Extreme fuzzy ideals and its applications on De Morgan residuated lattices

Abstract

The variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices (see L.C. Holdon [7]). X. Zhu, J. Yang and A. Borumand Saeid [16] used a special family of extreme fuzzy filters $F$ on a BL-algebra L, they constructed a uniform structure $(L, K),$ and then the part $K$ induced a uniform topology $τ_{F}$ in L . Also, they proved that the pair $(L, τ_{F})$ is a topological BL-algebra, and some properties of $(L, τ_{F})$ were investigated. Inspired by their study, in this paper, we define the family of extreme fuzzy ideals $I$ on a De Morgan residuated lattice L, we construct a uniform structure (L, K) , and then the part K induce a uniform topology $τ_{I}$ in L . We prove that the pair $(L, τ_{I})$ is a Topological De Morgan Residuated lattice, and some properties of $(L, τ_{I})$ are investigated. In particular, we show that $(L, τ_{I})$ is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, we give some characterizations of topological properties of $(L, τ_{I}) .$ We note that, since ideals and filters are dual in BL-algebras (see C. Lele and J. B. Nganou [12]), a study on extreme fuzzy ideals in BL-algebras follows by duality, but in the framework of De Morgan residuated lattices, which is a larger class than BL-algebras, the duality between ideals and filters does not hold, so the study of extreme fuzzy ideals in De Morgan residuated lattices becomes interesting from algebraic and topological point of view, and the results of X. Zhu, J. Yang and A. Borumand Saeid [16] become particular cases of our theory.

Keywords

De Morgan residuated lattice obstinate ideal extreme fuzzy ideal

1 Introduction

The concept of residuated lattice is a fundamental concept of ordered structures and categories with its origin in Mathematical Logic without contraction. The structural theory of a residuated lattice was given by Galatos et al. [5] in 2007. The concept of De Morgan residuated lattice was investigated from the viewpoint of ideal theory by L.C. Holdon [7] in 2017. The variety of De Morgan residuated lattices includes important subvarieties of residuated lattices such as Boolean algebras, MV-algebras, BL-algebras, Stonean residuated lattices, MTL-algebras and involution residuated lattices.

Ideals theory play an important rule in studying residuated lattices. From logical point of view, various ideals correspond to various sets of provable formula. From a purely algebraic point of view, the ideals have a proper meaning in residuated lattices. In particular, in [3 , 17] some types of ideals in subclasses of residuated lattices were introduced, and some of their characterizations and relations were investigated. One conclusion is that the duality between ideals and filters does not hold in the variety of De Morgan residuated lattices (see [7]). The concept of fuzzy sets was introduced by Zadeh [15] in 1965. Fuzzification of special types of ideals and filters on several different algebras of many-valued logics has been popular in recent years (see [1 , 16]). The concept of extreme fuzzy filter in a BL-algebra was defined by X. Zhu, J. Yang and A. Borumand Saeid [16] in 2017, with the reason to induce a fuzzified version of the uniform topology. As observed above, to induce uniform topology in their corresponding algebraic structure, authors used special family of filters which is closed under intersection.

In this paper, motivated by the previous research we think it will be interesting to give an answer to the following question: How to give a fuzzified version of the uniform topology for De Morgan residuated lattices (based on ideal theory)?

We show by examples that the notions of fuzzy ideals and fuzzy filters are not dual in residuated lattices, hence the concept of fuzzy ideals has a proper meaning and studies are required. We define the family of extreme fuzzy ideals $I$ on the larger class of De Morgan residuated lattices denoted by L, in order to construct a uniform structure (L, K) . The part K induce a uniform topology $τ_{I}$ in L . We prove that the pair $(L, τ_{I})$ is a topological De Morgan residuated lattices, and some properties of $(L, τ_{I})$ are investigated. In particular, we show that $(L, τ_{I})$ is a first-countable, zero-dimensional, disconnected and completely regular space. Finally, we give some characterizations of topological properties of $(L, τ_{I}) .$ We mention that the duality between concepts of extreme fuzzy ideals and extreme fuzzy filters failed in the variety of De Morgan residuated lattices, hence the concept of extreme fuzzy ideals has a proper meaning and studies are required. Because of that, we think it is a new direction to study the ideals by the concept of extreme fuzzy ideals, which will enrich and develop the theory of ideals in varieties of residuated lattices.

The paper is organized as follows: In Section 2, we review some basic definitions and results about ideals and filters in residuated lattices. In Section 3, we study some properties of fuzzy ideals in residuated lattices and introduce the concepts of extreme fuzzy ideal and extreme obstinate fuzzy ideal. The Section 3 consists of three subsections and it is organized as follows: in Subsection 3.1, we give some general informations on fuzzy filters and ideals in residuated lattices, in Theorem 3, we present a representation theorem. We note that fuzzy ideals and filters are dual notions in BL-algebras, but in the general case of residuated lattices they are not dual. In Subsection 3.2, we introduce the notion of an extreme fuzzy ideal of a residuated lattice L, we present some characterizations of extreme fuzzy ideals in Theorem 4. In Theorem 5, we present the expansion principal based on extreme ideals, and we show that the expansion of extreme fuzzy ideal is not unique. In Subsection 3.3, we define and study the notion of extreme obstinate fuzzy ideal.

In Section 4, as an application, we use extreme fuzzy ideals in order to induce an uniform topology on De Morgan residuated lattices and to study some interesting topological properties. The Section 4 consists of three subsections and it is organized as follows: in Subsection 41, we give some general informations on De Morgan residuated lattices and in Theorem 7 we show a congruence relation on L, based on extreme fuzzy ideals. After that, we present the direct product of residuated lattices and we construct the uniform topology in Theorem 9, Corollaries 5 and 6.

In Subsection 42, we present interesting topological properties of the space $(L, τ_{I}) .$ For example, in Theorem 16, we show that the topological space $(L, τ_{I})$ is a first-countable, zero-dimensional, disconnected and completely regular space, but, in general, it is not always connected. Finally, we naturally construct the relationship between isomorphism (algebraic invariant) and homeomorphism (topological invariant) in topological De Morgan 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. [5] 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. [5], Hájek [6], D. Buşneag et al. [3] and Iorgulescu [10]. 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 [6], D. Buşneag et al. [3], Galatos et al. [5]- Turunen [14] 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 ifandonlyif 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:

(i) Divisible if L verifies (i₁);

(ii) Semi-divisible if L verifies (i₂);

(iii) MTL-algebra if L verifies (i₃);

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

(v) Stonean if L verifies (i₄) ;

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

(vi) Involution if L verifies (i₆) ;

(vii) semi-G-algebra if L verifies (i₇) ;

(viii) De Morgan if L verifies (i₈) ;

(ix) Boolean algebra if L verifies (i₉) .

A filter ([6]) 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 .

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. [3, 7] 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 .

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

Definition 2. [3, 7] 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 .

It is known [3, 7] 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.

We note that, in MV-algebras, the notions of ideals defined in Definition 1.2 [4] and Definition 2 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. [7] 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. [3] Let S ⊆ L be a nonempty subset, a ∈ L and $I \in I (L) .$ Then:

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

(ii) (a] = {x∈ L : x ≤ na forsome n ≥ 1} ;

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

(iv) $(I (L), \subseteq)$ 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} .

At the end of this section, let us recall some basic notions of general topology which will be needed in the sequel.

Recall [8, 16] that a set X with a family τ of its subsets is called a topological space, denoted by (X, τ) , if:

(τ₁) X, ∅ ∈ τ,

(τ₂) the intersection of any finite members of τ is in τ,

(τ₃) the arbitrary union of members of τ is in τ .

The members of τ are called open sets of X, and the complement of an open set U, that is, X \ U, is called a closed set. For Λ an index set, a subfamily {U_i} _i∈Λ of τ is called a base of τ if for each x ∈ U ∈ τ there is an i ∈ Λ such that x ∈ U_i ⊆ U . A subset W of X is a neighborhood of x ∈ X, if there exists an open set U such that x ∈ U ⊆ W . Let τ_x denote the totality of all neighborhoods of x in X, then subfamily $V_{x}$ of τ_x is a fundamental system of neighborhoods of x, if for each U_x in τ_x, there exists a $V_{x} \in V_{x}$ such that V_x ⊆ U_x . If every point x in X has a countable fundamental system of neighborhoods, then we say that the space (X, τ) satisfies the first axiom of countability or is first-countable. A topological space (X, τ) is a zero-dimensional space if τ has a clopen base.

A topological space (X, τ) is called a regular space if for any closed subset C of X and x ∈ X such that x ∉ C, then there exist disjoint open sets U, V such that x ∈ U and C ⊆ V, or equivalently, for any open subset U containing x, there exists open subset V such that x ∈ V ⊆ U . Let (X, τ₁) and (Y, τ₂) be two topological spaces, a mapping f : X → Y is continuous if f^-1 (U) ∈ τ₁ for any U ∈ τ₂ .

A topological space (X, τ) is called a completely regular space, if for every x ∈ X and every closed set I ⊂ X such that x ∉ I there exists a continuous function f : X → [0, 1] such that f (x) =0 and f (y) =1 for y ∈ I .

The mapping f : (X, τ₁) → (Y, τ₂) is called a homeomorphism if f is bijective, and f and f^-1 are continuous, or equivalently, if f is bijective, continuous and open(closed). The mapping f : (X, τ₁) → (Y, τ₂) is called a quotient map if f is surjective, and V ∈ τ₂ if and only if f^-1 (V) ∈ τ₁ . A topological space (X, τ) is compact if each open cover of X is reducible to a finite open subcover. Let (X, τ) be a topological space. We have the following separation axioms in (X, τ) :

(T₀) : For each x, y ∈ X and x ≠ y, there is at least one in an open neighborhood excluding the other.

(T₁) : For each x, y ∈ X and x ≠ y, each has an open neighborhood not containing the other.

(T₂) : For each x, y ∈ X and x ≠ y, both have disjoint open neighborhoods U, V such that x ∈ U and y ∈ V .

A topological space satisfying T_i is called a T_i-space for any i ∈ {0, 1, 2} . A T₂-space is also known as a Hausdorff space.

3 Fuzzy ideals of a residuated lattice

In this section, we define the concept of fuzzy ideal in residuated lattices and we study their properties in details.

3.1 General information

We recall that [9, 16] 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 [13, 16]. In what follows, we extend the study of fuzzy ideals in residuated lattices.

Definition 3. 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] . 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 [13, 16]).

Theorem 1. The following statements are equivalent in any residuated lattice L :

(i) f is a fuzzy filter of L ;

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

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

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

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

Definition 4. [9] 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] . In general, it is not difficult to obtain that the intersection of a family of fuzzy ideals of L is a fuzzy ideal.

Example 1. Let L = {0, a, b, c, 1} with 0 < a, b < c < 1, but a and b are incomparable.

Then (L, ∧ , ∨ , ⊙ , → , 0, 1) is a residuated lattice ([10], page 187), where → and ⊙ are defined as in the tables:

$\begin{matrix} \to & 0 & a & b & c & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 \\ a & b & 1 & b & 1 & 1 \\ b & a & a & 1 & 1 & 1 \\ c & 0 & a & b & 1 & 1 \\ 1 & 0 & a & b & c & 1 \end{matrix} \begin{matrix} ⊙ & 0 & a & b & c & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ a & 0 & a & 0 & a & a \\ b & 0 & 0 & b & b & b \\ c & 0 & a & b & c & c \\ 1 & 0 & a & b & c & 1 \end{matrix}$

It is easy to ascertain that I₁ = {0, a} and I₂ = {0, b} are proper ideals of L . We define f a fuzzy ideal of L by: $f_{t}^{-} = {\begin{matrix} \emptyset, & t < 0.2, \\ {0, a}, & 0.2 \leq t \leq 0.6, \\ L, & t > 0.6 . \end{matrix}$ (2)

We define g a fuzzy ideal of L by: $g_{t}^{-} = {\begin{matrix} \emptyset, & t \leq 0.6, \\ {0, b}, & 0.6 < t < 0.8, \\ L, & t \geq 0.8 . \end{matrix}$ (3) By routine calculation follows that f ∩ g = g and f ∪ g = f .

Example 2. We consider L = {0, n, a, b, c, d, e, f, m, 1} with the Hasse diagram:

Then ([10], page 242) L is a residuated lattice with respect to the following operations:

$\begin{matrix} \to & 0 & n & a & b & c & d & e & f & m & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ n & m & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ a & f & f & 1 & f & 1 & f & 1 & f & 1 & 1 \\ b & e & e & e & 1 & 1 & e & e & 1 & 1 & 1 \\ c & d & d & e & f & 1 & d & e & f & 1 & 1 \\ d & c & c & c & c & c & 1 & 1 & 1 & 1 & 1 \\ e & b & b & c & b & c & f & 1 & f & 1 & 1 \\ f & a & a & a & c & c & e & e & 1 & 1 & 1 \\ m & n & n & a & b & c & d & e & f & 1 & 1 \\ 1 & 0 & n & a & b & c & d & e & f & m & 1 \end{matrix} \begin{matrix} ⊙ & 0 & n & a & b & c & d & e & f & m & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ n & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & n \\ a & 0 & 0 & a & 0 & a & 0 & a & 0 & a & a \\ b & 0 & 0 & 0 & b & b & 0 & 0 & b & b & b \\ c & 0 & 0 & a & b & c & 0 & a & b & c & c \\ d & 0 & 0 & 0 & 0 & 0 & d & d & d & d & d \\ e & 0 & 0 & a & 0 & a & d & e & d & e & e \\ f & 0 & 0 & 0 & b & b & d & d & f & f & f \\ m & 0 & 0 & a & b & c & d & e & f & m & m \\ 1 & 0 & n & a & b & c & d & e & f & m & 1 \end{matrix}$

It is easy to ascertain that the sets I₁ = {0} , I₂ = {0, n} are proper ideals 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 .

Example 3. We consider the residuated lattice L from Example 2. It is easy to ascertain that the sets I₁ = {0} , I₂ = {0, n} are 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 proper filters of L .

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 .

Theorem 2. [9] The following statements are equivalent in any residuated lattice L :

(i) f is a fuzzy ideal of L ;

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

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

Proof. The proof is straightforward. □

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]).

We note that fuzzy ideals and fuzzy filters are not dual notions. Indeed, if we consider the fuzzy ideal f defined in Example 2, 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 3, 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 .

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

By duality with Corollary 1, we have:

Corollary 2. Let f be a fuzzy ideal of L . Then $f_{t}^{-} = \emptyset$ iff t < f (0) .

Theorem 3. (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] , $X_{⋁_{a \in M} a} = ⋂_{a \in M} X_{a} .$

Proof. The proof follows easy by Theorem 2.4 [2]. □

Corollary 3. By Theorem 3, it follows that:

(i) X₀ = L ;

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

3.2 Extreme fuzzy ideals

Definition 5. A fuzzy ideal f of a residuated lattice L is called an extreme fuzzy ideal of L (EF-ideal for short) if f (0) =0 .

It is clear that I is an ideal of a residuated lattice L iff λ_I is a fuzzy ideal in L, where λ_I is the complementary function of the characteristic function of I (that is, λ_I (x) =0 if x ∈ I, and λ_I (x) =1 if x ∉ I). Clearly, λ_I is an EF-ideal if I is an ideal of L . We denote Id (L) , FId (L) and EFId (L) the sets of all ideals, fuzzy ideals and extreme fuzzy ideals of L, respectively. Then Card (Id (L)) ≤ Card (EFId (L)) ≤ Card (FId (L)) , where Card (X) is cardinality of the set of X .

Theorem 4. Let f be a fuzzy set of a residuated lattice L . The following statements are equivalent:

(i) f is an extreme fuzzy ideal of L,

(ii) f_t≠ ∅ for all t ∈ [0, 1] ,

(iii) f₀ ≠ ∅ ,

(iv) f₀ is an ideal of L,

(v) f_t is an ideal of L for all t ∈ [0, 1] .

Proof. (i) ⇒ (ii) . Assume that the item (i) is satisfied. Then f_t≠ ∅ for all t ∈ [0, 1] . Otherwise there exists t ∈ [0, 1] such that $f_{t}^{-} = f_{t} = \emptyset .$ By Corollary 2, 0 < f (0) =0, which is a contradiction.

(ii) ⇒ (iii) and (iii) ⇒ (iv) are straightforward.

(iv) ⇒ (v) . Assume that the item (iv) is satisfied. It follows that ∅ ≠ f₀ ⊆ f_t for all t ∈ [0, 1] . Since f is a fuzzy ideal, then f_t is an ideal of L .

(v) ⇒ (i) . Assume that the item (v) is satisfied. Since f is a fuzzy ideal of L and 0 ∈ f₀ . It follows that f₀ = 0 . □

Interestingly, in residuated lattices every fuzzy ideal can be expanded into an extreme fuzzy ideal (Expansion principle).

Theorem 5. (Expansion principle) Let f be a fuzzy ideal of L . Then there is an extreme fuzzy ideal $\bar{f}$ such that $\bar{f} \leq f .$

Proof. Let f be a fuzzy ideal of L . We define $\bar{f}$ by $\bar{f} (x) = {\begin{matrix} 0, & x = 0, \\ f (x), & otherwise . \end{matrix}$ (4)

Clearly, $\bar{f} \leq f .$ One can easily check that for all t ∈ [0, 1] , $\bar{f} (x) = {\begin{matrix} { 0}, & f (0) < x \leq 0, \\ f_{t}, & t \geq f (0) . \end{matrix}$ (5) As a consequence $\bar{f}$ is an extreme fuzzy ideal of L . □

The conclusion of Theorem 5 is that for every fuzzy ideal f there is an extreme fuzzy ideal $\bar{f}$ such that $\bar{f} \leq f .$ In this case, f is called an expansion of $\bar{f} .$ An easy consequence of Theorems 4 and 5 is the following result.

Corollary 4. Let f be a fuzzy ideal of L . Then f is an extreme fuzzy ideal iff $f = \bar{f} .$ In particular, if I = {0} is a maximal ideal of L, then $f = \bar{f} .$

Example 4. In this example, we present a residuated lattice L in which the property from Corollary 4 holds. Let L = {0, a, b, c, d, e, f, m, 1} with the Hasse diagram:

Then ([10], page 252) L is a residuated lattice with respect to the following operations: $\begin{matrix} \to & 0 & a & b & c & d & e & f & m & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ a & m & 1 & m & 1 & m & 1 & m & 1 & 1 \\ b & m & m & 1 & 1 & m & m & 1 & 1 & 1 \\ c & m & m & m & 1 & m & m & m & 1 & 1 \\ d & m & m & m & m & 1 & 1 & 1 & 1 & 1 \\ e & m & m & m & m & m & 1 & m & 1 & 1 \\ f & m & m & m & m & m & m & 1 & 1 & 1 \\ m & m & m & m & m & m & m & m & 1 & 1 \\ 1 & 0 & a & b & c & d & e & f & m & 1 \end{matrix} \begin{matrix} ⊙ & 0 & a & b & c & d & e & f & m & 1 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ a & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & a \\ b & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & b \\ c & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & c \\ d & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & d \\ e & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & e \\ f & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f \\ m & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & m \\ 1 & 0 & a & b & c & d & e & f & m & 1 \end{matrix}$ It is easy to ascertain that I = {0} is a maximal ideal of L . If we define the fuzzy set f in L by f (0) =0 and f (a) = f (b) = f (c) = f (e) = f (f) = f (m) = f (1) = t with t ∈ (0, 1] , then f is a fuzzy ideal of L and $f = \bar{f} .$ Moreover, f is an extreme fuzzy ideal of L .

Unfortunately, in general, the expansion of extreme fuzzy ideal is not unique.

Example 5. We consider the residuated lattice L from Example 2. It is easy to ascertain that the sets I₁ = {0} , I₂ = {0, n} , I₃ = {0, n, a, b, c} are proper ideals of L . If we define the fuzzy set f in L by f (0) = f (n) = f (a) = f (b) = f (c) =0.3, and f (e) = f (f) = f (m) = f (1) =0.4, then f is a fuzzy ideal of L . Define the fuzzy sets $\bar{f_{1}}$ and $\bar{f_{2}}$ in L by $\bar{f_{1}} (0) = 0,$ $\bar{f_{1}} (n) = 0.1,$ $\bar{f_{1}} (a) = \bar{f_{1}} (b) =$ $\bar{f_{1}} (c) =$ $\bar{f_{1}} (d) = \bar{f_{1}} (e) =$ $\bar{f_{1}} (f) = \bar{f_{1}} (m) =$ $\bar{f_{1}} (1) = 0.3$ and $\bar{f_{2}} (0) = 0,$ $\bar{f_{2}} (n) = \bar{f_{2}} (a) = \bar{f_{2}} (b) =$ $\bar{f_{2}} (c) = 0.2,$ $\bar{f_{2}} (d) = \bar{f_{2}} (e) =$ $\bar{f_{2}} (f) = \bar{f_{2}} (m) =$ $\bar{f_{2}} (1) = 0.4 .$ Routine calculation shows that f is an expansion of $\bar{f_{1}}$ and $\bar{f_{2}},$ but they are different.

3.3 Extreme obstinate fuzzy ideals

Definition 6. A proper ideal I of L is called an obstinate ideal if x, y ∉ I imply x^* ⊙ y ∈ I and x ⊙ y^* ∈ I for all x, y ∈ L .

Example 6. In the residuated lattice L from Example 4, we have that I = {0} is not an obstinate ideal because a, 1 ∉ I with a^* ⊙ 1 = m ⊙ 1 = m ∉ I and 1^* ⊙ a = 0 ⊙ a = 0 ∈ I .

In the residuated lattice L from Example 2, we have that I = {0, n, a, b, c} is an obstinate ideal of L .

Definition 7. Let f be a extreme fuzzy ideal of a residuated lattice L . Then f is called a extreme obstinate fuzzy ideal if $f_{t}^{-}$ is either empty or an obstinate ideal of L for all t ∈ [0, 1] .

Example 7. In the residuated lattice L from Example 2, we have that I = {0, n, a, b, c} is an obstinate ideal of L . If we define the fuzzy set f in L by f (0) = f (n) = f (a) = f (b) = f (c) =0, and f (e) = f (f) = f (m) = f (1) =0.1, then f is a extreme obstinate fuzzy ideal of L .

In the residuated lattice L from Example 2, we have that I = {0} is not an obstinate ideal. If we define the fuzzy set f in L by f (0) =0, f (n) = f (a) = f (b) = f (c) = f (e) = f (f) = f (m) = f (1) =0.1, then f is a extreme fuzzy ideal of L, but it is not extreme obstinate fuzzy ideal because for t = 0.05 we have $f_{0.05}^{-} = {0} = I .$

4 Fuzzy ideals of a De Morgan residuated lattice

In this section, we study the concept of fuzzy ideal and extreme fuzzy ideals in De Morgan residuated lattices, in order to obtain some topological properties likewise the uniform topology and separability properties.

4.1 General information

In Theorem 9 from [7] was 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 iff x^* ⊙ y ∈ I and x ⊙ y^* ∈ I) is a congruence on the reduct (L, ⊙ , ∧ , ∨ , → , 0, 1) of the De Morgan residuated lattice L . For x ∈ L we denote by x/I the congruence class of x modulo θ_I and L/I = {x/I : x ∈ L} . Define the binary operations ∨, ∧ , ⊙ and → on L/I 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 a De Morgan residuated lattice, which is called the quotient De Morgan residuated lattice of L with respect to I, where 0 = 0/I and 1 = 1/I . The order relation on L/I is defined by (x/I) ≤ (y/I) iff (x → y) ^* ∈ I . For a nonempty subset S of L we denote by S/I = {x/I : x ∈ S} . Clearly, for x ∈ L, x/I = 1 iff x^* ∈ I and x/I = 0 iff x ∈ I .

Theorem 6. Let I be a proper ideal of L and x, y ∉ I . Then (x, y) ∈ θ_I iff I is an obstinate ideal.

Proof. Let x, y ∉ I . Assume that (x, y) ∈ θ_I . We have that (x, y) ∈ θ_I iff x^* ⊙ y ∈ I and x ⊙ y^* ∈ I, that is, I is an obstinate ideal. Conversely, we assume that I is an obstinate ideal, that is, x^* ⊙ y ∈ I and x ⊙ y^* ∈ I . Hence, (x, y) ∈ θ_I . □

Let f be an extreme fuzzy ideal of a De Morgan residuated lattice L . We define a binary relation ≡_f on L by: x ≡ _fy iff f (x^* ⊙ y) ∨ f (x ⊙ y^*) =0 . Clearly, x ≡ _fy iff f (x^* ⊙ y) ∨ f (x ⊙ y^*) =0 iff f (x^* ⊙ y) =0 and f (x ⊙ y^*) =0 iff x^* ⊙ y = f^-1 (0) and x ⊙ y^* = f^-1 (0) iff (x^* ⊙ y) ∨ (x ⊙ y^*) ∈ f^-1 (0) . It is clear that ≡_f : = {(x, y) : x ≡ _fy} .

Theorem 7. Let f be an fuzzy extreme ideal of L . Then ≡_f is a congruence on L .

Proof. Let f be an fuzzy extreme ideal of L and x, y ∈ L \ {0} . Clearly, I = {0} is a proper ideal of L . Since I = {0} is a proper ideal of L, it follows that (x, y) ∈ θ_I iff x^* ⊙ y ∈ I and x ⊙ y^* ∈ I iff x^* ⊙ y ∈ {0} and x ⊙ y^* ∈ {0} iff f (x^* ⊙ y) ∨ f (x ⊙ y^*) =0 iff x ≡ _fy . □

Theorem 8. Let f be an extreme fuzzy ideal of L . Then for all x, y ∈ L \ {0} , x ≡ _fy iff {0} is an obstinate ideal.

Proof. Let f be an extreme fuzzy ideal of L and x, y ∈ L \ {0} . Since f is an extreme fuzzy ideal of L, it follows that f^-1 (0) =0 . We obtain successively x ≡ _fy iff f (x^* ⊙ y) ∨ f (x ⊙ y^*) =0 iff f (x^* ⊙ y) =0 and f (x ⊙ y^*) =0 iff x^* ⊙ y ∈ f^-1 (0) and x ⊙ y^* ∈ f^-1 (0) iff x^* ⊙ y = 0 and x ⊙ y^* = 0 iff x^* ⊙ y ∈ {0} and x ⊙ y^* ∈ {0} . Hence {0} is an obstinate ideal. □

Proposition 3. Let I be a proper ideal of L, f be an extreme fuzzy ideal and x, y ∈ L . If x ∈ I and x ≡ _fy, then y ∈ I .

Proof. Let I be a proper ideal of L, f be an extreme fuzzy ideal and x, y ∈ L, such that x ∈ I and x ≡ _fy . We obtain successively x ≡ _fy iff x^* ⊙ y = x ⊙ y^* = 0 iff x^* ≤ y^* and y^* ≤ x^* iff x^* = y^* . Hence x^** = y^** . By Proposition 1, x ∈ I iff x^** ∈ I . Since x^** ∈ I and x^** = y^**, it follows that y^** ∈ I . By Proposition 1, x ∈ I iff x^** ∈ I iff y^** ∈ I iff y ∈ I . Therefore, if x ∈ I and x ≡ _fy, then y ∈ I . □

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

for all x, y ∈ L and

Then L × L′ with the above operations is a residuated lattice called direct product of L and L′.

We recall [7] that for any residuated lattices L, L′: K is an ideal of L × L′ iff there exist $P \in I (L)$ and $Q \in I (L^{'})$ such that K = P × Q .

If X is a non-empty set, U and V are subsets of X × X. Then

U ∘ V = {(x, y) ∈ X × X: (z, y) ∈ U and (x, z) ∈ V forsome z ∈ X} ,

U^-1 = {(x, y) ∈ X × X: (y, x) ∈ U} ,

▵ = {(x, x) ∈ X × X: x ∈ X} .

In universal algebra if X is a non-empty set, then a non-empty collection K of subsets of X × X is called a uniformity on X if it satisfies the following conditions:

(U₁) ▵ ⊆ U for any U ∈ K,

(U₂) if U ∈ K, then U^-1 ∈ K,

(U₃) if U ∈ K, then there exists a set V ∈ K such that V ∘ V ⊆ U,

(U₄) if U, V ∈ K, then U ∩ V ⊆ K,

(U₅) if U ∈ K and U ⊆ V ⊆ X × X, then V ∈ K .

The pair (X, K) is called a uniform structure.

For the rest of this paper we always suppose that $I$ is a family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection, unless otherwise stated. In the following we use $I$ to induce uniform structures. In [8], the uniform topology, based on filter theory, is investigated.

Theorem 9. Let L be a De Morgan residuated lattice, f belong to $I,$ U_f = {(x, y) ∈ L × L : x ≡ _fy} and $K^{*} = {U_{f} : f \in I} .$ Then K^* satisfies the conditions (U₁) - (U₄) .

Proof. (U₁) . Since f is a fuzzy ideal of L, we have x ≡ _fx, for any x ∈ L . Hence ▵ ⊆ U_f for all U_f ∈ K^* .

(U₂) . For any U_f ∈ K^*, we have (x, y) ∈ (U_f) ^-1 iff (y, x) ∈ U_f iff y ≡ _fx iff x ≡ _fy iff (x, y) ∈ U_f .

(U₃) . For any U_f ∈ K^*, the transitivity of ≡_f implies that U_f ∘ U_f ⊆ U_f .

(U₄) . For any U_f, U_g ∈ K^*, we claim that U_f ∩ U_g = U_f∩g . If (x, y) ∈ U_f ∩ U_g, then f (x ⊙ y^*) = f (x^* ⊙ y) =0 and g (x ⊙ y^*) = g (x^* ⊙ y) =0 . It follows that (f ∩ g) (x ⊙ y^*) = (f ∩ g) (x^* ⊙ y) =0 . Hence (x, y) ∈ U_f∩g . Conversely, let (x, y) ∈ U_f∩g . Then (f ∩ g) (x ⊙ y^*) = (f ∩ g) (x^* ⊙ y) =0 . It follows that f (x^* ⊙ y) ∧ g (x^* ⊙ y) = f (x ⊙ y^*) ∧ g (x ⊙ y^*) =0 . We can get (f (x^* ⊙ y) ∧ g (x^* ⊙ y)) ∨ f (x^* ⊙ y) = 0 ∨ f (x^* ⊙ y) , this implies that f (x^* ⊙ y) =0 . Using similar approach, we get f (x ⊙ y^*) =0 . Hence (x, y) ∈ U_f . Similarly, we can get (x, y) ∈ U_g . Thus U_f∩g ⊆ U_f ∩ U_g . Since $f, g \in I,$ we have $f \cap g \in I,$ U_f ∩ U_g ∈ K^* . □

Corollary 5. Let L be a De Morgan residuated lattice and K = {U ⊆ L × L : ∃ U_f ∈ K^*suchthat U_f ⊆ U} , where K^* comes from Theorem 9. Then K is a uniformity on L and the pair (L, K) is a uniform structure.

Proof. By Theorem 9, the collection K satisfies conditions (U₁) - (U₄) . It suffices to show that K satisfies (U₅) . Let U ∈ K and U ⊆ V ⊆ L × L . Then there exists U_f ⊆ U ⊆ V, which means that V ∈ K . Let x ∈ L and U ∈ K . Define U [x] : = {y ∈ L : (x, y) ∈ U} . Clearly, if V ⊆ U, then V [x] ⊆ U [x] . □

Corollary 6. Let L be a De Morgan residuated lattice. Then τ = {G ⊆ L : (∀ x ∈ G) (∃ U ∈ K) suchthat U [x] ⊆ G} is a topology on L, where K comes from Corollary 5.

Proof. Clearly, ∅ and the set L belong to τ . It is clear that τ is closed under arbitrary union. Finally to show that τ is closed under finite intersection, let G, H ∈ τ and suppose that x ∈ G ∩ H . Then there exist U, V ∈ K such that U [x] ⊆ G and V [x] ⊆ H .

If W = U ∩ V, then W ∈ K . Also W [x] ⊆ U [x] ∩ V [x] and so W [x] ⊆ G ∩ H, hence G ∩ H ∈ τ . Thus τ is a topology on L . Note that for any x in L, U [x] is a neighborhood of x . □

Definition 8. Let $I$ be an arbitrary family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection. Then the topology $τ_{I}$ comes from Corollary 6 is called a uniform topology on L induced by $I .$ We denote the uniform topology obtained by an arbitrary family of extreme fuzzy ideals $I$ by $τ_{I},$ and if $I = {f},$ we denote it by τ_f .

Example 8. In Example 1 we define a fuzzy set f by f (0) = f (a) =0, f (b) = f (c) = f (1) =0.2 . It is easily check that f is an extreme fuzzy ideal. Consider $I = {f} .$ By Corollary 5, we construct K^* = {U_f} = {(x, y) : x ≡ _fy} = {(0, 0) , (a, a) , (b, b) , (c, c) , (1, 1) , (b, 1) , (1, b) , (b, c) , (c, b) , (0, a) , (a, 0) , (c, 1) , (1, c)} . We can check that (L, K) is a uniform space, where K = {U : U_f ⊆ U} . Open neighborhoods U_f [x] : = {y ∈ L : (x, y) ∈ U_f} are U_f [0] = {0, a} , U_f [a] = {0, a} , U_f [b] = {b, c, 1} , U_f [c] = {b, c, 1} , U_f [1] = {b, c, 1} . By simple calculation, we get τ_f = {∅ , {0, a} , {b, c, 1} , {0, a, b, c, 1}} .

4.2 Topological properties of the space $(L, τ_{I})$

Note that from Corollary 6, giving $I$ the family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection, we can induce a uniform topology $τ_{I}$ on L . In this section we study topological properties of $(L, τ_{I}) .$

Let τ be a topology of L . Then (L, τ) is called a topological De Morgan residuated lattice (TDM residuated lattice for short) if the operations ∨, ∧ , ⊙ , → are continuous. Note that the operation * ∈ {∨ , ∧ , ⊙ , →} is continuous iff for any x, y ∈ L and any neighborhood W of x * y there exist two neighborhoods U and V of x and y, respectively, such that U * V ⊆ W .

Theorem 10. Let $I$ be a family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection. Then the space $(L, τ_{I})$ is a TDM residuated lattice.

Proof. By definition, it suffices to show that * is continuous, where * ∈ {∨ , ∧ , ⊙ , →} . Indeed, assume that x * y ∈ G, where x, y ∈ L and G is an open subset of L . Then there exist U ∈ K, U [x * y] ⊆ G, and an extreme fuzzy ieal f such that U_f ∈ K^* and U_f ⊆ U . We claim that the following relation holds: U_f [x] * U_f [y] ⊆ U_f [x * y] ⊆ U [x * y] . Let h * k ∈ U_f [x] * U_f [y] . Then h ∈ U_f [x] and k ∈ U_f [y] we get that x ≡ _fh and y ≡ _fk . Hence x * y ≡ _fh * k . It follows that (x * y, h * k) ∈ U_f ⊆ U . Hence h * k ∈ U_f [x * y] ⊆ U [x * y] . Then h * k ∈ G . Clearly, U_f [x] and U_f [y] are neighborhoods of x and y, respectively. Therefore, the operation * is continuous. □

Theorem 11. Consider the space $(L, τ_{I})$ in Theorem 10. If x ∈ L and $f \in I,$ then U_f [x] is a clopen subset of L for the topology $τ_{I} .$

Proof. Let x ∈ L and $f \in I .$ Now we show that (U_f [x]) ^c is open. Let y ∈ (U_f [x]) ^c . We claim that U_f [y] ⊆ (U_f [x]) ^c . Assume z ∈ U_f [y] . Then f (z^* ⊙ y) = f (z ⊙ y^*) =0 . If z ∈ U_f [x] , then we have f (z^* ⊙ x) = f (z ⊙ x^*) =0 . It follows that 0 = f (z ⊙ y^*) ∨ f (z^* ⊙ x) ≥ f (x ⊙ y^*) and 0 = f (x^* ⊙ z) ∨ f (z^* ⊙ y) ≥ f (x^* ⊙ y) . Hence f (x^* ⊙ y) = f (y^* ⊙ x) =0 and y ∈ U_f [x] which is a contradiction. Hence U_f [y] ⊆ (U_f [x]) ^c for all y ∈ (U_f [x]) ^c, and so U_f [x] is closed. It is clear that U_f [x] is open. So U_f [x] is a clopen subset of L . □

A topological space X is connected if and only if X has only X and ∅ as clopen subsets. Therefore, by Theorem 11, we have the following result.

Corollary 7. The space $(L, τ_{I})$ is not, in general, a connected space.

We recall that for any ideal I of L, λ_I is a fuzzy ideal in L, where λ_I is the complementary function of the characteristic function of I (that is, λ_I (x) =0 if x ∈ I, and λ_I (x) =1 if x ∉ I).

Theorem 12. Let I be a proper ideal of L . Then I is a clopen subset of L .

Proof. Let I be an ideal of L . Then $λ_{I} \in I .$ It suffices to show that I^c = ⋃ {U_{λ
_I} [y] : y ∈ I^c} and I = ⋃ {U_{λ
_I} [y] : y ∈ I} . To do this, suppose y ∈ I^c . Since y ∈ U_{λ
_I} [y] , then y ∈ ⋃ {U_{λ
_I} [y] : y ∈ I^c} . Conversely, assume y ∈ ⋃ {U_{λ
_I} [y] : y ∈ I^c} . Then there exists z ∈ I^c such that y ∈ U_{λ
_I} [z] . Hence λ_I (y^* ⊙ z) = λ_I (y ⊙ z^*) =0 . It follows that y^* ⊙ z, y ⊙ z^* ∈ I . If y ∈ I, then z ∈ I (by Proposition 3), which is a contradiction. Hence y ∉ I and thus I^c = ⋃ {U_{λ
_I} [y] : y ∈ I^c} . By similar way we can show that I = ⋃ {U_{λ
_I} [y] : y ∈ I} . □

The following result is clear.

Lemma 3. Let f and g be extreme fuzzy filters of a De Morgan residuate lattice L . If g ≤ f, then U_g ⊆ U_f .

Theorem 13. Let f and g be extreme fuzzy ideals of a De Morgan residuated lattice L . If g ≤ f, then τ_f ⊆ τ_g .

Proof. Let $I = f,$ K^* = U_f and K = {U ⊆ L × L : U_f ⊆ U} . Let O ∈ τ_f . Then for all x ∈ O, there exists U ∈ K such that U [x] ∈ O and so U_f [x] ⊆ U [x] ⊆ O . By Lemma 3, U_g ⊆ U_f . It follows that U_g [x] ⊆ U_f [x] ⊆ O . Therefore, O ∈ τ_g and so τ_f ⊆ τ_g . □

Theorem 14. Let $I$ be a family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection. If $g = {f : f \in I},$ then $τ_{I} = τ_{g} .$

Proof. Let $I = {g},$ K^* = U_g and K = {U ⊆ L × L : U_g ⊆ U} . Let O ∈ τ_g . Then for all x ∈ O, there exists U ∈ K such that U [x] ⊆ O . So U_g [x] ⊆ U [x] . Since $I$ is closed under intersection, then $g \in I .$ Thus we have U_g ∈ K and hence $O \in τ_{I} .$ So $τ_{g} \subseteq τ_{I} .$ Conversely, it directly follows from Theorem 13. □

Proposition 4. For $I$ a family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection. If $g = {f : f \in I},$ then we have the following statements:

(i) By Theorem 4.8, we know that $τ_{I} = τ_{g} .$ For any U ∈ K, x ∈ L, we can get that U_g [x] ⊆ U [x] . Hence $τ_{I}$ is equivalent to {A ⊆ L : ∀ x ∈ A, U_g [x] ⊆ A} . So A ⊆ L is an open set if and only if for all x ∈ A, U_g [x] ⊆ A if and only if A =⋃ _x∈AU_g [x] ;

(ii) For all x ∈ L, by (i) , we know that U_g [x] is the smallest open neighborhood of x ;

(iii) Let β_g = {U_g [x] : x ∈ L} . By (i) and (ii) , it is easy to check that β_g is a base of τ_g;

(iv) For all x ∈ L, {U_g [x]} is a denumerable fundamental system of neighborhoods of x .

Theorem 15. Let $I$ be a family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection. If $g = {f : f \in I},$ then for all x ∈ L, U_g [x] is a clopen compact set in the topological space $(L, τ_{I}) .$

Proof. By Theorem 11, it is enough to show that U_g [x] is a compact set. Let U_g [x] ⊆ ⋃ _i∈ΛO_i, where each O_i is an open set of L and Λ is an index set. Since x ∈ U_g [x] , there exists i ∈ Λ such that x ∈ O_i . Then U_g [x] ⊆ O_i . Hence U_g [x] is compact. Therefore, U_g [x] is a clopen compact set in the topological space $(L, τ_{I}) .$ □

Theorem 16. The topological space $(L, τ_{I})$ is a first-countable, zero-dimensional, disconnected and completely regular space.

Proof. By Theorem 14, it suffices to show that (L, τ_g) is a first-countable, zero-dimensional, disconnected and completely regular space. Let x ∈ L . By Proposition 4(iv) , {U_g [x]} is a denumerable fundamental system of neighborhoods of x, so (L, τ_g) is first-countable. Let β_g = {U_g [x] : x ∈ L} . By Proposition 4(iii) and Theorem 11, we get that β_g is a clopen basis of (L, τ_g) , hence (L, τ_g) is a zerodimensional space. By Corollary 7, we get (L, τ_g) is a disconnected space. By Theorem 15 and Proposition 4(ii) , U_g [x] is a compact neighborhood of x, hence (L, τ_g) is a locally compact space. Let x ∈ L and V be an open neighborhood of x . By Proposition 4(ii) and Theorem 15, there exists closed neighborhood U_g [x] of x such that U_g [x] ⊆ V . Therefore, (L, τ_g) is a regular space. Since (L, τ_g) is a locally compact space, it follows that it is completely regular. □

Theorem 17. The space $(L, τ_{I})$ is a discrete space if and only if there exists $f \in I$ such that U_f [x] = {x} for all x ∈ L .

Proof. Let $τ_{I}$ be a discrete topology on L . If for any $f \in I,$ there exists x ∈ L such that U_f [x] ≠ {x} . Let $g = \cap I .$ Then $g \in I,$ there exists x₀ ∈ L such that U_g [x₀] ≠ {x₀} . It follows that there exists y₀ ∈ U_g [x₀] and x₀ ≠ y₀ . By Proposition 4(ii) , U_g [x₀] is the smallest open neighborhood of x₀ . Hence {x₀} is not an open subset of L, which is a contradiction. Conversely, for any x ∈ L, there exists $f \in I$ such that U_f [x] = {x} . Hence {x} is an open set of L . Therefore, $(L, τ_{I})$ is a discrete space. □

Theorem 18. Let $I$ be a family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection. If $g = ⋂ I,$ then the following statements are equivalent:

(i) $(L, τ_{I})$ is a discrete space,

(ii) g₀ = {0} , where g₀ is a level subset.

Proof. (i) ⇒ (ii) . By Theorem 17, there exist $f \in I$ such that U_f [0] = {0} . Since 0 ∈ U_g [0] ⊆ U_f, we have U_g [0] = {0} . It suffices to show g₀ ⊆ U_g [0] . Let x ∈ g₀ . Then g (x) =0, this implies that g (x^* ⊙ 0) = g (x ⊙ 0^*) =0 . Hence x ∈ U_g [0] .

(ii) ⇒ (i) . Let g₀ = {0} . By Theorem 17, it suffices to show that U_g [0] = {0} . Suppose x ∈ U_g [0] , then g (x) =0 . It follows that x = 0 . Clearly, {0} ⊆ U_g [0] . Therefore, U_g [0] = {0} . □

Theorem 19. Let $I$ be a family of extreme fuzzy ideals of a De Morgan residuated lattice L which is closed under intersection. If $g = ⋂ I,$ then the space $(L, τ_{I})$ is a Hausdorff space if and only if g₀ = {0} .

Proof. Let $(L, τ_{I})$ be a Hausdorff space. First we show that for any x ∈ L, U_g [x] = {x} . If not, then there exist x ≠ y ∈ U_g [x] such that y ∈ U_g [x] ∩ U_g [y] . By Proposition 4(ii) , U_g [x] and U_g [y] are the smallest open neighborhoods of x and y, respectively. Hence for any open neighborhood U of x and open neighborhood V of y, we have that y ∈ U_g [x] ∩ U_g [y] ⊆ U ∩ V ≠ ∅ , which is a contradiction. Hence, by Theorems 17 and 18, g₀ = {0} . The remainder of the proof directly follows from Theorem 18. □

We recall [6] that for two De Morgan residuated lattices L₁ and L₂, the map ψ : L₁ → L₂ is called a morphism (for short, DMRL-morphism) if and only if it satisfies the following conditions, for all x, y ∈ L₁ :

(i) ψ (0) =0, ψ (1) =1 ;

(ii) ψ (x ⊙ y) = ψ (x)⊙ ψ (y) ;

(iii) ψ (x → y) = ψ (x) → ψ (y) .

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

Proposition 5. Let L₁ and L₂ be De Morgan residuated lattices and ψ : L₁ → L₂ be a DMRL-morphism. Then the following statements hold:

(i) if g is an extreme fuzzy ideal of L₂, then g ∘ ψ is an extreme fuzzy ideal of L₁,

(ii) if ψ is a DMRL-morphism and f is an extreme fuzzy ideal of L₁, then f ∘ ψ^-1 is an extreme fuzzy ideal of L₂ .

Proof. (i) . Let ψ : L₁ → L₂ be a DMRL-morphism and g be an extreme fuzzy ideal of L₂ . Clearly, (g ∘ ψ) (0) =0 . By Theorem 2, it suffices to show that g ∘ ψ is order-preserving and (g ∘ ψ) (x ⊕ y) ≤ (g ∘ ψ) (x) ∨ (g ∘ ψ) (y) for all x, y ∈ L₁ . Since g and ψ are order-preserving, then g ∘ ψ is order-preserving. We easily have that (g∘ ψ) (x ⊕ y) = g (ψ (x ⊕ y)) ≤ g (ψ (x) ∨ ψ (y)) ≤ g (ψ (x)) ∨ g (ψ (y)) = (g ∘ ψ) (x) ∨ (g ∘ ψ) (y) . Therefore, g ∘ ψ is an extreme fuzzy ideal of L₁ .

(ii) By similar way, we prove (ii) holds. □

The following result is clear.

Theorem 20. Let L₁ and L₂ be De Morgan residuated lattices and g be an extreme fuzzy ideal of L₂ . If ψ : L₁ → L₂ is a DMRL-morphism, then we have (a, b) ∈ U_g∘ψ if and only if (ψ (a) , ψ (b)) ∈ U_g for any a, b ∈ L₁ .

Lemma 4. Let L₁ and L₂ be De Morgan residuated lattices and g be an extreme fuzzy ideal of L₂ . If ψ : L₁ → L₂ is a DMRL-morphism, then we have the following statements:

(i) if ψ is surjective, then ψ (U_g∘ψ [a]) = U_g [ψ (a)] for all a ∈ L₁,

(ii) if ψ is a DMRL-isomorphism, then ψ^-1 (U_g [b]) = U_g∘ψ [ψ^-1 (b)] for all b ∈ L₂ .

Proof. (i) . Let b ∈ ψ (U_g∘ψ [a]) . Then there exists c ∈ U_g∘ψ [a] such that ψ (c) = b . It follows that (g ∘ ψ) (c^* ⊙ a) = (g ∘ ψ) (c ⊙ a^*) =0 . Since ψ is a DMRL-morphism, we have g (ψ^* (c) ⊙ ψ (a)) = g (ψ (c) ⊙ ψ^* (a)) =0 . Since ψ (c) = b, we get g (b^* ⊙ ψ (a)) = g (b ⊙ ψ^* (a)) =0 . Hence b ∈ U_g [ψ (a)] . Conversely, let b ∈ U_g [ψ (a)] . Then we have g (b^* ⊙ ψ (a)) = g (b ⊙ ψ^* (a)) =0 . Since ψ is surjective, there exists c ∈ L₁ such that ψ (c) = b . It follows that g (ψ^* (c) ⊙ ψ (a)) = g (ψ^* (a) ⊙ ψ (c)) =0 . Hence (g ∘ ψ) (c^* ⊙ a) = (g ∘ ψ) (c ⊙ a^*) =0, namely, c ∈ U_g∘ψ [a] and so b = ψ (c) ∈ ψ (U_g∘ψ [a]) .

(ii) . We have successively a ∈ ψ^-1 (U_g [b]) iff ψ (a) ∈ U_g [b] iff g (ψ^* (a) ⊙ b) = g (ψ (a) ⊙ b^*) =0 iff g (ψ^* (a) ⊙ ψ (ψ^-1 (b))) = g (ψ^* (ψ^-1 (b)) ⊙ ψ (a)) =0 iff (g ∘ ψ) (a^* ⊙ ψ^-1 (b)) = (g ∘ ψ) ((ψ^-1) ^* (b) ⊙ a) =0 iff a ∈ U_g∘ψ [ψ^-1 (b)] . □

Theorem 21. Let L₁ and L₂ be De Morgan residuated lattices and g be an extreme fuzzy ideal of L₂ . If ψ : L₁ → L₂ is a DMRL-isomorphism, then ψ is a continuous map from (L₁, τ_g∘ψ) to (L₂, τ_g) .

Proof. Let A ∈ τ_g . By Proposition 4(i) , we can get that A = ⋃ _a∈AU_g [a] . It follows that ψ^-1 (A) = ψ^-1 (⋃ _a∈AU_g [a]) = ⋃ _a∈Aψ^-1 (U_g [a]) . By Lemma 4, we have that ψ^-1 (U_g [a]) = U_g∘ψ [ψ^-1 (a)] . Hence ψ^-1 (A) = ⋃ _a∈AU_g∘ψ [ψ^-1 (a)] ∈ τ_g∘ψ . Therefore, ψ is continuous. □

Theorem 22. Let L₁ and L₂ be De Morgan residuated lattices and g be an extreme fuzzy ideal of L₂ . If ψ : L₁ → L₂ is a DMRL-isomorphism, then ψ is a quotient map from (L₁, τ_g∘ψ) to (L₂, τ_g) .

Proof. By Theorem 21, we get ψ is a continuous surjective map. It is enough to show that ψ is an open map. Let A be an open set of (L₁, τ_g∘ψ) . We claim that ψ (A) is an open set in (L₂, τ_g) . Let b ∈ ψ (A) . We shall show that U_g [b] ⊆ ψ (A) . Since ψ is surjective, there exists a ∈ A such that ψ (a) = b . Hence U_g [b] = U_g [ψ (a)] and so U_g [ψ (a)] = ψ (U_g∘ψ [a]) by Lemma 4(ii) . Since A is open and a ∈ A, we get U_g∘ψ [a] ⊆ A . It follows that U_g [b] = U_g [ψ (a)] = ψ (U_g∘ψ [a]) ⊆ ψ (A) . Therefore, $U_{I} [a] \subseteq ψ (A) .$ So ψ is a quotient map. □

The next follows from Theorem 22.

Corollary 8. Let L₁ and L₂ be De Morgan residuated lattices and g be an extreme fuzzy ideal of L₂ . If ψ : L₁ → L₂ is a DMRL-isomorphism, then ψ is a homeomorphic map from (L₁, τ_g∘ψ) to (L₂, τ_g) .

5 Conclusions

In this paper, we introduced the concepts of fuzzy ideals, fuzzy obstinate ideals and extreme fuzzy ideals in residuated lattices, we proposed some characterizations and we showed some important properties of those. The goal is to develop a study of fuzzy ideals in varieties of residuated lattices and to put in evidence their important role. Inspired by the study of X. Zhu, J. Yang and A. Borumand Saeid ([16]), after we defined the family of extreme fuzzy ideals $I$ on a De Morgan residuated lattice L, we constructed a uniform structure (L, K) , and then the part K induced a uniform topology $τ_{I}$ in L . We proved that the pair $(L, τ_{I})$ is a topological De Morgan residuated lattices, and some interesting properties of $(L, τ_{I})$ were investigated. In particular, we proved that $(L, τ_{I})$ is a first-countable, zero-dimensional, disconnected and completely regular space. From the category point of view, the role of isomorphism in algebras is the same as role of homeomorphism in topologies, and all of them are invariants in their fields, respectively. Naturally, we investigated the roles of these invariants in topological De Morgan residuated lattices.

Some important issue for future work are:

(i) developing the properties of different types of fuzzy ideals and congruence relations in residuated lattices,

(ii) finding useful results on other structures;

(iii) introducing the new concepts of fuzzy ideals and investigating its applications on residuated lattices. For example, in [11] we found the notions of r-fuzzy ideal separation axioms, we think that based on these notions we can develop a study on residuated lattices;

(iv) taking as a guide line the work on Fuzzy Lie Algebras [1] we can find more properties for the classes of fuzzy ideal in (fuzzy) residuated lattices;

(v) constructing a topology based on these new concepts of fuzzy 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.

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Extreme fuzzy ideals and its applications on De Morgan residuated lattices

Abstract

Keywords

1 Introduction

2 Preliminaries

3.1 General information

4 Fuzzy ideals of a De Morgan residuated lattice

4.1 General information

4.2 Topological properties of the space ( L , τ I )

5 Conclusions

Footnotes

Acknowledgments

References

4.2 Topological properties of the space $(L, τ_{I})$