hence ynleqx . Since x is a node, it follows that x < y . Hence (x] ⊆ (y] ⊆ I . We conclude that x ∈ I, that is a contradiction. Therefore, if x ∉ I, then I ⊆ (x] . □

Example 7. In this example, we show that the converse of Proposition 8 may not always hold. Indeed, in the residuated lattice L from Example 5 we have that (c] = L is a nodal ideal of L, but c is not a node.

Remark 8. We note that if L is a chain, then every ideal of L is a nodal ideal, but the converse may not hold. Indeed, if we consider the residuated lattice L from Example 1, we have that I₁ = {0} and I₂ = L are the only ideals of L . Clear, they are nodal ideals, but L is not a chain.

The next result is a characterization for nodal ideals in residuated lattices.

Theorem 4. Let I be an ideal of L . The following assertions are equivalent:

Let J be an ideal of L . Then we have either I ⊆ J or J⊆ I ;

I is a node of the lattice $(I (L), \subseteq);$

Let x ∈ I and y ∉ I . Then x ≨ y .

Proof.

⇒ (ii) . It follows immediate from the definition of a node.

⇒ (iii) . If I is a node of the lattice $(I (L), \subseteq),$ then for every x ∈ I and every y ∉ I we have (y] ⊈ I . Hence (x] ⊆ I ⊆ (y] . Therefore, x ≨ y .

⇒ (i) . Assume that there is an ideal J incomparable with I . Then there are elements x, y ∈ L such that x ∈ I \ J, y ∈ J \ I . By hypothesis, we have that x ≨ y or y ≨ x . Hence I ⊆ J or J ⊆ I .

We note that (iii) ⇒ (ii) follows from Proposition 7.□

Definition 8. An ideal I of L is called distributive if I is distributive as an element of the lattice $(I (L), \subseteq) .$

Theorem 5. Every nodal ideal of L is a distributive element of the lattice $(I (L), \subseteq) .$

Proof. Let I be a nodal ideal of L and $P, Q \in I (L) .$ We consider the following cases:

Case 1. If P, Q ⊆ I, then P ∧ Q ⊆ I . We conclude that I ∨ (P ∧ Q) = I and (I ∨ P) ∧ (I ∨ Q) = I .

Case 2. If I ⊆ P, Q, then I ⊆ P ∧ Q . We conclude that I ∨ (P ∧ Q) = P ∧ Q and (I ∨ P) ∧ (I ∨ Q) = P ∧ Q .

Case 3. If P ⊆ I ⊆ Q, then I ∨ (P ∧ Q) = I and (I ∨ P) ∧ (I ∨ Q) = I ∧ Q = I .

Case 4. If Q ⊆ I ⊆ P, then I ∨ (P ∧ Q) = I ∨ Q = I and (I ∨ P) ∧ (I ∨ Q) = P ∧ I = I .□

We recall [9] that the lattice $(I (L), \subseteq)$ need not be distributive. However, if L is a De Morgan residuated lattice, then $(I (L), \subseteq)$ is a distributive lattice. The next result is a direct consequence of Theorem 5.

Corollary 1. If every ideal of L is a nodal ideal, then $I (L)$ is a distributive lattice.

Example 8. In this example, we show that the converse of Corollary 1 may not always hold. Indeed, in the residuated lattice L from Example 5 we have I₁ = {0} , I₂ = {0, a} , I₃ = {0, b} , I₄ = L . Hence $I (L)$ is a distributive lattice, but I₂ and I₃ are not nodal ideals.

We note that the class Nod_Id (L) of all nodal ideals of a residuated lattice L is a totally ordered structure, consequently, the set of all nodal ideals of L, ordered by inclusion, is a chain with the greatest element L and smallest element {0} . If I is a nodal ideal of L, then I satisfies the intersection and union properties. If I and J are nodal ideals of L such that I ∩ J = {0} , then I = {0} or J = {0} . It is easy to ascertain that the lattice (Nod_Id (L) , ∧ , ∨ , {0} , L) is a complete distributive Brouwerian lattice.

For every ideals I, J ∈ Nod_Id (L) , we define the operator I → J : = {x ∈ L : I ∩ (x] ⊆ J} . We denote by I^* : = I → {0} = {x ∈ L : I ∩ (x] ⊆ {0}} . Clearly, {0} ^* = L and L^* = {0} .

Theorem 6. If L is a chain and ⊙ : = ∧ , then $(I (L), \land, \lor, \to, ⊙, {0}, L)$ is a G-algebra.

Proof. Let L be a chain and for every two ideals I and J . We consider I ∧ J : = I ⊙ J . Since L is a chain, it follows that every ideal of L is a nodal ideal, that is, $I (L) = {Nod}_{Id} (L) .$ We prove that the residuation property hold for all $I, J, Q \in I (L),$ equivalently, I ∧ J ⊆ Q if and only if I ⊆ J → Q if and only if J ⊆ I → Q . Let x ∈ I . We assume that I ∧ J ⊆ Q . Since x ∈ I, it follows that (x] ⊂ I, (x] ∩ J ⊆ I ∩ J = I ∧ J ⊆ Q . Hence x ∈ J → Q .

Conversely, let x ∈ I ∧ J . Thus x ∈ I and x ∈ J . We assume that I ⊆ J → Q . Since I ⊆ J → Q, it follows that x ∈ J → Q, equivalently, (x] ∧ J ⊆ Q . Since x ∈ (x] and x ∈ J, it follows that x ∈ Q . Hence I ∧ J ⊆ Q . The rest of the residuation property follows in a similar manner. Since I² = I ⊙ I = I ∧ I = I, it follows that (Nod_Id (L) , ∧ , ∨ , → , ⊙ , {0} , L) is a G-algebra. □

Example 9. The converse of Theorem 6 may not hold. Indeed, if we consider the residuated lattice L from Example 2, then we have that I₁ = {0} , I₂ = {0, n} and I₃ = L are all the ideals of L . It is easy to ascertain that $(I (L), \land, \lor, \to, ⊙, {0}, L)$ is a G-algebra, but L is not a chain.

Theorem 7. If I is a nodal ideal of L and x is a node of L, then I (x) is a nodal ideal of L, too.

Proof. Let I be a nodal ideal of L and x be a node of L . If x ∈ I, then I (x) = I is a nodal ideal of L . By Proposition 8, since x is a node of L, it follows that (x] is a nodal ideal of L . We obtain successively I (x) = (I ∪ {x}] = (I ∪ (x]] = I ∪ (x] (as the union of nodal ideals is a nodal ideal). Therefore, I (x) is a nodal ideal of L . □

Example 10. In this example, we show that if I is a nodal ideal of L and the element a is not a node of L, then I (a) possible is not a nodal ideal of L . Indeed, in the residuated lattice L from Example 5, we have that I = {0} is a nodal ideal of L and the element a is not a node of L . We conclude that I (a) = (I ∪ {a}] = (I ∪ (a]] = (a] = {0, a} , but (a] is not a nodal ideal of L (because the ideals (a] = {0, a} and (b] = {0, b} are incomparable).

Example 11. In this example, we show that the extension property may not hold for any nodal ideals. Indeed, in the residuated lattice L from Example 5, we have the ideals I₁ = {0} , I₂ = {0, a} , I₃ = {0, b} , where I₁ is a nodal ideal, I₁ ⊆ I₂, I₁ ⊆ I₃, but I₂ and I₃ are incomparable (that is, I₂ and I₃ are not nodal ideals).

We recall [9] that for two residuated lattices L₁ and L₂, the map f : L₁ → L₂ is called a homomorphism if and only if it satisfies the following conditions, for all x, y ∈ L₁ :

f (0) =0, f (1) =1 ;

f (x ⊙ y) = f (x)⊙ f (y) ;

f (x → y) = f (x) → f (y) .

Moreover, if I is an ideal of L₂, then f^-1 (I) is an ideal of L₁ .

Corollary 2. Let L₁ and L₂ be two residuated lattices. If the map f : L₁ → L₂ is a homomorphism of residuated lattices and I is a nodal ideal of L₂, then the inverse image of I is a nodal ideal of L₁ .

Proof. Let I be a nodal ideal of L₂ and x, y ∈ L₁ . We need to prove that for every x ∈ f^-1 (I) and y ∉ f^-1 (I) implies x ≨ y, in order to apply Theorem 4. Let x, y ∈ L be such that x ∈ f^-1 (I) and y ∉ f^-1 (I) . Thus f (x) ∈ I and f (y) ∉ I . By Theorem 4, since I is a nodal ideal of L₂, it follows that that f (x) ≨ f (y) . By (c₂) , we obtain successively f (x) → f (y) =1, f (x → y) =1, x → y = 1, x ≤ y . If x = y, then x ∈ f^-1 (I) and y ∈ f^-1 (I) , that is a contradiction. Hence x ≨ y . We conclude that for every x ∈ f^-1 (I) and y ∉ f^-1 (I) implies x ≨ y, and applying Theorem 4, we obtain that f^-1 (I) is a nodal ideal of L₁ . □

In what follows, we will provide some examples with the goal to establish the relationship between nodal ideals and other types of ideals in residuated lattices. The conclusion is that the class of nodal ideals in residuated lattices is a new class of ideals.

Example 12. In this example, we show that a nodal ideal may not be a Boolean ideal and a Boolean ideal may not be a nodal ideal, in general. In Example 1, we have that I = {0} is a nodal ideal, but it is not a Boolean ideal because a ∧ a^* = a ∧ m = a ∉ I .

On the other hand, in Example 2, we have that J = {0, n, a, b, c} is a Boolean ideal, but it is not a nodal ideal because there are two ideals P = {0, n, b, d, f} and Q = {0, n, a, d, e} incomparable with I .

Proposition 9. Let L be a G-algebra. Then every nodal ideal of L is a Boolean ideal.

Proof. Let L be a G-algebra and x, y ∈ L . Clearly, x ⊙ y ≤ x, y . To prove x ⊙ y = x ∧ y, let t ∈ L be such that t ≤ x and t ≤ y . Since L is a G-algebra, we have that t² = t . Then t = t² ≤ x ⊙ y, that is, x ⊙ y = x ∧ y . Let I be a nodal ideal of L . Since x ∧ x^* = x ⊙ x^* = 0 ∈ I, it follows that x ∧ x^* ∈ I, for all x ∈ L . Therefore, I is a Boolean ideal of L . □

Remark 9. We note that, in residuated lattices, Boolean ideals have the extension property, that is, if I is a Boolean ideal and J is an ideal of L such that I ⊆ J, then J is a Boolean ideal, too. Indeed, if I is a Boolean ideal, then x ∧ x^* ∈ I ⊆ J, for all x ∈ L . Therefore, J is a Boolean ideal.

Example 13. It is easy to ascertain that in the residuated lattice L from Example 5, we have that I₁ = {0} is a nodal ideal. Since I₂ = {0, a} , I₃ = {0, b} are ideals of L and I₁ ⊆ I₂, I₃, by Remark 9, we get that I₂ and I₃ are Boolean ideals, too. Moreover, L is a G-algebra and I₁ is the only nodal ideal and it is a Boolean ideal of L, that is, the condition from Proposition 9 holds. The converse of Proposition 9 may not always hold, indeed, I₂ and I₃ are Boolean ideals and L is a G-algebra, but they are not nodal ideals of L .

Proposition 10. If I is a proper ideal such that for any x ∈ L \ {1} , x ∉ I implies x^* ∉ I is satisfied, then I is a prime ideal of L .

Proof. Let I be a proper ideal of L and x, y ∈ L \ {1} . By contrary, we assume that x ∧ y ∈ I with x ∉ I and y ∉ I . By hypothesis, x ∉ I implies x^* ∉ I and y ∉ I implies y^* ∉ I are satisfied. Since x ∧ y ∈ I, it is necessary that (x ∧ y) ^* ∉ I, and by hypothesis, it follows that (x ∧ y) ^** ∉ I . By Proposition 1, x ∧ y ∈ I iff (x ∧ y) ^** ∈ I, that is a contradiction. Therefore, I is a prime ideal of L . □

Example 14. It is easy to ascertain that in the residuated lattice L from Example 2, I = {0} is a prime ideal of L and the condition from Proposition 10 holds. The converse of Proposition 10 may not always hold, indeed, in the residuated lattice L from Example 5, I = {0, a} is a prime ideal of L, but b ∉ I and b^* = a ∈ I .

Theorem 8. If I is a non-principal nodal ideal of L, then I is a prime ideal of L .

Proof. Let I be a non-principal nodal ideal of L and x, y ∈ L . By contrary, we assume that x ∧ y ∈ I with x ∉ I and y ∉ I . Since I is a nodal ideal of L and x, y ∉ I, by Theorem 4, it follows that t ≨ x and t ≨ y, for all t ∈ I . Then t ≤ x ∧ y, for all t ∈ I . Since x ∧ y ∈ I, we conclude that I = (x ∧ y] , that is a contradiction with the fact that I is a non-principal ideal of L . □

The converse of Theorem 8 may not always hold, indeed, in the residuated lattice from Example 2, we have I = {0} is a nodal prime ideal of L, but it is a principal ideal (because I = (0]).

Example 15. In this example, we show that a nodal ideal may not be a prime ideal and a prime ideal may not be a nodal ideal, in general. In Example 1, we have that I = {0} is a nodal ideal, but it is not a prime ideal because a ∧ d = 0 ∈ I and a, d ∉ I .

On the other hand, in Example 2, we have that J = {0, n, a, b, c} is a prime ideal, but it is not a nodal ideal because there are two ideals P = {0, n, b, d, f} and Q = {0, n, a, d, e} incomparable with I .

Example 16. In this example, we show that a nodal ideal may not be a maximal ideal and a maximal ideal may not be a nodal ideal, in general. In Example 5, we have that I₁ = {0} is a nodal ideal, but it is not a maximal ideal because there are two ideals I₂ = {0, a} and I₃ = {0, b} which include it.

On the other hand, in Example 2, we have that J = {0, n, a, b, c} , P = {0, n, b, d, f} and Q = {0, n, a, d, e} are three maximal ideals of L, but they are not nodal ideals because they are incomparable two-by-two.

Corollary 3. Let L be a G-algebra. Then every non-principal nodal ideal of L is a primary ideal.

Proof. Let L be a G-algebra and x, y ∈ L . By the proof of Proposition 9, we have that x ⊙ y = x ∧ y . Let I be a nodal ideal of L . By Theorem 8, I is a prime ideal of L . By contrary, we assume that x ⊙ y ∈ I with xⁿ ∉ I and yⁿ ∉ I, for all integers n ≥ 1 . Since L is a G-algebra, it follows that xⁿ = x and yⁿ = y, for all integers n ≥ 1 . Since I is a prime ideal and x ∧ y = x ⊙ y ∈ I, it follows that x ∈ I or y ∈ I . Hence, for n = 1 we have xⁿ = x¹ ∈ I or yⁿ = y¹ ∈ I, that is a contradiction. Therefore, I is a primary ideal of L . □ The converse of Proposition 3 may not always hold, indeed, in the G-algebra L from Example 5, we have I = {0, a} is a primary ideal of L, but it is a principal ideal (because I = (a]).

Example 17. In this example, we show that a nodal ideal may not be a primary ideal and a primary ideal may not be a nodal ideal, in general. In Example 2, we have that I₁ = {0, n} is a nodal ideal, but it is not a primary ideal because a ⊙ d = 0 ∈ I₁, and for all n ≥ 1, aⁿ = a, dⁿ = d and a, d ∉ I₁ .

On the other hand, in Example 2, we have that J = {0, n, a, b, c} , P = {0, n, b, d, f} and Q = {0, n, a, d, e} are three primary ideals of L, but they are not nodal ideals because they are incomparable two-by-two.

Corollary 4. Let L be a G-algebra. Then every non-principal nodal ideal of L is a implicative ideal.

Proof. Let L be a G-algebra and x, y, z ∈ L . By the proof of Proposition 9, we have that x ⊙ y = x ∧ y . Let I be a nodal ideal of L . By contrary, we assume that z ⊙ (y^* ⊙ x^*) ∈ I and y ⊙ x^* ∈ I with z ⊙ x^* ∉ I . Since L is a G-algebra, it follows that z ⊙ (y^* ⊙ x^*) = z ∧ (y^* ∧ x^*) ∈ I, y ⊙ x^* = y ∧ x^* ∈ I and z ⊙ x^* = z ∧ x^* ∉ I . By Theorem 8, I is a prime ideal of L . If z ∧ (y^* ∧ x^*) ∈ I, then z ∈ I or y^* ∈ I or x^* ∈ I . We have the following three situations: If z ∈ I, then z ⊙ x^* ≤ z ∈ I . If x^* ∈ I, then z ⊙ x^* ≤ x^* ∈ I . If y^* ∈ I, then y ∉ I, on the other hand y ∧ x^* ∈ I, so there is necessary that x^* ∈ I, and so z ⊙ x^* ≤ x^* ∈ I . We conclude that in all three situations we get contradictions, so I is an implicative ideal of L . □

Example 18. In this example, we show that a nodal ideal may not be an implicative ideal, in general. For that we consider the lattice L = {0, a, b, c, d, e, f, g, 1} with the Hasse diagram:

Then ([13], page 166) L is a residuated lattice with respect to the following operations:

It is easy to ascertain that I₁ = {0} , I₂ = {0, a, b} and I₃ = {0, c, f} are all proper ideals of L . We have that b ⊙ (a^* ⊙ g^*) = b ⊙ (g ⊙ a) = b ⊙ 0 =0 ∈ I₁ and a ⊙ g^* = a ⊙ a = 0 ∈ I₁, but b ⊙ g^* = b ⊙ a = a ∉ I₁ . Hence I₁ is a nodal ideal, but it is not an implicative ideal.

Example 19. In this example, we show that an implicative ideal may not be a nodal ideal, in general. For that we consider the lattice L={0, a, b, c, d, e, f, 1} with the Hasse diagram:

Then ([13], page 164) L is a residuated lattice with respect to the following operations:

It is easy to ascertain that I₁ = {0, b, d, f} , I₂ = {0, a, b, c} and I₃ = {0, a, d, e} are implicative ideals, but they are not nodal ideals because they are incomparable. Moreover, I₁ = (f] , I₂ = (c] and I₃ = (e] are principal implicative ideals and L is a G-algebra, so the converse of Corollary 4 may not hold.

Example 20. In this example, we show that a nodal ideal may not be an obstinate ideal and an obstinate ideal may not be a nodal ideal, in general. In Example 2, we have that I = {0} is a nodal ideal and c, f ∉ I, but c^* ⊙ f = d ⊙ f = d ∉ I and c ⊙ f^* = c ⊙ a = a ∉ I, it follows that I is a nodal ideal, but it is not an obstinate ideal.

On the other hand, in Example 2, we have that J = {0, n, a, b, c} is an obstinate ideal, but it is not a nodal ideal.

Example 21. In this example, we show that a nodal ideal may not be an ⊙-prime ideal and an ⊙-prime ideal may not be a nodal ideal, in general. In Example 5, we have that I = {0} is a nodal ideal, but it is not an ⊙-prime ideal because a ⊙ b = 0 ∈ I and a, b ∉ I .

On the other hand, in Example 5, we have that J = {0, a} is an ⊙-prime ideal, but it is not a nodal ideal.

3.1. An equivalence relation

Definition 9. We say that two elements x and y of L are connected (in symbol (x, y) ∈ R) if there is no nodal ideal which separates them, let us notice that (x, y) ∉ R implies either x ≨ y or y ≨ x .

Example 22. In Example 2, we have that I = {0, n} is a nodal ideal of L . Clearly, (a, n) ∉ R and a, d are connected (in symbol (a, d) ∈ R).

Proposition 11. The binary relation R is an equivalence relation.

Proof. We prove that the binary relation R is symmetric, reflexive and transitive. By Definition 9, we have that (x, y) ∈ R iff (y, x) ∈ R and (x, x) ∈ R, hence R is symmetric and reflexive. Let (x, y) ∈ R, (y, z) ∈ R and (x, z) ∉ R . Then there is a nodal ideal I such that x ∈ I and z ∉ I . Hence y ∈ I and (y, z) ∉ R, that is a contradiction. Hence R is transitive. Therefore, R is an equivalence relation. □

Theorem 9. The relation R has the following properties:

Any R-class contains at most one node;

Any R-class is totally ordered if and only if it is a singleton;

L/R is a chain dually isomorphic to Nod_Id (L) .

Proof.

Let x and y be connected nodes of L . We have either x ≨ y or y ≨ x . By Proposition 8, we conclude that in the first case, x and y are separated by the nodal ideal (x] , and in the second case by (y] .

We assume that (x] R (the R-class of x) is totally ordered and (y, x) ∈ R, x ≠ y . Any element a ∈ L is comparable with x and y . Hence both x and y are nodes of L, that is a contradiction.

We define the mapping φ : L/R → Nod_Id (L) by φ (t) = I_t, where I_t is a nodal ideal generated by the R-class t . In fact, I_t = {x ∈ L : x ≤ y forsome y ∈ t} . Obviously, φ is bijective and in L/R if and only if in Nod_Id (L) .□

Example 23. We consider the residuated lattice L from Example 5. Clearly, Nod_Id (L) = {{0} , L} and R = {(0, 0) , (a, a) , (a, b) , (b, a) , (b,b),(a,n),(n,a), (b, n) , (n, b) , (n, n) , (a, c) , (c, a) , (c, n) , (n, c) , (c, c) , (a, d) , (d, a) , (d, b) , (b, d) , (d, n) , (n, d) , (c, d) , (d, c) , (1, a) , (a, 1) , (1, b) , (b, 1) , (1, n) , (n, 1) , (1, c) , (c, 1) , (1, d) , (d, 1) , (1, 1)} . We have L/R = {[0] , [1]} . So L/R is a chain dually isomorphic to Nod_Id (L) . Moreover, it is clear that R is not antisymmetric because (c, d) ∈ R and (d, c) ∈ R, but c ≠ d .

3.2. The direct product

Let L₁, L₂ be residuated lattices. On L₁ × L₂ we consider the order relation (x, y) ≤ (x′, y′) iff x ≤ x′ and y ≤ y′ and the operations

(x, y) ∧ (x′, y′) = (x ∧ x′, y ∧ y′) ,

(x, y) ∨ (x′, y′) = (x ∨ x′, y ∨ y′) ,

(x, y) ⊙ (x′, y′) = (x ⊙ x′, y ⊙ y′) ,

(x, y) → (x′, y′) = (x → x′, y → y′) for all x, y ∈ L₁ and x′, y′ ∈ L₂ .

The L₁ × L₂ with the above operations is a residuated lattice called the direct product of L₁ and L₂ .

Theorem 10. Let L₁, L₂ be residuated lattices. Then Q is an nodal ideal of L₁ × L₂ iff there exist I ∈ Nod_Id (L₁) and J ∈ Nod_Id (L₂) such that Q = I × J .

Proof. “⇒”. Let Q ∈ Nod_Id (L₁ × L₂) . We consider the sets I = {x ∈ L₁ : (x, x′) ∈ Q forsome x′ ∈ L₂} and J = {x′ ∈ L₂ : (x, x′) ∈ Q forsome x ∈ L₁} .

Clearly, Q = I × J . Since (0, 0) ∈ Q, it follows that 0 ∈ I . Let x, y ∈ I . Thus, there are x′, y′ ∈ L₂ such that (x, x′) , (y, y′) ∈ Q . Hence (x, x′) ⊕ (y, y′) = (x ⊕ y, x′ ⊕ y′) ∈ Q . We conclude that x ⊕ y ∈ I .

Let x ≤ y and y ∈ I . Thus, there is x′ ∈ L₂ such that (x, x′) ∈ Q . Since (x, x′) ≤ (y, x′) , it follows that x ∈ I . So $I \in I (L_{1}) .$ Similarly, $J \in I (L_{2}) .$ The fact that I ∈ Nod_Id (L₁) and J ∈ Nod_Id (L₂) follows from Theorem 4.

“⇐”. Let Q = I × J for some I ∈ Nod_Id (L₁) and J ∈ Nod_Id (L₂) . Clearly, Q ⊆ L₁ × L₂ . We consider (x, y) , (p, q) ∈ Q . Hence x, p ∈ I and y, q ∈ J, that is x ⊕ p ∈ I, y ⊕ q ∈ J . Therefore, (x, y) ⊕ (p, q) = (x ⊕ p, y ⊕ q) ∈ I × J = Q .

Now, let (p, q) ∈ Q be such that (x, y) ≤ (p, q) . Then x ≤ p with p ∈ I and y ≤ q with q ∈ J . Since I ∈ Nod_Id (L₁) and J ∈ Nod_Id (L₂) , it follows that x ∈ I and y ∈ J, that is (x, y) ∈ Q . Hence Q ∈ Nod_Id (L₁ × L₂) . □

By mathematical induction relative to n we obtain the following result:

Corollary 5. Let $\prod_{i = 1}^{n} L_{i}$ be a finite direct product of residuated lattices. Then J is a nodal ideal of $\prod_{i = 1}^{n} L_{i}$ iff there exist I_i ∈ Nod_Id (L_i) such that $J = \prod_{i = 1}^{n} I_{i},$ for every 1 ≤ i ≤ n .

3.3. Conodal ideals in residuated lattices

We recall that an element a of a resiuated lattice L is called a conode of L, if there exist elements of L which are incomparable with a . We denote the set of all conodal elements of L by CoNod (L) . In the residuated lattice L from Example 2, the set of all conodes of L is CoNod (L) = {a, b, c, d, e, f} . Clearly, nodal and conodal elements of L are dual notions.

Definition 10. An ideal I of L, will be called a conodal ideal of L, if I is a conode of $I (L) .$ In the residuated lattice L from Example 2, there are three conodal ideals of L : I = {0, n, a, c, d} , J = {0, n, b, d, f} and Q = {0, n, a, d, e} . Clearly, nodal and conodal ideals of L are dual notions. By duality, we obtain the following results:

Theorem 11. Let I be an ideal of L . Then I is a conodal ideal of L if and only if for some x ∈ I and for some y ∉ I, the relation xnleqny, for all n ≥ 1, is satisfied.

Proof. Let I be a conodal ideal of L . Then there is an ideal J of L such that I ⊈ J and J ⊈ I . Hence there are x ∈ I \ J and y ∈ J \ I . We conclude that xnleqny, for all n ≥ 1 . Conversely, we assume that I is not a conodal ideal of L . Since nodal and conodal ideals of L are dual notions, it follows that I ⊆ J or J ⊆ I, for all $J \in I (L) .$ So, for some x ∈ I and for some y ∉ I, we conclude that I ⊆ (y] or (y] ⊆ I . Since (y] ⊈ I and I is a nodal ideal, it follows that I ⊆ (y] . Hence, x ∈ (y] . By Proposition 2(ii), it follows that x ≤ ny, for some n ≥ 1, that is a contradiction. Therefore, I is a conodal ideal of L . □

Corollary 6. Let L₁ and L₂ be two residuated lattices. If the map f : L₁ → L₂ is a homomorphism of residuated lattices and I is a conodal ideal of L₂, then the inverse image of I is a conodal ideal of L₁ .

Proof. Let I be a conodal ideal of L₂ and x, y ∈ L₁ . We need to show that for some x ∈ f^-1 (I) and for some y ∉ f^-1 (I) implies xnleqny, for all n ≥ 1, in order to apply Theorem 11. We consider the elements x, y ∈ L₁ be such that x ∈ f^-1 (I) and y ∉ f^-1 (I) . Since I is an ideal of L₂ and f a homomorphism, it follows that f^-1 (I) is an ideal of L₁ . By Proposition 1(ii) , it follows that y ∉ f^-1 (I) if and only if ny ∉ f^-1 (I) , for all n ≥ 1 . Then f (x) ∈ I and f (ny) ∉ I, for all n ≥ 1 . By Theorem 11, since I is a conodal ideal of L₂, we conclude that f (x) nleqf (ny) . By (c₂) , we obtain successively f (x) → f (ny) ≠1, f (x → ny) ≠1, x → ny ≠ 1, xnleqqny . Hence xnleqny . We conclude that for some x ∈ f^-1 (I) and for some y ∉ f^-1 (I) implies xnleqny, for all n ≥ 1, and using Theorem 11, we conclude that f^-1 (I) is a conodal ideal of L₁ . □

Example 24. In this example, we show that if x is a conode of L, then (x] may not be a conodal ideal of L . Indeed, in the residuated lattice L from Example 5, we have that d is a conode of L, but (d] = L is a nodal ideal of L . Moreover, since I = {0, a} is a conodal ideal, Q = L is a nodal ideal and I ⊆ Q, we conclude that the extension property does not hold.

On the other hand, we show that if I is a conodal ideal of L, and x a conode of L, then I (x) may not be a conodal ideal of L . Indeed, in residuated lattice L from Example 5, we have that I = {0, a} and J = {0, b} are conodal ideals of L and d is a conode of L, but I (d) = L and J (d) = L are nodal ideals of L (because a ⊕ d = b ⊕ d = 1).

In what follows, we will provide some examples with the goal to establish the relationship between conodal ideals and other types of ideals in residuated lattices. The conclusion is that the class of conodal ideals in residuated lattices is a new class of ideals.

Example 25. In this example, we show that a conodal ideal may not be a Boolean ideal and a Boolean ideal may not be a conodal ideal, in general. In Example 18, we have that I = {0, a, b} is a conodal ideal, but it is not a Boolean ideal because e ∧ e^* = e ∧ c = c ∉ I .

For the converse, we consider the lattice L = {0, n, a, b, c, d, 1} with the Hasse diagram:

Then ([13], page 247) L is a residuated lattice with respect to the following operations:

It is easy to ascertain that I = {0} is a Boolean ideal, but it is not a conodal ideal.

Example 26. In this example, we show that a conodal ideal may not be a prime ideal and a prime ideal may not be a conodal ideal, in general. It is easy to ascertain that in Example 19, I = {0, a} is a conodal ideal, but it is not a prime ideal because c ∧ e = a ∈ I and c, e ∉ I .

On the other hand, in Example 2, we have that J = {0} is a prime ideal, but it is not a conodal ideal.

Example 27. In this example, we show that a conodal ideal may not be a maximal ideal and a maximal ideal may not be a conodal ideal, in general. In Example 25, we have that I₁ = {0} is a maximal ideal, but it is not a conodal ideal.

On the other hand, in Example 19, we have that J = {0, a} is a conodal ideal, but it is not maximal because there is Q = {0, a, b, c} an ideal such that J ⊂ Q .

Example 28. In this example, we show that a conodal ideal may not be a primary ideal and a primary ideal may not be a conodal ideal, in general. In Example 25, we have that I = {0} is a primary ideal, but it is not a conodal ideal.

On the other hand, in Example 19, we have that J = {0, a} is a conodal ideal, but it is not a primary ideal because c ⊙ d = 0 ∈ J and for all n ≥ 1, cⁿ = c ∉ J and dⁿ = d ∉ J .

Example 29. In this example, we show that a conodal ideal may not be an implicative ideal and an implicative ideal not be a conodal ideal, in general. In Example 1, we have that I = {0} is an implicative ideal, but it is not a conodal ideal. Another Example of an implicative ideal which is not a conodal ideal is I = {0} from Example 25.

On the other hand, in Example 18, we have that J = {0, a, b} is a conodal ideal, but it is not an implicative ideal because f ⊙ (c^* ⊙ d^*) = f ⊙ (e ⊙ d) = f ⊙ a = 0 ∈ J and c ⊙ d^* = c ⊙ d = 0 ∈ J, but f ⊙ d^* = f ⊙ d = c ∉ J .

Example 30. In this example, we show that a conodal ideal may not be an obstinate ideal and an obstinate ideal may not be a conodal ideal, in general. In Example 18, we have that I = {0, a, b} is a conodal ideal, but it is not an obstinate ideal because e, g ∉ I and e^* ⊙ g = c ⊙ g = c ∉ I .

On the other hand, in Example 25, we have that I = {0} is an obstinate ideal, but it is not a conodal ideal.

Example 31. In this example, we show that a conodal ideal may not be an ⊙-prime ideal and an ⊙-prime ideal may not be a conodal ideal, in general. In Example 19, we have that I = {0, a} is a conodal ideal, but it is not an ⊙-prime ideal because c ⊙ e = a ∈ I and c, e ∉ I .

On the other hand, in Example 25, we have that I = {0} is an ⊙-prime ideal, but it is not a conodal ideal.

4. Applications

The variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean residuated lattices, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices (see [11]).

In Theorem 9 from [11] is proved: If I is an ideal of a De Morgan residuated lattice L, then the binary relation θ_I on L ((x, y) ∈ θ_I if and only if x^* ⊙ y ∈ I and x ⊙ y^* ∈ I) is a congruence relation. For x ∈ L, we denote by [x] _{θ
_I} : = x/I the congruence class of x modulo θ_I and L/I = {x/I : x ∈ L} . On L/I, we define the binary operations ∨, ∧ , ⊙ and → by (x/I) ∨ (y/I) = (x ∨ y)/I, (x/I) ∧ (y/I) = (x ∧ y)/I, (x/I) ⊙ (y/I) = (x ⊙ y)/I and (x/I) → (y/I) = (x → y)/I for all x, y ∈ L . Then (L/I, ∨ , ∧ , ⊙ , → , 0, 1) is the quotient De Morgan residuated lattice of L with respect to I. Clearly, 0 = 0/I and 1 = 1/I. The order relation on L/I is defined by (x/I) ≤ (y/I) if and only if (x → y) ^* ∈ I. For a nonempty subset S of L, we denote by S/I = {x/I : x ∈ S}. Clearly, for x ∈ L, we have that x/I = 1 if and only if x^* ∈ I and x/I = 0 if and only if x ∈ I .

Remark 10. We note that for every congruence θ on L, we have that [0] _θ is an ideal of L . We shall denote it by I_θ . Conversely, if I is any ideal of L, then the relation θ_I on L defined by (x, y) ∈ θ_I if and only if x^* ⊙ y ∈ I and x ⊙ y^* ∈ I is a congruence relation. In other words, the correspondence between ideals and congruences is one-to-one in De Morgan residuated lattices.

Theorem 12. Let θ be a congruence on L . Then the following assertions are equivalent:

[0] _θ is a nodal ideal of L;

θ is a node of con (L) , the congruence lattice of L .

Proof. The mapping θ → I_θ is an isomorphism of con (L) on to $I (L)$ and I_θ is a nodal ideal if and only if it is a node of $I (L) .$ □

Lemma 3. If L is a De Morgan residuated lattice and x, y ∈ L, then:

(mx) ∧ (ny) ≤ (mn) (x ∧ y) , for all m, n≥ 1 ;

(x]∩ (y] = (x ∧ y] ;

If I is a non-principal nodal ideal, then I is a prime ideal.

Proof. (i) . See Corollary 5, from [11].

(ii) . Let x, y ∈ L . Since x ∧ y ≤ x, y, it follows that (x ∧ y] ⊆ (x] and (x ∧ y] ⊆ (y] , and so (x ∧ y] ⊆ (x] ∩ (y] . Conversely, by item (i) , we have (mx) ∧ (ny) ≤ (mn) (x ∧ y) , for all m, n ≥ 1 . We obtain successively (x] ∩ (y] = {a∈ L : a ≤ mx forsome m ≥ 1} ∩ {a∈ L : a ≤ ny forsome n ≥ 1} ⊆ {a∈ L : a ≤ (mx) ∧ (ny) forsome m, n ≥ 1} ⊆ {a ∈ L : a ≤ (mn) (x ∧ y) forsome n ≥ 1} = (x ∧ y] . Therefore, (x] ∩ (y] = (x ∧ y] .

(iii) . Let I be a non-principal nodal ideal of L, such that x ∧ y ∈ I and x ∉ I and y ∉ I . We have that (x ∧ y] ⊆ I . Since x ∉ I and y ∉ I, it follows that (x] ⊈ I and (y] ⊈ I, so I ⊂ (x] and I ⊂ (y] . By item (ii) , it follows that I ⊆ (x] ∩ (y] = (x ∧ y] . We conclude that I = (x ∧ y] , that is a contradiction. Therefore, x ∈ I or y ∈ I, that is, I is a prime ideal.□

Theorem 13. Let L be a MTL-algebra and I a non-principal nodal ideal of L . Then (L/I, ∧ , ∨ , → , ⊙ , 0, 1) is a chain.

Proof. Let L be an MTL-algebra. It is known that [11] the class of MTL-algebras is a subclass of De Morgan residuated lattices, so Lemma 3 hold for L . Let I be a non-principal nodal ideal of L and a/I, b/I be such that a/Inleqb/I . Then (a → b) ^* ∉ I . Since L is an MTL-algebra and by (c₁₁) , it follows that (a → b) ^* ∧ (b → a) ^* = [(a → b) ∨ (b → a)] ^* = 1^* = 0 ∈ I . Since I is a non-principal nodal ideal, by Lemma 3(iii), it follows that I is a prime ideal, so (b → a) ^* ∈ I . Hence, b/I ≤ a/I . Therefore, L/I is a chain. □

Example 32. The converse of Theorem 13 may not hold. Indeed, if we consider the chain L₅ = {0, a, 1} with the Hasse diagram:

Organized as a residuated lattice by natural ordering, with the operations → and ⊙ as in the following tables:

It is easy to ascertain that L is a MTL-algebra and I = {0} is the only proper ideal of L . We have successively 0/I ≤ a/I iff (0 → a) ^* = 0 ∈ I and a/I ≤ 1/I iff (a → 1) ^* = 0 ∈ I . Hence L/I = {0/I, a/I, 1/I} is a chain, but I is a principle nodal ideal of L .

Open problem: Does the assertion from Theorem 13 hold for every residuated lattice L .

Next, we study nodal (conodal) ideals in special classes of residuated lattices such as involution residuated lattices and semi-G-algebras. It is known that the classes of involution residuated lattices and semi-G-algebras are different [11]. We present a characterization for nodal ideals in involution residuated lattices. We propose a new characterization for Boolean residuated lattices, in this sense we prove that a residuated lattice L is a Boolean algebra if and only if L is an involution semi-G-algebra. That is, the class of Boolean residuated lattices is exactly the intersection between the classes of involution residuated lattices and semi-G-algebras. We consider this new characterization of Boolean residuated lattices because of its applications in Graph Theory.

In the next result, we present the relationship between ideals and filters in the case of involution residuated lattices (that is, x^** = x for every x ∈ L). We recall that the set of complemented elements (with respect to X) is defined by D (X) : = {x ∈ L : x^* ∈ X} .

Lemma 4. Let L be an involution residuated lattice. Then I is an ideal of L if and only if D (I) is a filter of L .

Proof. ⇒ . Let I be an ideal of L . Since 1 ∉ I, it follows that 0 ∉ D (I) . Since 1^* = 0 ∈ I, it follows that 1 ∈ D (I) . Let x ≤ y and x ∈ D (I) . Since x ∈ D (I) , it follows that x^* ∈ I . By (c₄) , y^* ≤ x^* ∈ I, and so y^* ∈ I (as I is an ideal of L). Since y^* ∈ I, it follows that y ∈ D (I) . Hence, D (I) is an upper set. Now, let x, y ∈ D (I) , we need to prove that x ⊙ y ∈ D (I) . Since x, y ∈ D (I) , we have x^* ∈ I and y^* ∈ I . Then x^* ⊕ y^* ∈ I . By Proposition 4, x^** ⊙ y^** ∈ D (I) iff (x^** ⊙ y^**) ^* ∈ I iff x^* ⊕ y^* ∈ I . Hence, x^** ⊙ y^** ∈ D (I) . Since L is an involution residuated lattice, we conclude that x ⊙ y ∈ D (I) . Therefore, D (I) is a filter of L .

⇐ . Let I be a subset of L such that D (I) is a filter of L . Since 0 ∉ D (I) , it follows that 0^* = 1 ∉ I . Since 1 ∈ D (I) , it follows that 1^* = 0 ∈ I . So I is nonempty. Let x, y ∈ L be such that x ≤ y and y ∈ I . We need to prove that x ∈ I . By Proposition 4, it follows that y ∈ I iff y^* ∈ D (I) iff y^** ∈ I . Hence y^* ∈ D (I) . By (c₄) , since x ≤ y, it follows that y^* ≤ x^* . Since D (I) is a filter of L, y^* ∈ D (I) and y^* ≤ x^*, it follows that x^* ∈ D (I) , and applying Proposition 4, we conclude that x^* ∈ D (I) iff x ∈ I . Hence x ∈ I (that is, I is a down set). Now, let x, y ∈ I . We need to prove that x ⊕ y ∈ I . By Proposition 4, it follows that x^* ∈ D (I) and y^* ∈ D (I) . Since D (I) is a filter of L, x^* ∈ D (I) , y^* ∈ D (I) , it follows that x^* ⊙ y^* ∈ D (I) (as D (I) is a filter). Since x^* ⊙ y^* ∈ D (I) , by definition of D (I) , we conclude that x ⊕ y = (x^* ⊙ y^*) ^* ∈ I . Therefore, I is an ideal of L . □

By Lemma 4, there is a strong connection between ideals and filters in involution residuated lattices. An ideal $I \in I (L)$ is called proper if I ≠ L.

Theorem 14. If L is an involution residuated lattice, then I is a nodal ideal of L if and only if D (I) is a nodal filter of L .

Proof. ⇒ . Let L be an involution residuated lattice and I is a nodal ideal of L . Then x^** = x, for all x ∈ L . By Lemma 4, D (I) is a filter of L . It remains to prove that D (I) is a nodal filter of L . Let x ∈ D (I) and y ∉ D (I) . That is, x^* ∈ I and y^* ∉ I . We need to prove that x ≩ y, in order to apply Theorem 1. Since I is a nodal ideal of L, x^* ∈ I and y^* ∉ I, by Theorem 4(iii), it follows that x^* ≨ y^* . By (c₅) , y^** ≤ x^**, so y ≤ x (as L is an involution residuated lattice). If x = y, then y ∈ D (I) , that is a contradiction. Hence x ≩ y . We conclude that for all x ∈ D (I) and for all y ∉ D (I) , the relation x ≩ y is satisfied, and applying Theorem 1, we conclude that D (I) is a nodal filter of L .

⇐ . Let I be a nonempty set of L and D (I) a nodal filter of L . By Lemma 4, I is an ideal of L . It remains to prove that I is a nodal ideal of L . Let x ∈ I and y ∉ I . By Proposition 4, x^* ∈ D (I) and y^* ∉ D (I) . We need to prove that x ≨ y, in order to apply Theorem 4. Since D (I) is a nodal filter of L, x^* ∈ D (I) and y^* ∉ D (I) , by Theorem 1(iii), it follows that x^* ≩ y^* . By (c₅) , x^** ≤ y^**, so x ≤ y (as L is an involution residuated lattice). If x = y, then y ∈ I, that is a contradiction. Hence x ≨ y . We conclude that for all x ∈ I and for all y ∉ I, the relation x ≨ y is satisfied, and applying Theorem 4, we conclude that I is a nodal ideal of L . □

Proposition 12. If L is an involution residuated lattice, then L is a semi-G-algebra if and only if x ⊕ x = x, for all x ∈ L .

Proof. Let L be an involution semi-G-algebra, that is, x^** = x and (x²) ^* = x^*, for all x ∈ L . We obtain successively x ⊕ x = (x^* ⊙ x^*) ^* = [(x^*) ²] ^* = x^** = x . Conversely, since x^* = x^* ⊕ x^* = (x^** ⊙ x^**) ^* = (x ⊙ x) ^* = (x²) ^*, it follows that (x²) ^* = x^*, for all x ∈ L . Therefore, L is a semi-G-algebra. □

We recall [9] that an element x ∈ L is Boolean if and only if x ∨ x^* = 1 .

Theorem 15. L is a Boolean residuated lattice if and only if L is an involution semi-G-algebra.

Proof. ⇒ . Assume that L is a Boolean residuated lattice. Let x ∈ L . Then x ∈ B (L) and we conclude that x = x^** and x² = x . So (x²) ^* = x^* . Therefore, L is an involution semi-G-algebra.

⇐ . We assume that L is an involution semi-G-algebra. Let x ∈ L and t ∈ L be such that t ≤ x, t ≤ x^* . Since L is a semi-G-algebra, it follows that (t²) ^* = t^*, for all t ∈ L . We obtain successively t² ≤ x ⊙ x^* = 0, t² = 0, (t²) ^* = 0^* = 1, t^* = 1, t^** = 0, t ≤ t^** = 0, t = 0 . Hence, x ∧ x^* = 0 . Since L is an involution residuated lattice (x = x^**, for all x ∈ L) and using (c₁₁) , we obtain successively 1 = 0^* = (x ∧ x^*) ^* = (x^** ∧ x^*) ^* = [(x^* ∨ x) ^*] ^* = (x^* ∨ x) ^** . Hence, (x^* ∨ x) ^** = 1 . By (c₅) , we conclude that x^* ∨ x = 1, that is, x ∈ B (L) . Therefore, L is a Boolean residuated lattice. □

The following result is a directly consequences of Theorem 15.

Corollary 7. Let L be a Boolean residuated lattice. Then

(a] = {x ∈ L : x ≤ a} and (a] ∩ (b] = (a ∧ b] , for all a, b∈ L ;

I is a nodal ideal of L if and only if for every x, y ∈ L such that x ∈ I and y ∉ I, the relation x < y is satisfied;

if x ∈ L and (x] is a nodal ideal of L, then x is a node of L ;

$I (L) \cap {Nod}_{Id} (L) = {{0}, L};$

I is a nodal ideal if and only if D (I) is a nodal filter.

Proof. (i) . By Proposition 12, we have that x ⊕ x = x, for all x ∈ L . By mathematical induction relative to n we conclude that na = a . Therefore, (a] = {x ∈ L : x ≤ na forsome n ≥ 1} = {x ∈ L : x ≤ a} . The rest is obvious.

(ii) . Let I be a nodal ideal of L . Then for all x, y ∈ L such that x ∈ I and y ∉ I, we have that (x] ⊆ I and I ⊆ (y] . Hence, (x] ⊆ I ⊆ (y] , so x ∈ (y] . Since L is an Boolean residuated lattice, it follows that x < y . The converse follows from Proposition 7.

(iii) . Let (x] be a nodal ideal of L and x ∈ L . By contrary, we assume that the element x is not a node of L . Then there exists an element y ∈ L such that it is incomparable with x, that is, xnleqy and ynleqx . So x ∉ (y] and y ∉ (x] . We conclude that (x] ⊈ (y] and (y] ⊈ (x] , that is, (x] and (y] are incomparable, that is a contradiction. Therefore, x is a node of L .

(iv) . We need to prove that if L has n nodes, then it has at least n nodal ideals. Let a ∈ L be a node. By Proposition 8, we conclude that (a] is a nodal ideal of L . We conclude that (a] = {x ∈ L : x ≤ a} . We conclude that if L has n nodes, then it has at least n nodal ideals. Hence, if there are n nodal ideals, then there are at most n nodes. By Theorem 2, Nod (L) ∩ B (L) = {0, 1} . Therefore, $I (L) \cap {Nod}_{Id} (L) = {{0}, L} .$

(v) . It follows from Theorem 14. □

Example 33. In this example, we offer a Boolean residuated lattice with prime ideals which are not nodal ideals. It is easy to ascertain that the residuated lattice L from Example 19 is a Boolean residuated lattice (involution semi-G-algebra) and I₁ = {0, a, b, c} , I₂ = {0, b, d, f} , I₃ = {0, a, d, e} are prime ideals, but they are incomparable, so they are not nodal ideals.

5. Fuzzy nodal ideals and some applications

A fuzzy set in X is a mapping f : X → [0, 1] . Let f be a fuzzy set in X, t ∈ [0, 1] , the sets $f_{t}^{-} = {x \in X : f (x) \leq t}$ and $f_{t}^{+} = {x \in X : f (x) \geq t}$ are called level subsets of f . For any a, b ∈ [0, 1] , we denote a ∨ b : = max {a, b} and a ∧ b : = min {a, b} . For any fuzzy sets f, g in L, we define f ≤ g iff f (x) ≤ f (y) for all x ∈ L . The fuzzy sets f ∩ g and f ∪ g are defined as follows: (f ∪ g) (x) = f (x) ∨ g (x) , (f ∩ g) (x) = f (x) ∧ g (x) for all x ∈ L . Some results on fuzzy filters in BL-algebras can be found in [15, 22]. In what follows, we extend the study of fuzzy ideals in residuated lattices.

Definition 11. Let f be a fuzzy set of a residuated lattice L . Then f is called a fuzzy filter if $f_{t}^{+}$ is either empty or a filter of L for all t ∈ [0, 1] . Moreover, f is called a fuzzy nodal filter if $f_{t}^{+}$ is either empty or a nodal filter of L for all t ∈ [0, 1] . In general, it is not difficult to obtain that the intersection of a family of fuzzy filters of L is a fuzzy filter.

The proofs for the following results are similar as in the case of BL-algebras (see [15, 22]).

Theorem 16. The following statements are equivalent in any residuated lattice L :

f is a fuzzy filter of L ;

f is order-preserving and f (x ⊙ y) ≥ f (x) ∧ f (y) for any x, y∈ L ;

x ⊙ y ≤ z implies f (z) ≥ f (x) ∧ f (y) for all x, y, z∈ L ;

x → (y → z) =1 implies f (z) ≥ f (x) ∧ f (y) for all x, y, z ∈ L .

Corollary 8. Let f be a fuzzy filter of L . Then $f_{t}^{+} = \emptyset$ iff t > f (1) .

Corollary 9. The following statements are equivalent:

(i) f is a fuzzy nodal filter of L ;

(ii) $f_{t}^{+}$ is empty or, if $x \notin f_{t}^{+},$ then x < a for all $a \in f_{t}^{+} .$

Proof. The proof follows by Theorems 1 and 16. □

Definition 12. Let f be a fuzzy set of a residuated lattice L . Then f is called a fuzzy ideal if $f_{t}^{-}$ is either empty or an ideal of L for all t ∈ [0, 1] . Moreover, f is called a fuzzy nodal ideal if $f_{t}^{-}$ is either empty or a nodal ideal of L for all t ∈ [0, 1] . In general, it is not difficult to obtain that the intersection of a family of fuzzy ideals of L is a fuzzy ideal.

Example 34. We consider the residuated lattice L from Example 2. It is easy to ascertain that the sets I₁ = {0} , I₂ = {0, n} are all proper ideals of L and F₁ = {a, c, e, m, 1} , F₂ = {b, c, f, m, 1} , F₃ = {d, e, f, m, 1} , F₄ = {c, m, 1} , F₅ = {e, m, 1} , F₆ = {m, 1} , F₇ = {m, 1} , and F₈ = {1} are all proper filters of L . Define the fuzzy set f in L by f (0) =0.1, f (n) =0.2, f (a) = f (b) = f (c) = f (d) = f (e) = f (f) = f (m) = f (1) =0.9 . Then f is a fuzzy ideal of L . Since I₁ and I₂ are nodal ideals of L, it follows that f is a fuzzy nodal ideal.

Now, define the fuzzy set $\bar{f}$ by $\bar{f} (0) = \bar{f} (n) =$ $\bar{f} (n) =$ $\bar{f} (a) = \bar{f} (b) = \bar{f} (d) =$ $\bar{f} (c) = \bar{f} (e) = \bar{f} (f)) =$ $\bar{f} (m) = 0.8$ and $\bar{f} (1) = 0.9,$ it is easy to checked that $\bar{f}$ is a fuzzy filter of L . Moreover, $\bar{f}$ is a nodal fuzzy filter (because F₈ and L are nodal filters of L).

Theorem 17. The following statements are equivalent in any residuated lattice L :

f is a fuzzy ideal of L ;

f is order-preserving and f (x ⊕ y) ≤ f (x) ∨ f (y) for any x, y∈ L ;

x ⊕ y ≥ z implies f (z) ≤ f (x) ∨ f (y) for all x, y, z ∈ L .

Proof. The proof is straightforward. □

Corollary 10. The following statements are equivalent:

f is a fuzzy nodal ideal of L ;

$f_{t}^{-}$ is empty or, if $x \notin f_{t}^{-},$ then x > a for all $a \in f_{t}^{-} .$

Proof. The proof follows by Theorems 4 and 17. □

Notation: For a fuzzy set f of L (with $f_{t}^{-} : = {x \in L : f (x) \leq t}$ for all t ∈ [0, 1]) we denote its dual set by f^* (with $f_{t}^{+} : = {x \in L : f (x) \geq t}$ for all t ∈ [0, 1]).

Remark 11. We note that fuzzy ideals and fuzzy filters are not dual notions. Indeed, if we consider the fuzzy ideal f defined in Example 34, for t = 0.2, we get that $f_{t}^{+} = {n, a, b, c, d, e, f, m, 1}$ is not a filter of L because n ⊙ n = 0 ∉ f⁺ (t) . We conclude that f^* is not a fuzzy filter of L .

Now, if we consider the fuzzy filter $\bar{f}$ defined in Example 34, for t = 0.8, we get that $f_{t}^{-} = {0, n, a, b, c, d, e, f, m}$ is not an ideal of L because $m \oplus m = 1 \notin f_{t}^{-} .$ We conclude that ${\bar{f}}^{*}$ is not a fuzzy ideal of L .

Since the notions of fuzzy ideals and fuzzy filters are not dual in residuated lattices, from logic and algebraic point of view, we conclude that the concept of fuzzy ideals has a proper meaning and studies are required.

Theorem 18. Let L be an involution residuated lattice. Then for every fuzzy ideal f of L there exists a fuzzy filter $\bar{f}$ of L .

Proof. Let L be an involution residuated lattice. By Lemma 4, we have that each set I is an ideal of L iff D (I) is a filter of L . Let Λ be an index set. We consider (I_i) _i∈Λ the family of all ideals of L . Clearly, [D (I_i)] _i∈Λ is a family of filters of L . Let f be a fuzzy ideal of L such that $f_{t}^{-} = I_{i},$ for some t ∈ [0, 1] and for all i ∈ Λ . We can define a fuzzy filter $\bar{f}$ of L such that $f_{t}^{+} = D (I_{i}),$ for some t ∈ [0, 1] and for all i ∈ Λ . □

Corollary 11. Let L be an involution residuated lattice. Then for every fuzzy nodal ideal f of L there exists a fuzzy nodal filter $\bar{f}$ of L .

Proof. The proof follows by Theorems 14 and 18. □

Theorem 19. (Representation Theorem) Let {X_a : a ∈ [0, 1]} be a class of subsets of L . The necessary and sufficient condition for the existence of fuzzy subset I of L with I_a = X_a for all a ∈ [0, 1] , is that for every M ⊆ [0, 1] , $X \underset{a \in M}{\lor} a = \underset{a \in M}{\cap} X_{a} .$

Proof. The proof follows easy by Theorem 2.4 [3]. □

Corollary 12. By Theorem 19, it follows that:

X₀ = L ;

if a, b ∈ [0, 1] are such that a ≤ b, then X_b ⊆ X_a .

A function S : [0, 1] × [0, 1] → [0, 1] satisfying the following properties is called a t-conorm: for all x, y, z ∈ [0, 1] ,

S (x, 0) = x ;

if x ≤ y, then S (z, x)≤ S (z, y) ;

S (x, y) = S (y, x) ;

S (x, S (y, z)) = S (S (x, y) , z) .

The following three kinds of continuous t-conorms are customary in literature.

The standard max operator, which is defined as S_M (x, y) = x ∨ y . It is the smallest t-conorm.

The probabilistic sum, which is defined as S_P (x, y) = x + y - x * y .

The bold sum, which is defined as S_L (x, y) =1 ∨ (x + y) .

Now, following Theorem 17, we generalize the concept of fuzzy ideal in residuated lattices as follows.

Definition 13. A fuzzy subset μ of L is said to be a S-fuzzy ideal iff μ is order-preserving and μ (x ⊕ y) ≤ S (μ (x) , μ (y)) for any x, y ∈ L . Subsequently, in this section, any S_M-fuzzy ideal is called a fuzzy ideal.

Theorem 20. (Representation Theorem) Let {I_a : a ∈ [0, 1]} be a class of ideals of L . The necessary and sufficient condition for the existence of a S-fuzzy ideal μ of L such that for all a ∈ [0, 1] , I_a = μ_a is that for all M ⊆ [0, 1] , $(5.1) I \underset{a \in M}{\lor} a = \underset{a \in M}{\cap} I_{a} .$

Proof. Suppose that (5.1) holds, and define a fuzzy subset μ of L by $μ (x) \underset{a \in M}{\land} a χ_{I a}, \forall x \in L,$ where χ_{I
_a} (x) , ∀ x ∈ L is the characteristic function. We shall prove that μ is a S-fuzzy ideal. Let x ∈ L . Since every nonempty ideal contains 0, {a∈ [0, 1] : x ∈ I_a} ⊆ {a ∈ [0, 1] :0 ∈ I_a} . This implies that μ (x) =⋀ _a∈[0,1]aχ_{I
_a} (x) ≥ ⋀_a∈[0,1]aχ_{I
_a} (0) = μ (0) . Hence μ (0) ≤ μ (x) for all x ∈ L . Now, let x, y ∈ L be such that x ≤ y . We shall prove that μ (x) ≤ μ (y) . By assumption x ≤ y, so {a∈ [0, 1] : x ∈ I_a} ⊇ {a ∈ [0, 1] : y ∈ I_a} . This implies that μ (x) =⋀ _a∈[0,1]aχ_{I
_a} (x) ≤ ⋀_a∈[0,1]aχ_{I
_a} (y) = μ (y) . Now, let x, y ∈ L and S (⋀ _a∈[0,1]aχ_{I
_a} (x) , ⋀ _b∈[0,1]bχ_{I
_b} (y)) = S (μ (x) , μ (y)) = t ∈ [0, 1] . Then μ (x) = ⋀ _a∈[0,1]aχ_{I
_a} (x) ≤ t and μ (y) = ⋀ _b∈[0,1]bχ_{I
_b} (y) ≤ t . Now, let M = {a ∈ [0, 1] : x ∈ I_a} . Then x ∈ ⋂ _m∈MI_m = I_{⋁_m∈Mm}⊆ I_t . Similarly, we get that y ∈ ⋂ _n∈MI_n = I_{⋁_n∈Mn}⊆ I_t . Hence x, y ∈ I_t . Since I_t is an ideal of L, we conclude that x ⊕ y ∈ I_t . Hence μ (x ⊕ y) ≤ t = S (μ (x) , μ (y)) . Now, we show that μ_a = I_a, for all a ∈ [0, 1] . Let x ∈ I_p, for some p ∈ [0, 1] . Then μ (x) = ⋀_a∈[0,1]aχ_{I
_a} (x) ≤ p, that is, x ∈ μ_p and so I_p ⊆ μ_p . Now, if x ∈ μ_p, then ⋀_a∈[0,1]aχ_{I
_a} (x) = μ (x) ≤ p, and so for M = {a ∈ [0, 1] : x ∈ I_a} we have that x ∈ ⋂ _a∈MI_a = I_{⋁_a∈Ma}⊆ I_p . Therefore, we proved that μ_a = I_a, for all a ∈ [0, 1] .

Conversely, we assume that there exists a S-fuzzy ideal I of L such that for all t ∈ [0, 1] , I_t = μ_t . We prove the validity of (5.1) . For this, let M ⊆ [0, 1] and x ∈ I_{⋁_a∈Ma} . Then x ∈ μ_{⋁_a∈Ma} and so μ (x) ≤ a, for all a ∈ M . This implies that x ∈ μ_a = I_a, for all a ∈ M, and so x ∈ ⋂ _a∈MI_a . Therefore, I_{⋁_a∈Ma} ⊆ ⋂ _a∈MI_a .

The converse inequality is proved similarly. □

An interesting application on decision-making of the above results on fuzzy (nodal) ideals in residuated lattices can be obtained in a similar manner as in J. Zhan et al. [20]: if we replace the non-empty and finite set U by L (a residuated lattice, not necessarily finite) and the collection of non-empty subsets C = {C_i ⊆ U : ⋃ _{C_i∈C}C_i = U} by the family of all nodal ideals of L, we get that the neighborhood of any x ∈ L denoted by N_C(x) is exactly the principle nodal ideal (x] generated by element x . Moreover, for any subset X of L, the lower and upper approximations of X with respect to C are exactly the ideal generated by X, denoted by (X] , so X is an exact set (see Definition 2.1 [20]). By Definition 2.3 [20] we get that any fuzzy set A on L is definable. In conclusion, we can apply the study developed by J. Zhan et al. [20] with the previous assumption on residuated lattices and the results follow in a simple and particular way.

6. Conclusions

In this paper, we introduced the concepts of nodal and conodal ideals in residuated lattices, we proposed some characterizations and many important properties of nodal and conodal ideals in residuated lattice. In addition, we showed that if L is an MTL-algebra and I is a non-principal nodal ideal of L, then L/I is a chain. We investigated the relationships of nodal and conodal ideals with other types of ideals in residuated lattices, likeness Boolean ideals, primary ideals, prime ideals, implicative ideals, maximal ideals and ⊙-prime ideals. Based on nodal ideals we defined an equivalence relation: the elements x and y of L are connected (in symbol (x, y) ∈ R) if there is no nodal ideal which separates them) and we showed that L/R is dually isomorphic with Nod_Id (L) (see Theorem 9). We studied the inverse image of a nodal (conodal) ideal under a homomorphism. Finally, we applied our theory on De Morgan, involution and Boolean residuated lattices in order to present some interesting results. As an application to fuzzy set theory, we pay attention to characterization theorems for fuzzy (nodal) filters and fuzzy (nodal) ideals, also, we introduced the concept of a fuzzy ideal of a residuated lattice, with respect to a t-conorm briefly, S-fuzzy ideals, and we proved the Representation Theorem in residuated lattices.

Some important issue for future work are:

developing the properties of different types of ideals and congruence relations in residuated lattices,

finding useful results on other structures,

introducing the concept of extreme fuzzy (nodal) ideals and investigating its applications on residuated lattices (inspired by [22]),

constructing a topology based on extreme fuzzy (nodal) ideals in residuated lattices.

Footnotes

Acknowledgments

The author is extremely grateful to the editors and the anonymous reviewers for giving him many valuable comments and helpful suggestions which helps to improve the presentation of this paper. This research work was supported by the Dept. of Systems Engineering-Automatic and Informatics Applied, Technical University of Civil Engineering, Bucharest, Romania.

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