On ( L,M )-double fuzzy remote neighborhood systems in ( L,M )-DFTML 1

Abstract

The aim of this paper is to introduce and study the notions of (L, M)-double fuzzy topological molecular lattice, (L, M)-double fuzzy remote neighborhood space, and to construct the category (L, M)-DFRN which contains (L, M)-TDFRN as a full subcategory. (L, M)-TDFRN is shown as an isomorphic with (L, M)-DFTML. Finally, the notion of (L, M)-double fuzzy ideal was introduced and the theory of (L, M)-double fuzzy ideal convergence in an (L, M)-double fuzzy topological molecular lattice was established.

Keywords

isomorphic category

1 Introduction

Kubiak [24] and Šostak [33] introduced the notion of (L-) fuzzy topological space as a generalization of L-topological spaces (originally called (L-) fuzzy topological spaces by Chang [9] and Goguen [19]). It is the grade of openness of an L-fuzzy set. A general approach to the study of topological-type structures on fuzzy powersets was developed in [20 , 24].

As a generalization of fuzzy sets, the notion of intuitionistic fuzzy sets was introduced by Atanassov [4, 5]. Recently, Çoker and his colleagues [14, 15] introduced the notion of intuitionistic fuzzy topological space using intuitionistic fuzzy sets. Samanta and Mondal [30, 31], introduced the notion of intuitionistic gradation of openness as a generalization of intuitionistic fuzzy topological spaces [15] and L-fuzzy topological spaces.

Working under the name “intuitionistic” did not continue because doubts were thrown about the suitability of this term, especially when working in the case of complete lattice L. These doubts were quickly ended in 2005 by Gutierrez Garcia and Rodabaugh [16]. They proved that this term is unsuitable in mathematics and applications. They concluded that they work under the name “double”, and under this name many works have been launched [1 , 40].

Wang [36] introduced the concept of topological molecular lattice (briefly, TML) by the tool of remote neighborhood and established the theory of remote neighborhood systems, which played an important role in TML. Based on this, a series of profound research works have been launched [10–13 , 37]. Topological molecular lattices are categorically isomorphic to singleton topological spaces in which the base lattice changes with each other space, a category part of larger category of variable-basis topology (or point-set lattice-theoretic topology) in the sense of Rodabaugh [27, 28]. On the other hand, certain lattice-theoretic aspects of general topology and L-topology —viewed co-topologically— are captured by TML. So TML is an important category in fuzzy topology. Yue and Fang [38], studied Wang’s TML in a Kubiak-Šostak sense and constructed a category Kubiak- Šostak extension of TML (denoted by FTML). Also, they defined the concept of fuzzy remote neighborhood systems and constructed the category of topological fuzzy remote neighborhood systems (denoted by TFRNS) and they proved that it is isomorphic with FTML.

In general topology, by introducing the notion of ideal, Kuratowsiki [25], Vaidyanathaswamy [34, 35] and several other authors carried out such analysis. Recently, there has been an extensive study on the importance of the ideal in general topology in the paper of Jankovic and Hamlett [23]. Sarkar [32] introduced the notion of fuzzy ideal in fuzzy set theory. In [26], Ramadan et al. investigated some properties of smooth ideals. Yue and Fang [39] studied fuzzy limit structures using fuzzy ideals in the frame work of FTML.

In this paper, we introduce the notions of (L, M)-double fuzzy topological molecular lattice, (L, M)-double fuzzy remote neighborhood space, and construct the category (L, M)-DFRN which contains (L, M)-TDFRN as a full subcategory. Also, we can prove that the category (L, M)-TDFRN is isomorphic with (L, M)-DFTML and (L, M)-DFRN is a topological category over CD^op. Finally, we introduce the notion of (L, M)-double fuzzy ideal and establish the theory of (L, M)-double fuzzy ideal convergence in (L, M)-double fuzzy topological molecular lattice.

2 Preliminaries

An element a in a complete lattice M is said to be coprime if a ≤ b ∨ c implies that a ≤ b or a ≤ c. The set of all coprimes of M is denoted by c (M). We say a is a wedge below b, in symbol, a ⊲ b or b ⊳ a, if for every arbitrary subset D ⊆ M, ⋁D ≥ b implies a ≤ d for some d ∈ D. The lattice M is called completely distributive if every element a ∈ M is the supremum of all elements wedge below it. For more details about completely distributive lattice, please refer to [18]. Let e ∈ c (L) and e|a denote the set {b ∈ L : enotleb, b ≥ a} and β^* (e) denote the standard minimal set of e. 0_L (1_L) and 0_M (1_M) denotes the smallest (largest) elements of lattice L and M respectively. Throughout this paper L and M are completely distributive lattices with an order-reversing involution ^′, if not otherwise stated.

Definition 2.1. [36] Let L₁ and L₂ be complete lattices and f : L₁ → L₂. f will be called a generalized order-homomorphism(briefly, GOH), if

f (a) =0_L
₂ if and only if a = 0_L
₁;

f is union-preserving;

f^⊢ is union-preserving, where f^⊢ (b) = ⋁ {a ∈ L₁ : f (a) ≤ b} for all b ∈ L₂.

Theorem 2.2. [36] Let f : L₁ → L₂ be a GOH. Then

f and f^⊢ are order-preserving.

f^⊢ (f (a)) ≥ a, for each a ∈ L₁.

f (f^⊢ (b)) ≤ b, for each b ∈ L₂.

f (a) ≤ b if and only if a ≤ f^⊢ (b).

f (a) = ⋀ {b ∈ L₂ : f^⊢ (a) ≥ a}, for each a ∈ L₁.

f^⊢ : L₂ → L₁ is intersection-preserving.

Definition 2.3. [36] For a completely distributive lattice M, the CD laws are

⋀_i∈Γ (⋁ _{j∈J_i}a_i,j) = ⋁ _{f∈ΠJ_i} (⋀ _i∈Γa_i,f(i)) ,

⋁_i∈Γ (⋀ _{j∈J_i}a_i,j) = ⋀ _{f∈ΠJ_i} (⋁ _i∈Γa_i,f(i)) ,

where, for each i ∈ Γ and each j ∈ J, a_i,j ∈ M and f ∈ ΠJ_i means f is a mapping f : Γ → ∪ J_i such that for each i ∈ Γ, f (i) ∈ J_i. As is well known, (CD1) is equivalent to (CD2).

Definition 2.4. [3] (1) Category A is said to be a subcategory of category B provided that the following conditions are satisfied:

Ob (A) ⊆ Ob (B);

for each , A,A′ ∈ ob(A), hom_A (A,A′) ⊆ hom_B(A,A′);

for each A-object A, the B-identity on A is the A-identity on A;

the composition law in A is the restriction of the composition law in B to the morphisms of A.

(2) A is called a full subcategory of B if, in addition to the above, for each , hom_A(A,A′) = hom_B(A,A′).

Definition 2.5. [3, 29] Category A is topological w.r.t. category X and functor V : A → X if and only if each V-structured source in X has a unique, initial V-lift in A.

Remark 2.6. In [29], Rodabaugh also gives a longer definition of topological category to make it absolutely clear what is the intention of [3] in this definition w.r.t. each of the terms “V-structured source”, “V-lift”, “initial”, and “unique”.

For undefined notions about categories, we refer to [3 , 29].

Definition 2.7. [38] Let L and M be completely distributive lattice. An (L, M)-fuzzy co-topology is a map η : L → M such that

η (1_L) = η (0_L) =1_M;

η (u ∨ v) ≥ η (u) ∧ η (v) for all u, v ∈ L;

η (⋀ _i∈Γu_i) ≥ ⋀ _i∈Γη (u_i), for any {u_i} _i∈Γ⊆L.

If η is an (L, M)-fuzzy co-topology, then we say (L, M, η) is a fuzzy topological molecular lattice (briefly, FTML). The value of η (u) can be interpreted as the degree of closeness of u ∈ L. A continuous map between FTMLs (L₁, M, η) and (L₂, M, δ) is a GOH f : L₁ → L₂ such that η (f^⊢ (u)) ≥ δ (u) for all u ∈ L₂. The category of FTMLs and their continuous maps is called Kubiak-Šostak extension of TML denoted by FTML.

Definition 2.8. [38] Let L and M be completely distributive lattices. An (L, M)-fuzzy remote neighborhood system is a set R = {R_e : e ∈ c (L)} of maps R_e : L → M such that for each u, v ∈ L, we have

R_e (1_L) =0_M, R_e (0_L) =1_M;

R_e (u) >0_M implies enotleu;

R_e (u ∨ v) = R_e (u) ∧ R_e (v).

The pair (L, M, R) is called a fuzzy remote neighborhood space (briefly, FRNS). If it also satisfies the following equation:

R_e (u) = ⋁ _v∈e|u ⋀ _snotlevR_s (v),

we call it a topological fuzzy remote neighborhood space (briefly, TFRNS).

A continuous map between FRNSs (L₁, M, R) and (L₂, M, S) is a GOH f : L₁ → L₂ such that S_f(e) (u) ≤ R_e (f^⊢ (u)), for all e ∈ c (L₁) and u ∈ L₂.

Definition 2.9. [2] Let L and M be completely distributive lattices. A map $I : L \to M$ is called an (L, M)-fuzzy ideal on L if it satisfies the following conditions:

$I (0_{L}) = 1_{M}$ , $I (1_{L}) = 0_{M}$ ;

$I (u \lor v) \geq I (u) \land I (v)$ for each u, v ∈ L;

u ł ev implies $I (u) \geq I (v)$ .

3 (L, M)-Double fuzzy remote neighborhood spaces

Definition 3.1. Let L and M be completely distributive lattices. The pair (η, η^*) of maps η, η^* : L → M is called an (L, M)-double fuzzy co-topology on L if it satisfies the following conditions:

η (u) ≤ (η^* (u)) ′, for each u ∈ L,

η (0_L) = η (1_L) =1_M,η^* (0_L) = η^* (1_L) =0_M,

η (u ∨ v) ≥ η (u) ∧ η (v), and η^* (u ∨ v)≤η^* (u) ∨ η^* (v), for any u, v ∈ L.

η (⋀ _i∈Γu_i) ≥ ⋀ _i∈Γη (u_i), and η^* (⋀ _i∈Γu_i) ≤ ⋁ _i∈Γη^* (u_i), for any family {u_i : i ∈ Γ} ⊆ L.

If (η, η^*) is an (L, M)-double fuzzy co-topology, then we say that (L, η, η^*) is an (L, M)-double fuzzy topological molecular lattice (briefly, (L, M)-DFTML). The value of η (u) can be interpreted as the degree of closeness of u ∈ L, and the value of η^* (u) can be interpreted as the degree of non-closeness of u ∈ L. If $(η_{1}, η_{1}^{*})$ and $(η_{2}, η_{2}^{*})$ are two (L, M)-double fuzzy co-topologies on L, we say that $(η_{1}, η_{1}^{*})$ is finer than $(η_{2}, η_{2}^{*})$ (or $(η_{2}, η_{2}^{*})$ is coarser than $(η_{1}, η_{1}^{*})$ ), denoted by $(η_{2}, η_{2}^{*}) \leq (η_{1}, η_{1}^{*})$ if and only if η₂ (u) ≤ η₁ (u) and $η_{2}^{*} (u) \geq η_{1}^{*} (u)$ , for each u ∈ L.

A continuous map between two (L, M)-DFTMLs (L₁, η, η^*) and (L₂, δ, δ^*) is a GOH f : L₁ → L₂ with η (f^⊢ (u)) ≥ δ (u) and η^* (f^⊢ (u)) ≤ δ^* (u) , ∀ u ∈ L₂ .

The category of (L, M)-DFTMLs with their continuous maps is denoted by (L, M)-DFTML.

Remark 3.2. Let η : L → M be an (L, M)-fuzzy co-topology and η^* : L → M be a mapping defined by, η^* (u) = (η (u)) ′, ∀u ∈ L. Then the pair (η, η^*) is an (L, M)-double fuzzy co-topology. Therefore, (L, M)-double fuzzy co-topology is a generalization of (L, M)-fuzzy co-topology.

Definition 3.3. Let L and M be completely distributive lattices. An (L, M)-double fuzzy remote neighborhood system is defined to be a set $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : e \in c (L)}$ of maps $R_{e}, R_{e}^{*} : L \to M$ such that ∀u, v ∈ L,

$R_{e} (u) \leq (R_{e}^{*} (u))^{'}$ .

R_e (0_L) =1_M, R_e (1_L) =0_M, and $R_{e}^{*} (0_{L})$ = 0_M, $R_{e}^{*} (1_{L}) = 1_{M}$ .

R_e (u) ≠0_M, $R_{e}^{*} (u) \neq 1_{M}$ implies enotleu.

R_e (u ∨ v) = R_e (u) ∧ R_e (v) and $R_{e}^{*} (u \lor v) = R_{e}^{*} (u) \lor R_{e}^{*} (v)$ .

The triplet (L, R, R^*) is called an (L, M)-double fuzzy remote neighborhood space (briefly, (L, M)-DFRNS), and it will be called (L, M)-topological double fuzzy remote neighborhood space (briefly, (L, M)-TDFRNS) if it satisfies moreover, for all e ∈ c (L), u ∈ L,

R_e (u) = ⋁ _v∈e|u ⋀ _snotlevR_s (v), and $R_{e}^{*} (u) =$ $⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*} (v)$ .

If (R, R^*) and (S, S^*) are two (L, M)-double fuzzy remote neighborhood systems, we say that (R, R^*) is finer than (S, S^*) (or (S, S^*) is coarser than (R, R^*)), denoted by (S, S^*) ≤ (R, R^*) if and only if S_e (u) ≤ R_e (u) and $S_{e}^{*} (u) \geq R_{e}^{*} (u)$ , for each u ∈ L.

A continuous map between two (L, M)-DFRNSs (L₁, R, R^*) and (L₂, S, S^*) is a GOH f : L₁ → L₂ such that for each e ∈ c (L₁) and u ∈ L₂, $S_{f (e)} (u) \leq R_{e} (f^{⊢} (u)), S_{f (e)}^{*} (u) \geq R_{e}^{*} (f^{⊢} (u)) .$

The category of (L, M)-DFRNSs with their continuous maps is denoted by (L, M)-DFRN, and (L, M)-TDFRN the full subcategory of (L, M)-DFRN consisting of (L, M)-TDFRNSs.

Remark 3.4. (i) Let R_e : L → M be an (L, M)-fuzzy remote neighborhood system and $R_{e}^{*} : L \to M$ be a mapping defined by, $R_{e}^{*} (u) = (R_{e} (u))^{'}$ , ∀u ∈ L. Then the pair $(R_{e}, R_{e}^{*})$ is an (L, M)-double fuzzy remote neighborhood system. Therefore, (L, M)-double fuzzy remote neighborhood system is a generalization of (L, M)-fuzzy remote neighborhood system.

(ii) For all e ∈ c (L) and u ∈ L, R_e (u) can be thought as the degree of u being a remote neighborhood of e and $R_{e}^{*} (u)$ can be thought as the degree of u being non remote neighborhood of e.

Suppose that (η, η^*) be an (L, M)-double fuzzy co-topology, where η, η^* : L → M. Define $R_{e}^{η}, R_{e}^{*^{η^{*}}} : L \to M$ as: $R_{e}^{η} (u) = {\begin{matrix} ⋁ {η (v) : v \in e | u}, & if e ≰ u \\ 0_{M}, & if e \leq u \end{matrix}$ $R_{e}^{*^{η^{*}}} (u) = {\begin{matrix} ⋀ {η^{*} (v) : v \in e | u}, & if e ≰ u \\ 1_{M}, & if e \leq u \end{matrix}$ for each e ∈ c (L) and u ∈ L. Then we have:

Proposition 3.5. The set of $(R^{η}, R^{*^{η^{*}}}) = {(R_{e}^{η}$ , $R_{e}^{*^{η^{*}}}) : e \in c (L)}$ is an (L, M)-topological double fuzzy remote neighborhood system, and called induced (L, M)-topological double fuzzy remote neighborhood system from (η, η^*).

Proof. (DFR1): By (DFCT1), we have $\begin{matrix} R_{e}^{η} (u) = ⋁ {η (v) : v \in e | u} \leq ⋁ {(η^{*} (v))^{'} : v \in e | u} \\ = (⋀ {η^{*} (v) : v \in e | u})^{'} = (R_{e}^{*^{η^{*}}} (u))^{'} . \end{matrix}$

(DFR2) and (DFR3) are true trivially.

(DFR4): From the definition of $R_{e}^{η}$ and $R_{e}^{*^{η^{*}}}$ , we have: if u, v ∈ L such that u ≤ v, then $R_{e}^{η} (u) \geq R_{e}^{η} (v)$ and $R_{e}^{*^{η^{*}}} (u) \leq R_{e}^{*^{η^{*}}} (v)$ . Thus for each u, v ∈ L we have: $\begin{matrix} R_{e}^{η} (u \lor v) \leq R_{e}^{η} (u) \land R_{e}^{η} (v), \\ R_{e}^{*^{η^{*}}} (λ \lor μ) \geq R_{e}^{*^{η^{*}}} (λ) \lor R_{e}^{*^{η^{*}}} (μ) . \end{matrix}$

On the other hand, for each α ∈ c (M) such that $α ⊲ R_{e}^{η} (u) \land R_{e}^{η} (v)$ , we have $α ⊲ R_{e}^{η} (u)$ and $α ⊲ R_{e}^{η} (v)$ . Thus there exist u₁ ∈ e|u, v₁ ∈ e|v such that α ≤ η (u₁) and α ≤ η (v₁), respectively. Therefore $α \leq η (u_{1}) \land η (v_{1}) \leq η (u_{1} \lor v_{1}) .$

It is clear that u₁ ∨ v₁ ∈ e| (u ∨ v). Then, by the definition of $R_{e}^{η}$ , $α \leq η (u_{1} \lor u_{1}) \leq R_{e}^{η} (u \lor v) .$

Hence, from arbitrariness of α, we have $R_{e}^{η} (λ \lor μ) \geq R_{e}^{η} (λ) \land R_{e}^{η} (μ) .$

It remains to prove that: $R_{e}^{*^{η^{*}}} (λ \lor μ) \leq R_{e}^{*^{η^{*}}} (λ) \lor R_{e}^{*^{η^{*}}} (μ) .$ So let $β ⊲ R_{e}^{*^{η^{*}}} (u \lor v)$ , β ∈ c (M). Then, β ≤ η^* (w), for each w ∈ L with e≰w, w ≥ u ∨ v. Since w ∈ e|u and w ∈ e|v, then $β \leq R_{e}^{*^{η^{*}}} (u)$ and $β \leq R_{e}^{*^{η^{*}}} (v)$ . Therefore, $β \leq R_{e}^{*^{η^{*}}} (u) \lor R_{e}^{*^{η^{*}}} (v)$ . Hence, from arbitrariness of α, we have $R_{e}^{*^{η^{*}}} (u \lor v) \leq R_{e}^{*^{η^{*}}} (u) \lor R_{e}^{*^{η^{*}}} (v) .$ (DFR5) ∀ v ∈ e|u, we have $\begin{matrix} η (v) \leq ⋀_{s ≰ v} R_{s}^{η} (v) \leq R_{e}^{η} (v) \leq R_{e}^{η} (u), \\ η^{*} (v) \geq ⋁_{s ≰ v} R_{s}^{*^{η^{*}}} (v) \geq R_{e}^{*^{η^{*}}} (v) \geq R_{e}^{*^{η^{*}}} (u) . \end{matrix}$ Therefore, $\begin{matrix} R_{e}^{η} (u) & = ⋁_{v \in e | u} η (v) \leq ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s}^{η} (v) \\ \leq R_{e}^{η} (u), \\ R_{e}^{*^{η^{*}}} (u) & = ⋀_{v \in e | u} η^{*} (v) \geq ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*^{η^{*}}} (v) \\ \geq R_{e}^{*^{η^{*}}} (u) . \end{matrix}$

This means that: $\begin{matrix} R_{e}^{η} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s}^{η} (v), \\ R_{e}^{*^{η^{*}}} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*^{η^{*}}} (v) . \end{matrix}$

It is easy to verify the following lemma.

Lemma 3.6. For t, s ∈ c (L), if t ≤ s then, $(R_{t}^{η}$ , $R_{t}^{*^{η^{*}}}) \leq (R_{s}^{η}, R_{s}^{*^{η^{*}}})$ .

Lemma 3.7. ∀ u ∈ L, $η (u) = ⋀_{e ≰ u} R_{e}^{η} (u)$ and $η^{*} (u) = ⋁_{e ≰ u} R_{e}^{*^{η^{*}}} (u)$ .

Proof. Obviously, ∀ u ∈ L, $η (u) \leq ⋀_{e ≰ u} R_{e}^{η} (u) and η^{*} (u) \geq ⋁_{e ≰ u} R_{e}^{*^{η^{*}}} (u) .$

So, it suffices to prove that: $η (u) \geq ⋀_{e ≰ u} R_{e}^{η} (u) and η^{*} (u) \leq ⋁_{e ≰ u} R_{e}^{*^{η^{*}}} (u) .$

In fact, we have $\begin{matrix} ⋀_{e ≰ u} R_{e}^{η} (u) = ⋀_{e ≰ u} ⋁_{v \in e | u} η (v) \\ = ⋁_{f \in Π_{e ≰ u} e | u} ⋀_{e ≰ u} η (f (e)) (by CD - Laws) \\ \leq ⋁_{f \in Π_{e ≰ u} e | u} η (⋀_{e ≰ u} f (e)) = η (u) . \end{matrix}$

The last equality is due to ⋀_e≰uf (e) = u, ∀f ∈ Π_e≰ue|u. Also, $\begin{matrix} ⋁_{e ≰ u} R_{e}^{*^{η^{*}}} (u) = ⋁_{e ≰ u} ⋀_{v \in e | u} η^{*} (v) \\ = ⋀_{f \in Π_{e ≰ u} e | u} ⋁_{e ≰ u} η^{*} (f (e)) (by CD - Laws) \\ \geq ⋀_{f \in Π_{e ≰ u} e | u} η^{*} (⋀_{e ≰ u} f (e)) = η^{*} (u) . \end{matrix}$

Proposition 3.8. (i) If $(η_{1}, η_{1}^{*})$ and $(η_{2}, η_{2}^{*})$ are two (L, M)-double fuzzy co-topologies which determine the same (L, M)-topological double fuzzy remote neighborhood system, then $(η_{1}, η_{1}^{*})$ = $(η_{2}, η_{2}^{*})$ .

(ii) Suppose that $f : (L_{1}, η_{1}, η_{1}^{*}) \to (L_{2}, η_{2}, η_{2}^{*})$ is a continuous map between (L, M)-double fuzzy co-topological spaces. Then, $f : (L_{1}, R^{η_{1}}, R^{*^{η_{1}^{*}}}) \to (L_{2}, R^{η_{2}}, R^{*^{η_{2}^{*}}})$ is also continuous with respect to induced (L, M)-topological double fuzzy remote neighborhood systems.

Proof. (i) Hold by Lemma 3.7. (ii) Since $f : (L_{1}, η_{1}, η_{1}^{*}) \to (L_{2}, η_{2}, η_{2}^{*})$ is continuous, then for each u ∈ L₂ we have, $η_{2} (u) \leq η_{1} (f^{⊢} (u)) and η_{2}^{*} (u) \geq η_{1}^{*} (f^{⊢} (u)) .$

Notice that, $R_{e}^{η} (u) = ⋁_{v \in e | u} η (v) and R_{e}^{*^{η^{*}}} (u) = ⋀_{v \in e | u} η^{*} (v) .$

It follows that: $\begin{matrix} R_{f (e)}^{η_{2}} (u) = ⋁_{v \in f (e) | u} η_{2} (v) \\ \leq ⋁_{f^{⊢} (v) \in e | f^{⊢} (u)} η_{1} (f^{⊢} (v)) \leq R_{e}^{η_{1}} (f^{⊢} (u)), \end{matrix}$ $\begin{matrix} R_{f (e)}^{*^{η_{2}^{*}}} (u) = ⋀_{v \in f (e) | u} η_{2}^{*} (v) \\ \geq ⋀_{f^{⊢} (v) \in e | f^{⊢} (u)} η_{1}^{*} (f^{⊢} (v)) \geq R_{e}^{*^{η_{1}^{*}}} (f^{⊢} (u)) . \end{matrix}$

From the above proposition, we have obtained a functor from (L, M)-DFTML to (L, M)-DFRN which is injective on objects.

Let $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : e \in c (L)}$ be an (L, M)-topological fuzzy remote neighborhood system on X. Define a pair (η^R, η^{*^{R
^*}}) of maps η^R, η^{*^{R
^*}} : L → M as: $η^{R} (u) = {\begin{matrix} ⋀_{e ≰ u} R_{e} (u), & if u \neq 0_{L} \\ 1_{M}, & if u = 0_{L} \end{matrix}$ $η^{*^{R^{*}}} (u) = {\begin{matrix} ⋁_{e ≰ u} R_{e}^{*} (u), & if u \neq 0_{L} \\ 0_{M}, & if u = 0_{L} . \end{matrix}$

Then we have:

Proposition 3.9.(i) (η^R, η^{*^{R
^*}}) defined above is an (L, M)-double fuzzy co-topology, called induced (L, M)-double fuzzy co-topology from (R, R^*). Moreover, if (P, P^*) and (R, R^*) are two (L, M)-topological double fuzzy remote neighborhood systems which determine the same (L, M)-double fuzzy co-topology, then (P, P^*)= (R, R^*).

(ii) If a map f : (L₁, P, P^*) → (L₂, R, R^*) is a continuous map between (L, M)-topological double fuzzy remote neighborhood spaces, then f is continuous with respect to induced (L, M)-double fuzzy co-topologies.

Proof. (i) (DFCT1) and (DFCT2) are easily proved. (DFCT3) For each u, v ∈ L, we have: $\begin{matrix} η^{R} (u \lor v) & = ⋀_{e ≰ (u \lor v)} R_{e} (u \lor v) \\ = ⋀_{e ≰ (u \lor v)} (R_{e} (u) \land R_{e} (v)) \\ \geq (⋀_{e ≰ u} R_{e} (u)) \land (⋀_{e ≰ v} R_{e} (v)) \\ = η^{R} (u) \land η^{R} (v), \\ η^{*^{R^{*}}} (u \lor v) & = ⋁_{e ≰ (u \lor v)} R_{e}^{*} (u \lor v) \\ = ⋁_{e ≰ (u \lor v)} (R_{e}^{*} (u) \lor R_{e}^{*} (v)) \\ \leq (⋁_{e ≰ u} R_{e}^{*} (u)) \lor (⋁_{e ≰ v} R_{e}^{*} (v)) \\ = η^{*^{R^{*}}} (u) \lor η^{*^{R^{*}}} (v) . \end{matrix}$

(DFT4) For each {u_i : i ∈ Γ} ⊆ L, we have: $\begin{matrix} η^{R} (⋀_{i \in Γ} u_{i}) = ⋀_{e ≰ (⋀_{i \in Γ} u_{i})} R_{e} (⋀_{i \in Γ} u_{i}) \\ = ⋀_{i \in Γ} ⋀_{e ≰ u_{i}} R_{e} (⋀_{i \in Γ} u_{i}) \geq ⋀_{i \in Γ} ⋀_{e ≰ u_{i}} R_{e} (u_{i}) \\ = ⋀_{i \in Γ} η^{R} (u_{i}), \end{matrix}$ $\begin{matrix} η^{*^{R^{*}}} (⋀_{i \in Γ} u_{i}) = ⋁_{e ≰ (⋀_{i \in Γ} u_{i})} R_{e}^{*} (⋀_{i \in Γ} u_{i}) \\ = ⋁_{i \in Γ} ⋁_{e ≰ u_{i}} R_{e}^{*} (⋀_{i \in Γ} u_{i}) \leq ⋁_{i \in Γ} ⋁_{e ≰ u_{i}} R_{e}^{*} (u_{i}) \\ = ⋁_{i \in Γ} η^{*^{R^{*}}} (u_{i}) . \end{matrix}$

Hence, (η^R, η^{*^{R
^*}}) is an (L, M)-double fuzzy co-topology.

In addition, suppose that (η, η^*) is the same (L, M)-double fuzzy co-topology. Then, for each e ∈ c (L) and u ∈ L, $\begin{matrix} P_{e} (u) & = ⋁_{v \in e | u} ⋀_{s ≰ v} P_{s} (v) = ⋁_{v \in e | u} η (v) \\ = ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s} (v) = R_{e} (u), \\ P_{e}^{*} (u) & = ⋀_{v \in e | u} ⋁_{s ≰ v} P_{s}^{*} (v) = ⋀_{v \in e | u} η^{*} (v) \\ = ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*} (v) = R_{e}^{*} (u) . \end{matrix}$

Hence, (P, P^*) = (R, R^*).

(ii) Since e≰f^⊢ (v) if and only if f (e) ≰v, ∀ v ∈ L₂, and $\begin{matrix} {s \in c (L_{2}) : s ≰ v} \\ \supseteq {f (e) \in c (L_{2}) : e \in c (L_{1}), and f (e) ≰ v} \end{matrix}$

we have: ∀v ∈ L₂, $\begin{matrix} η^{R} (v) = ⋀_{s ≰ v} R_{s} (v) \leq ⋀_{f (e) ≰ v} R_{f (e)} (v) \\ \leq ⋀_{e ≰ f^{⊢} (v)} P_{e} (f^{⊢} (v)) = η^{P} (f^{⊢} (v)), \\ (since, f : (L_{1}, P, P^{*}) \to (L_{2}, R, R^{*}) is continuous) \\ η^{*^{R^{*}}} (v) = ⋁_{s ≰ v} R_{s}^{*} (v) \geq ⋁_{f (e) ≰ v} R_{f (e)}^{*} (v) \\ \geq ⋁_{e ≰ f^{⊢} (v)} P_{e}^{*} (f^{⊢} (v)) = η^{*^{P^{*}}} (f^{⊢} (v)) . \\ (since, f : (L_{1}, P, P^{*}) \to (L_{2}, R, R^{*}) is continuous) \end{matrix}$

Hence, f : (L₁, η^P, η^{*^{P
^*}}) → (L₂, η^R, η^{*^{R
^*}}) is continuous.

The next results follow from Propositions 3.5, 3.8- 3.9.

Theorem 3.10. (L, M)-TDFRNis isomorphic to (L, M)-DFTML.

Let CD denote the category of completely distributive lattices with complete lattice morphisms as morphisms. We know that the category of completely distributive lattices with GOHs as morphisms is the dual category of CD. The following two theorems are valid.

Theorem 3.11. (L, M)-DFRNis a topological category overCD^op.

Proof. We need to check the conditions of fibre-smallness and existence of initial structures for this category. The fibre-smallness condition is trivial. We need to prove that it fulfills the existence of initial structures. Let {f_j : L → (L_j, R^j, R^{*^j})} _j∈J (f_j is GOH for all j ∈ J) be a source in (L, M)-DFRN and $(R, R^{*}) = {(R_{e}, R_{e}^{*}) : R_{e}, R_{e}^{*} : L \to M, e \in c (L)}$ defined by: ∀e ∈ c (L), u ∈ L, $\begin{matrix} R_{e} (u) & = ⋁_{F \in J^{< w}} {⋀_{j \in F} R_{f_{j} (e)}^{j} (u_{j}) : \\ u_{j} \in L_{j}, u \leq ⋁_{j \in F} f_{j}^{⊢} (u_{j})}, \\ R_{e}^{*} (u) & = ⋀_{F \in J^{< w}} {⋁_{j \in F} R_{f_{j} (e)}^{*^{j}} (u_{j}) : \\ u_{j} \in L_{j}, u \leq ⋁_{j \in F} f_{j}^{⊢} (u_{j})} . \end{matrix}$

We will show that (R, R^*) is the unique (L, M)-DFRN-structure on L which is initial with respect to the source {f_j : L → (L_j, R^j, R^{*^j})} _j∈J.

Step 1. (V-lift): (R, R^*) is (L, M)-DFRN-structure on L, i.e., (R, R^*) is an (L, M)-double fuzzy remote neighborhood system on L and (R, R^*) makes f_j continuous for each j ∈ J.

(DFR1) and (DFR2) are trivial, (DFR3) is routine.

(DFR4) From the definition of R_e and $R_{e}^{*}$ , we know that R_e (u) ≥ R_e (v) and $R_{e}^{*} (u) \leq R_{e}^{*} (v)$ , when u ≤ v. Therefore, for each u, v ∈ L, we have: $\begin{matrix} R_{e} (u \lor v) \leq R_{e} (u) \land R_{e} (v), \\ R_{e}^{*} (u \lor v) \geq R_{e}^{*} (u) \lor R_{e}^{*} (v) . \end{matrix}$

On the other hand, let α ∈ c (M) such that α ⊲ R_e (u) ∧ R_e (v) . Then, α ⊲ R_e (u) and α ⊲ R_e (v) . By the definition of R_e, there exist {u_j} _j∈F₁ with $u \leq ⋁_{j \in F_{1}} f_{j}^{⊢} (u_{j})$ , and there exist {v_j} _j∈F₂ with $v \leq ⋁_{j \in F_{2}} f_{j}^{⊢} (v_{j})$ , such that: $α \leq ⋀_{j \in F_{1}} R_{f_{j} (e)}^{j} (u_{j}) and α \leq ⋀_{j \in F_{2}} R_{f_{j} (e)}^{j} (v_{j}),$ respectively, where F₁ and F₂ are two finite subsets of J. Let F = F₁ ∪ F₂ and $w_{j} = {\begin{matrix} u_{j}, & j \in F_{1}, j \notin F_{2} \\ v_{j}, & j \notin F_{1}, j \in F_{2} \\ u_{j} \lor v_{j}, & j \in F_{1} \cap F_{2} . \end{matrix}$

Then F is a finite subset of J and we have: $\begin{matrix} u \lor v & \leq ⋁_{j \in F_{1}} f_{j}^{⊢} (u_{j}) \lor ⋁_{j \in F_{2}} f_{j}^{⊢} (v_{j}) \\ = ⋁_{j \in F} f_{j}^{⊢} (w_{j}), \end{matrix}$ and $α \leq ⋀_{j \in F} R_{f_{j} (e)}^{j} (w_{j})$ . Thus α ≤ R_e (u ∨ v). From the arbitrariness of α, we have: $R_{e} (u \lor v) \geq R_{e} (u) \land R_{e} (v) .$

It remains to prove that: $R_{e}^{*} (u \lor v) \leq R_{e}^{*} (u) \lor R_{e}^{*} (v) .$ So, let β ∈ c (M) such that $β ⊲ R_{e}^{*} (u \lor v) .$ Then, there exists {u_j} _j∈F₃, where F₃ is a finite subset of Γ, with $u \lor v \leq ⋁_{j \in F_{3}} f_{j}^{⊢} (u_{j}) and β \leq ⋁_{j \in F_{3}} R_{f_{j} (e)}^{*^{j}} (u_{j}) .$

Then, $\begin{matrix} u \leq ⋁_{j \in F_{3}} f_{j}^{⊢} (u_{j}), β \leq ⋁_{j \in F_{3}} R_{f_{j} (e)}^{*^{j}} (u_{j}), \\ v \leq ⋁_{j \in F_{3}} f_{j}^{⊢} (u_{j}), β \leq ⋁_{j \in F_{3}} R_{f_{j} (e)}^{*^{j}} (u_{j}) . \end{matrix}$

Thus, $β \leq R_{e}^{*} (u)$ and $β \leq R_{e}^{*} (v)$ , this implies that $β \leq R_{e}^{*} (u) \lor R_{e}^{*} (v)$ .

Since β is arbitrary, $R_{e}^{*} (u \lor v) \leq R_{e}^{*} (u) \lor R_{e}^{*} (v) .$

Furthermore, from the definition of R_e and $R_{e}^{*}$ we have: $\begin{matrix} R_{e} (f_{j}^{⊢} (u_{j})) \geq R_{f_{j} (e)}^{j} (u_{j}), \\ R_{e}^{*} (f_{j}^{⊢} (u_{j})) \leq R_{f_{j} (e)}^{*^{j}} (u_{j}), \end{matrix}$ for each u_j ∈ L_j. Then, f_j : (L, R, R^*) → (L_j, R^j, R^{*^j}) is continuous for each j ∈ J.

Step 2. (initial V-lift): Now, we show that (R, R^*) is initial V-lift, i.e., for an (L, M)-DFRN-object (N, S, S^*) a map f : (N, S, S^*) → (L, R, R^*) is continuous if and only if f_j ∘ f : (N, S, S^*) → (L_j, R^j, R^{*^j}) is continuous, for each j ∈ J. Suppose that f : (N, S, S^*) → (L, R, R^*) is continuous. Then, for each u ∈ L_j and e ∈ c (N), we have: $\begin{matrix} S_{e} ((f_{j} \circ f)^{⊢} (u)) = S_{e} (f^{⊢} (f_{j}^{⊢} (u))) \geq R_{f (e)} (f_{j}^{⊢} (u)) \\ = ⋁_{F \in J^{< w}} {⋀_{k \in F} R_{f_{k} (f (e))}^{k} (u_{k}) : u_{k} \in L_{k}, \\ f_{j}^{⊢} (u) \leq ⋁_{k \in F} f_{k}^{⊢} (u_{k})} \\ \geq R_{f_{k} (f (e))}^{k} (u), \end{matrix}$ $\begin{matrix} S_{e}^{*} ((f_{j} \circ f)^{⊢} (u)) = S_{e}^{*} (f^{⊢} (f_{j}^{⊢} (u))) \leq R_{f (e)}^{*} (f_{j}^{⊢} (u)) \\ = ⋀_{F \in J^{< w}} {⋁_{k \in F} R_{f_{k} (f (e))}^{*^{k}} (u_{k}) : u_{k} \in L_{k}, \\ f_{j}^{⊢} (u) \leq ⋁_{k \in F} f_{k}^{⊢} (u_{k})} \\ \leq R_{f_{k} (f (e))}^{*^{k}} (u) . \end{matrix}$ Thus f_j ∘ f : (N, S, S^*) → (L_j, R^j, R^{*^j}) is continuous, for each j ∈ J.

Conversely, we want to show that: $S_{e} (f^{⊢} (u)) \geq R_{f (e)} (u), and S_{e}^{*} (f^{⊢} (u)) \leq R_{f (e)}^{*} (u),$

∀ e ∈ c (N) and ∀ u ∈ L. By, $\begin{matrix} R_{f (e)} (u) & = ⋁_{F \in J^{< w}} {⋀_{j \in F} R_{f_{j} (f (e))}^{j} (u_{j}) : \\ u_{j} \in L_{j}, u \leq ⋁_{j \in F} f_{j}^{⊢} (u_{j})}, \\ R_{f (e)}^{*} (u) & = ⋀_{F \in J^{< w}} {⋁_{j \in F} R_{f_{j} (f (e))}^{*^{j}} (u_{j}) : \\ u_{j} \in L_{j}, u \leq ⋁_{j \in F} f_{j}^{⊢} (u_{j})} . \end{matrix}$

It is only to prove that: $\begin{matrix} ⋀_{j \in F} R_{f_{j} (f (e))}^{j} (u_{j}) \leq S_{e} (f^{⊢} (u)), \\ ⋁_{j \in F} R_{f_{j} (f (e))}^{*^{j}} (u_{j}) \geq S_{e}^{*} (f^{⊢} (u)), \end{matrix}$ for all F ∈ J^<w and for all {u_j} _j∈F with $u \leq ⋁_{j \in F} f_{j}^{⊢} (u_{j})$ . Since f_j ∘ f is continuous for each j ∈ J, we have $R_{f_{j} (f (e))}^{j} (u_{j}) \leq S_{e} ((f_{j} \circ f)^{⊢} (u_{j})) .$

Then, $⋀_{j \in F} R_{f_{j} (f (e))}^{j} (u_{j}) \leq ⋀_{j \in F} S_{e} ((f_{j} \circ f)^{⊢} (u_{j})),$ but $\begin{matrix} ⋀_{j \in F} S_{e} ((f_{j} \circ f)^{⊢} (u_{j})) = S_{e} (⋁_{j \in F} (f_{j} \circ f)^{⊢} (u_{j})) \\ = S_{e} (⋁_{j \in F} (f^{⊢} (f_{j}^{⊢} (u_{j})))) = S_{e} (f^{⊢} (⋁_{j \in F} (f_{j}^{⊢} (u_{j})))) \\ \leq S_{e} (f^{⊢} (u)) . \end{matrix}$

Thus, $⋀_{j \in F} R_{f_{j} (f (e))}^{j} (u_{j}) \leq S_{e} (f^{⊢} (u)) .$

In the same manner, we can show that: $⋁_{j \in F} R_{f_{j} (f (e))}^{*^{j}} (u_{j}) \geq S_{e}^{*} (f^{⊢} (u)) .$

Step 3. (unique initial V-lift): Suppose that (L, P,P^*) is another initial V-lift with respect to the source {f_j : L → (L_j, R^j, R^{*^j})} _j∈J. Let id_L :(L, P, P^*) → (L, R, R^*). Since, (L, R, R^*) is initial V-lift and f_j ∘ id_L = f_j is continuous for all j ∈ J, then id_L is continuous. Hence, P_e (u) ≥ R_e (u) and $P_{e}^{*} (u) \leq R_{e}^{*} (u)$ , for each u ∈ L. Then, (P, P^*) ≥ (R, R^*). Using the similar argument, we have (P, P^*) ≤ (R, R^*). Therefore, (P, P^*) = (R, R^*). Hence, (L, M)-DFRN is a topological category over CD^op.

Theorem 3.12. (L, M)-TDFRNis bireflective in(L, M)-DFRN; hence (L, M)-TDFRNis a topological category overCD^op.

Proof. Assume that (L, R, R^*) is an (L, M)-double fuzzy remote neighborhood space, we assert that its (L, M)-TDFRN-reflection is defined by id_L : (L, R, R^*) → (L, R^H, R^{*^{H
^*}}) where, $\begin{matrix} (R^{H}, R^{*^{H^{*}}}) \\ = {(R_{e}^{H}, R_{e}^{*^{H^{*}}}) : R_{e}^{H}, R_{e}^{*^{H^{*}}} : L \to M, e \in c (L)}, \end{matrix}$ and for each e ∈ c (L), u ∈ L, $\begin{matrix} R_{e}^{H} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s} (v), \\ R_{e}^{*^{H^{*}}} (u) = ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*} (v) . \end{matrix}$

Step 1: (L, R^H, R^{*^{H
^*}}) is an (L, M)-topological double fuzzy remote neighborhood space, i.e., (L, R^H, R^{*^{H
^*}}) is (L, M)-TDFRN-object:

(DFR1), (DFR2) and (DFR3) are easy proved and verification of (FRN4) is routine.

(DFR5) For each e ∈ c (L) and u ∈ L, it is trivial that: $\begin{matrix} R_{e}^{H} (u) \geq ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s}^{H} (v), \\ R_{e}^{*^{H^{*}}} (u) \leq ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*^{H^{*}}} (v) . \end{matrix}$

On the other hand, let α ∈ c (M) such that $α ⊲ R_{e}^{H} (u)$ , then there exists some v_α ∈ e|u such that α ≤ ⋀ _{s≰v_α}R_s (v_α). Thus, α ≤ R_s (v_α), for all s≰v_α.Then, $\begin{matrix} ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s}^{H} (v) \\ = ⋁_{v \in e | u} ⋀_{s ≰ v} ⋁_{w \in s | v} ⋀_{m ≰ w} R_{m} (w) \\ \geq ⋀_{s ≰ v_{α}} ⋀_{m ≰ v_{α}} R_{m} (v_{α}) \geq α . \end{matrix}$

From the arbitrariness of α we have: $R_{e}^{H} (u) \leq ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s}^{H} (v) .$

It remains to prove that: $R_{e}^{*^{H^{*}}} (u) \geq ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*^{H^{*}}} (v) .$

So let β ∈ c (M) such that β ⊲ ⋀ _v∈e|u ⋁ _s≰v $R_{s}^{*^{H^{*}}} (v)$ . Then, $β \leq ⋁_{s ≰ v} R_{s}^{*^{H^{*}}} (v)$ , for each v ∈ e|u. Therefore there exists some v_β ∈ e|u with s≰v_β such that $β \leq R_{s}^{*^{H^{*}}} (v_{β})$ . Then, $\begin{matrix} β & \leq ⋀_{w \in s | v_{β}} ⋁_{m ≰ w} R_{m}^{*} (w) \\ \leq ⋀_{w \in e | u} ⋁_{m ≰ w} R_{m}^{*} (w) = R_{e}^{*^{H^{*}}} (u) . \end{matrix}$

From the arbitrariness of β we have: $R_{e}^{*^{H^{*}}} (u) \geq ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*^{H^{*}}} (v) .$

Step 2:id_L : (L, R, R^*) → (L, R^H, R^{*^{H
^*}}) is continuous: It is trivial.

Step 3: For each (L, M)-topological double fuzzy remote neighborhood space (N, P, P^*) and each map f : L → N, the continuity of f : (L, R, R^*) → (N, P, P^*) implies the continuity of f : (L, R^H, R^{*^{H
^*}}) → (N, P, P^*): It suffices to prove that: $\begin{matrix} R_{e}^{H} (f^{⊢} (u)) \geq P_{f (e)} (u), \\ R_{e}^{*^{H^{*}}} (f^{⊢} (u)) \leq P_{f (e)}^{*} (u) \end{matrix}$ for each e ∈ L and u ∈ N. Since (N, P, P^*) is an (L, M)-topological double fuzzy remote neighborhood space, we have: $\begin{matrix} P_{f (e)} (u) = ⋁_{v \in f (e) | u} ⋀_{s ≰ v} P_{s} (v), \\ P_{f (e)}^{*} (u) = ⋀_{v \in f (e) | u} ⋁_{s ≰ v} P_{s}^{*} (v) . \end{matrix}$ For each v ∈ f (e) |u, let w = f^⊢ (v), trivially w ∈ e|f^⊢ (u). By the continuity of f : (L, R, R^*) → (N, P, P^*) we get that ∀k ∈ L with k≰w, $\begin{matrix} R_{k} (w) \geq P_{f (k)} (v) \geq ⋀_{s ≰ v} P_{s} (v), \\ R_{k}^{*} (w) \leq P_{f (k)}^{*} (v) \leq ⋁_{s ≰ v} P_{s}^{*} (v) . \end{matrix}$

Moreover, $\begin{matrix} ⋀_{k ≰ w} R_{k} (w) \geq ⋀_{s ≰ v} P_{s} (v), \\ ⋁_{k ≰ w} R_{k}^{*} (w) \leq ⋁_{s ≰ v} P_{s}^{*} (v) . \end{matrix}$ Therefore, $\begin{matrix} R_{e}^{H} (f^{⊢} (u)) \geq P_{f (e)} (u), \\ R_{e}^{H^{*}} (f^{⊢} (u)) \leq P_{f (e)}^{*} (u), \end{matrix}$ since, $\begin{matrix} R_{e}^{H} (f^{⊢} (u)) = ⋁_{w \in e | f^{⊢} (u)} ⋀_{k ≰ w} R_{k} (w), \\ R_{e}^{*^{H^{*}}} (f^{⊢} (u)) = ⋀_{w \in e | f^{⊢} (u)} ⋁_{k ≰ w} R_{k}^{*} (w) . \end{matrix}$

4 (L, M)-double fuzzy ideal structures

Definition 4.1. The pair $(I, I^{*})$ of maps $I, I^{*} : L \to M$ is called an (L, M)-double fuzzy ideal on L if it satisfies the following conditions:

$I (u) \leq (I^{*} (u))^{'}$ , ∀ u ∈ L;

$I (0_{L}) = 1_{M}$ , $I (1_{L}) = 0_{M}$ and $I^{*} (0_{L}) = 0_{M}$ , $I^{*} (1_{L}) = 1_{M}$ ;

$I (u \lor v) \geq I (u) \land I (v)$ and $I^{*} (u \lor v) \leq I^{*} (u) \lor I^{*} (v)$ , for each u, v ∈ L;

If u ≤ v, then $I (u) \geq I (v)$ and $I^{*} (u) \leq I^{*} (v)$ .

If $(I_{1}, I_{1}^{*})$ and $(I_{2}, I_{2}^{*})$ are two (L, M)-double fuzzy ideals on L, we say that $(I_{1}, I_{1}^{*})$ is finer than $(I_{2}, I_{2}^{*})$ (or $(I_{2}, I_{2}^{*})$ is coarser than $(I_{1}, I_{1}^{*})$ ), denoted by $(I_{2}, I_{2}^{*}) \leq (I_{1}, I_{1}^{*})$ if and only if $I_{2} (u) \leq I_{1} (u)$ and $I_{2}^{*} (u) \geq I_{1}^{*} (u)$ , for each u ∈ L.

Example 4.2. For each e ∈ c (L), we define r (e) , r^* (e) : L → M as follows: $\forall u \in L, r (e) (u) = {\begin{matrix} 1_{M}, & if e ≰ u \\ 0_{M}, & if e \leq u . \end{matrix}$ $r^{*} (e) (u) = {\begin{matrix} 0_{M}, & if e ≰ u \\ 1_{M}, & if e \leq u . \end{matrix}$ Then, (r (e) , r^* (e)) is an (L, M)-double fuzzy ideal.

Remark 4.3. (i) Let $I : L \to M$ be an (L, M)-fuzzy ideal and $I^{*} : L \to M$ be a mapping defined by, $I^{*} (u) = (I (u))^{'}$ , ∀u ∈ L. Then the pair $(I, I^{*})$ is an (L, M)-double fuzzy ideal. Therefore, (L, M)-double fuzzy ideal is a generalization of (L, M)-fuzzy ideal.

(ii) If $(R_{e}, R_{e}^{*})$ is an (L, M)-double fuzzy remote neighborhood system, then from (DFRN1), (DFRN2) and (DFRN4), it is easy to show that $(R_{e}, R_{e}^{*})$ is an (L, M)-double fuzzy ideal.

(iii) Let (L, η, η^*) be an (L, M)-DFTML. Then by Proposition 3.5, $(R_{e}^{η}, R_{e}^{*^{η^{*}}})$ is an (L, M)-double fuzzy remote neighborhood system, therefore, it is an (L, M)-double fuzzy ideal and obviously, $(R_{e}^{η}, R_{e}^{*^{η^{*}}}) \leq (r (e), r^{*} (e))$ , for each e ∈ c (L).

Proposition 4.4.Let $(I, I^{*})$ be an (L, M)-double fuzzy ideal and f : L → L₁ be a GOH. Define the maps $f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}) : L_{1} \to M$ by: $\begin{matrix} f^{\Rightarrow} (I) (u) = ⋁_{f^{⊢} (u) \leq v} I (v) = I (f^{⊢} (u)), \\ f^{\Rightarrow} (I^{*}) (u) = ⋀_{f^{⊢} (u) \leq v} I (v) = I^{*} (f^{⊢} (u)), \end{matrix}$ for each u ∈ L₁. Then $(f^{\Rightarrow} (I), f^{\Rightarrow} (I^{*}))$ is an (L, M)-double fuzzy ideal on L₁ and is called the image of $(I, I^{*})$ under f.

Proof. (DFI1), (DFI2) and (DFI4) are easily checked.

(DFI3) For each u, v ∈ L₂, we have: $\begin{matrix} f^{\Rightarrow} (I) (u \lor v) = I (f^{⊢} (u \lor u)) \\ = I (f^{⊢} (u) \lor f^{⊢} (v)) (by Definition 2.1 (iii)) \\ \geq I (f^{⊢} (u)) \land I (f^{⊢} (v)) = f^{\Rightarrow} (I) (u) \land f^{\Rightarrow} (I) (v), \end{matrix}$ $\begin{matrix} f^{\Rightarrow} (I^{*}) (u \lor v) = I^{*} (f^{⊢} (u \lor u)) \\ = I^{*} (f^{⊢} (u) \lor f^{⊢} (v)) (by Definition 2.1 (iii)) \\ \leq I^{*} (f^{⊢} (u)) \lor I^{*} (f^{⊢} (v)) = f^{\Rightarrow} (I^{*}) (u) \lor f^{\Rightarrow} (I^{*}) (v) . \end{matrix}$

Lemma 4.5. Suppose that ${(I_{i}, I_{i}^{*})}_{i \in Γ}$ is the set of (L, M)-double fuzzy ideals. Then, (⋀ _i∈Γ $I_{i}, ⋁_{i \in Γ} I_{i}^{*})$ is an (L, M)-double fuzzy ideal where, $⋀_{i \in Γ} I_{i}, ⋁_{i \in Γ} I_{i}^{*} : L \to M$ and $(⋀_{i \in Γ} I_{i}) (u) = ⋀_{i \in Γ} I_{i} (u)$ , $(⋁_{i \in Γ} I_{i}^{*}) (u) = ⋁_{i \in Γ} I_{i}^{*} (u), \forall u \in L$ .

Proof. It is obvious.

Note 4.6. Let $B, B^{*} : L \to M$ be two maps and u ∈ L. We defined $〈 B 〉$ and $〈 B^{*} 〉$ as follows: $〈 B 〉 (u) = ⋁_{v \geq u} B (v)$ and $〈 B^{*} 〉 (u) = ⋀_{v \geq u} B^{*} (v)$ .

Definition 4.7. The pair $(B, B^{*})$ of maps $B, B^{*} : L \to M$ is called an (L, M)-double fuzzy ideal base if it satisfies the following conditions:

$B (u) \leq (B^{*} (u))^{'}$ , ∀ u ∈ L;

$B (1_{L}) = 0_{M}$ , $⋁_{u \in L} B (u) = 1_{M}$ and $B^{*} (1_{L}) = 1_{M}$ , $⋀_{u \in L} B^{*} (u) = 0_{M}$ ;

$〈 B 〉 (u \lor v) \geq B (u) \land B (v)$ and $〈 B^{*} 〉 (u \lor v) \leq B^{*} (u) \lor B^{*} (v)$ , ∀ u, v ∈ L.

Proposition 4.8. If $(B, B^{*})$ is an (L, M)-double fuzzy ideal base, then $(〈 B 〉, 〈 B^{*} 〉)$ is an (L, M)-double fuzzy ideal, and $(B, B^{*})$ is called an (L, M)-doublefuzzy ideal base of $(〈 B 〉, 〈 B^{*} 〉)$ .

Proof. (DFI1) For each u ∈ L we have $\begin{matrix} (〈 B^{*} 〉 (u))^{'} & = (⋀_{v \geq u} B^{*} (v))^{'} = ⋁_{v \geq u} (B^{*} (v))^{'} \\ \geq ⋁_{v \geq u} B (v) = 〈 B 〉 (u) . \end{matrix}$

(DFI2) and (DFI4) are easily checked.

(DFI3) Suppose that there exist u, v ∈ L such that $〈 B 〉 (u \lor v) ≱ 〈 B 〉 (u) \land 〈 B 〉 (v) .$

By the definition of $〈 B 〉$ and since M satisfies the distributive law, there exist u₁, v₁ ∈ L with u₁ ≥ u and v₁ ≥ v such that $〈 B 〉 (u \lor v) ≱ B (u_{1}) \land B (v_{1}) (4.1)$ Since $(B, B^{*})$ is an (L, M)-double fuzzy ideal base, $〈 B 〉 (u_{1} \lor v_{1}) \geq B (u_{1}) \land B (v_{1}) .$

Since u₁ ∨ v₁ ≥ u ∨ v, we have $〈 B 〉 (u \lor v) \geq 〈 B 〉 (u_{1} \lor v_{1}) \geq B (u_{1}) \land B (v_{1}) .$

It contradicts with (4.1). Thus, for each u, v ∈ L, $〈 B 〉 (u \lor v) \geq 〈 B 〉 (u) \land 〈 B 〉 (v) .$

Similarly, for each u, v ∈ L, $〈 B^{*} 〉 (u \lor v) \leq 〈 B^{*} 〉 (u) \lor 〈 B^{*} 〉 (v) .$

Proposition 4.9. Let (L, R, R^*) be an (L, M)-topological double fuzzy remote neighborhood space. Then, for each e ∈ c (L), the pair $(B_{e}, B_{e}^{*})$ of maps $B_{e}, B_{e}^{*} : L \to M$ defined by $B_{e} (u) = R_{e} (u) \land (⋀_{s ≰ u} R_{s} (u))$ and $B_{e}^{*} (u) = R_{e}^{*} (u) \lor (⋁_{s ≰ u} R_{s}^{*} (u))$ , for each u ∈ L, is an (L, M)-double fuzzy ideal base of $(R_{e}, R_{e}^{*})$ .

Proof. (DFIB1) It is clear.

(DFIB2) obviously, $B_{e} (1_{L}) = 0_{M}$ and $B_{e}^{*} (1_{L}) = 1_{M}$ $\begin{matrix} ⋁_{u \in L} B_{e} (u) = ⋁_{u \in L} (R_{e} (u) \land (⋀_{s ≰ u} R_{s} (u))) \\ \geq R_{e} (0_{L}) \land (⋀_{s ≰ 0_{L}} R_{s} (0_{L})) = 1_{L} . \end{matrix}$

Similarly, $⋀_{u \in L} B_{e}^{*} (u) = 0_{M}$ .

(DFIB3) If u, v ∈ L, then $\begin{matrix} 〈 B_{e} 〉 (u \lor v) = ⋁_{w \geq u \lor v} B_{e} (w) \\ \geq B_{e} (u \lor v) = R_{e} (u \lor v) \land (⋀_{s ≰ u \lor v} R_{s} (u \lor v)) \\ = R_{e} (u) \land R_{e} (v) \land ⋀_{s ≰ u, s ≰ v} (R_{s} (u) \land R_{s} (v)) \\ \geq B_{e} (u) \land B_{e} (v), \\ 〈 B_{e}^{*} 〉 (u \lor v) = ⋀_{w \geq u \lor v} B_{e}^{*} (w) \\ \leq B_{e}^{*} (u \lor v) = R_{e}^{*} (u \lor v) \lor (⋁_{s ≰ u \lor v} R_{s}^{*} (u \lor v)) \\ = R_{e}^{*} (u) \lor R_{e}^{*} (v) \lor ⋁_{s ≰ u, s ≰ v} (R_{s}^{*} (u) \lor R_{s}^{*} (v)) \\ \leq B_{e}^{*} (u) \lor B_{e}^{*} (v) . \end{matrix}$

Finally, $\begin{matrix} 〈 B_{e} 〉 (u) = ⋁_{v \geq u} B_{e} (v) \\ = ⋁_{v \geq u} (R_{e} (v) \land ⋀_{s ≰ v} R_{s} (v)) \\ = ⋁_{e ≰ v, v \geq u} ⋀_{s ≰ v} R_{s} (v) \\ = ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s} (v) = R_{e} (u), \\ 〈 B_{e}^{*} 〉 (u) = ⋀_{v \geq u} B_{e}^{*} (v) \\ = ⋀_{v \geq u} (R_{e}^{*} (v) \lor ⋁_{s ≰ v} R_{s}^{*} (v)) \\ = ⋀_{e ≰ v, v \geq u} ⋁_{s ≰ v} R_{s}^{*} (v) \\ = ⋀_{v \in e | u} ⋁_{s ≰ v} R_{s}^{*} (v) = R_{e}^{*} (u) . \end{matrix}$

Then, $(B_{e}, B_{e}^{*})$ is an (L, M)-double fuzzy ideal base of $(R_{e}, R_{e}^{*})$ .

In the rest of the paper, we assume that M = [0, 1].

Definition 4.10. Let (L, η, η^*) be an (L, M)-DFTML and $(I, I^{*})$ be an (L, M)-double fuzzy ideal on L. Then we have:

If $(R_{e}^{η}, R_{e}^{*^{η^{*}}}) \leq (I, I^{*})$ , then we say that e is a limit point of $(I, I^{*})$ (or $(I, I^{*})$ convergence to e): in symbols, $(I, I^{*}) ⟶ e$ .

If $⋁_{u \lor v = 1_{L}} (R_{e}^{η} (u) \land I (v)) = 0_{M}$ and $⋀_{u \lor v = 1_{L}} (R_{e}^{*^{η^{*}}} (u) \lor I^{*} (v)) = 1_{M}$ , then we say that e is a cluster point of $(I, I^{*})$ or $(I, I^{*})$ a accumulates to e (briefly, $(I, I^{*}) \propto e$ ).

We denote the union of all limit points of $(I, I^{*})$ by $\lim (I, I^{*})$ and the union of all cluster points of $(I, I^{*})$ by $adh (I, I^{*})$ .

Remark 4.11. From the above definition, we have:

$(R_{e}^{η}, R_{e}^{*^{η^{*}}}) ⟶ e$ ;

If 1_L ∈ c (L), then $(I, I^{*}) \propto e$ , for each (L, M)-double fuzzy ideal $(I, I^{*})$ and e ∈ c (L) i.e., $adh (I, I^{*}) = 1_{L}$ , for each (L, M)-double fuzzy ideal $(I, I^{*})$ .

Theorem 4.12. The following statements are true:

$(I, I^{*}) ⟶ e$ ⇔ $e \leq \lim (I, I^{*})$ ;

$(I, I^{*}) \propto e$ ⇔ $e \leq adh (I, I^{*})$ ;

If e ≥ u, then $(I, I^{*}) ⟶ e$ (resp. $(I, I^{*}) \propto e)$ ⇒ $(I, I^{*}) ⟶ u$ (resp. $(I, I^{*}) \propto u)$ ;

If $(I_{1}, I_{1}^{*}) \leq (I_{2}, I_{2}^{*})$ , then $\lim (I_{1}, I_{1}^{*}) \leq \lim (I_{2}, I_{2}^{*})$ and $adh (I_{2}, I_{2}^{*}) \leq adh (I_{1}, I_{1}^{*})$ ;

$(I, I^{*}) ⟶ e$ ⇒ $(I, I^{*}) \propto e$ .

Proof. We only prove (i) and (v), the others are trivial.

(i) $(I, I^{*}) ⟶ e$ ⇒ $e \leq \lim (I, I^{*})$ is obvious. Let $e \leq \lim (I, I^{*})$ .

First, we will prove that $R_{e}^{η} \leq I$ . Let $R_{e}^{η} (u) \geq α$ , where u ∈ L and α ∈ M. Since $R_{e}^{η} (u) = ⋁_{v \in e | u} ⋀_{s ≰ v} R_{s}^{η} (v)$ , there exists v ∈ L such that e≰v, v ≥ u and $α \leq ⋀_{s ≰ v} R_{s}^{η} (v)$ . Since e≰v, v ≥ u and e = ⋁ β^* (e), there exists r ∈ β^* (e) such that r≰v, v ≥ u. Hence, $α \leq R_{r}^{η} (v) \leq R_{r}^{η} (u)$ . Furthermore, by r ∈ β^* (e) and $e \leq \lim (I, I^{*})$ , there is some μ ∈ c (L) such that $(I, I^{*}) ⟶ μ$ and r ≤ μ. Thus, $R_{r}^{η} \leq R_{μ}^{η} \leq I$ . Therefore, $α \leq I (u)$ . Thus, $R_{e}^{η} (u) \leq I (u)$ , ∀ u ∈ L.

Second, we will prove that $R_{e}^{*^{η^{*}}} \geq I^{*}$ . Suppose that there exists k ∈ M and u ∈ L such that $I^{*} (u) > k \geq R_{e}^{*^{η^{*}}} (u)$ . Since $R_{e}^{*^{η^{*}}} (u) = ⋀_{v \in e | u}$ $⋁_{s ≰ v} R_{s}^{*^{η^{*}}} (v)$ , then there exists v ∈ e|u such that $⋁_{s ≰ v} R_{s}^{*^{η^{*}}} (v) \leq k$ . Since e≰v and e = ⋁ β^* (e), then there exists r ∈ β^* (e) such that r≰v, v ≥ u.Then $R_{r}^{*^{η^{*}}} (u) \leq R_{r}^{*^{η^{*}}} (v) \leq k$ . Since r ∈ β^* (e) and $e \leq \lim (I, I^{*})$ , there exists μ ∈ c (L) such that $(I, I^{*}) ⟶ μ$ and r ≤ μ. Thus, $R_{r}^{*^{η^{*}}} \geq R_{μ}^{*^{η^{*}}} \geq I^{*}$ . Therefore, $k \geq R_{r}^{*^{η^{*}}} (u) \geq I^{*} (u)$ . It is a contradiction. Hence, $R_{e}^{*^{η^{*}}} (u) \geq I^{*} (u)$ , ∀ u ∈ L.

(v) Let $(I, I^{*}) ⟶ e$ , i.e, $R_{e}^{η} \leq I$ and $R_{r}^{*^{η^{*}}} \geq I^{*}$ . Then, we have $\begin{matrix} ⋁_{u \lor v = 1_{L}} (R_{e}^{η} (u) \land I (v)) \leq ⋁_{u \lor v = 1_{L}} (I (u) \land I (v)) \\ \leq ⋁_{u \lor v = 1_{L}} I (u \lor v) = I (1_{L}) = 0_{M}, \\ ⋀_{u \lor v = 1_{L}} (R_{e}^{*^{η^{*}}} (u) \lor I^{*} (v)) \\ \geq ⋀_{u \lor v = 1_{L}} (I^{*} (u) \lor I^{*} (v)) \\ \geq ⋁_{u \lor v = 1_{L}} I^{*} (u \lor v) = I^{*} (1_{L}) = 1_{M} . \end{matrix}$ Then, $(I, I^{*}) \propto e$ .

Theorem 4.13. For any (L, M)-double fuzzy ideal $(I, I^{*})$ , $e \leq adh (I, I^{*})$ if and only if there exists an (L, M)-double fuzzy ideal $(G, G^{*})$ such that $(I, I^{*}) \leq (G, G^{*})$ and $(G, G^{*}) ⟶ e$ .

Proof. Let $e \leq adh (I, I^{*})$ . From Theorem 4.12(ii), we have $(I, I^{*}) \propto e$ , i.e., $⋁_{u \lor v = 1_{L}} (R_{e}^{η} (u) \land I (v)) = 0_{M}$ and $⋀_{u \lor v = 1_{L}} (R_{e}^{*^{η^{*}}} (u) \lor I^{*} (v)) = 1_{M}$ . Define the maps $G, G^{*} : L \to M$ as follows: ∀ u ∈ L, $\begin{matrix} G (u) = ⋁ {R_{e}^{η} (u_{1}) \land I (u_{2}) : u = u_{1} \lor u_{2}}, \\ G^{*} (u) = ⋀ {R_{e}^{*^{η^{*}}} (u_{1}) \lor I^{*} (u_{2}) : u = u_{1} \lor u_{2}} . \end{matrix}$ Then, we have $(G, G^{*})$ is an (L, M)-double fuzzy ideal on L, $(R_{e}^{η}, R_{e}^{*^{η^{*}}}) \leq (G, G^{*})$ and $(I, I^{*}) \leq (G, G^{*})$ . Hence, $(G, G^{*}) ⟶ e$ .

Conversely, suppose that there exists an (L, M)-double fuzzy ideal $(G, G^{*})$ such that $(I, I^{*}) \leq (G, G^{*})$ and $(G, G^{*}) ⟶ e$ . By Theorem 4.12(v), $(G, G^{*}) \propto e$ . By Theorem 4.12(ii), $e \leq adh (G, G^{*})$ . Also, by Theorem 4.12(iv), we have

$e \leq adh (G, G^{*}) \leq adh (I, I^{*})$ .

Theorem 4.14. If $(I, I^{*})$ is a maximal (L, M)-double fuzzy ideal on L, then $adh (I, I^{*}) = \lim (I, I^{*})$ .

Proof. From Theorem 4.12(v), we have $\lim (I, I^{*}) \leq adh (I, I^{*})$ . Now we will prove that $adh (I, I^{*}) \leq \lim (I, I^{*})$ . Let $e \leq adh (I, I^{*})$ . From Theorem 4.13, there exists an (L, M)-double fuzzy ideal $(G, G^{*})$ such that $(I, I^{*}) \leq (G, G^{*})$ and $(G, G^{*}) ⟶ e$ . Since $(I, I^{*})$ is a maximal (L, M)-double fuzzy ideal on L, we have $(G, G^{*}) = (I, I^{*})$ . Then, $(I, I^{*}) ⟶ e$ , i.e., $e \leq \lim (I, I^{*})$ . Therefore, $adh (I, I^{*}) \leq \lim (I, I^{*})$ .

From Theorem 4.13 and Theorem 4.14, we have the following corollary.

Corollary 4.15.For an (L, M)-DFTML (L, η, η^*), the following conditions are equivalent:

Every (L, M)-double fuzzy ideal has cluster points;

Every maximal (L, M)-double fuzzy ideal has limit points.

Definition 4.16. Let e ∈ c (L) and a ∈ L. If $R_{e}^{η} (a) = 0_{M}$ and $R_{e}^{*^{η^{*}}} (a) = 1_{M}$ , then e is called adherence point of a.

Theorem 4.17. e ∈ c (L) is called adherence point of a ∈ L if and only if there exists an (L, M)-double fuzzy ideal $(I, I^{*})$ such that $I (a) = 0_{M}$ , $I^{*} (a) = 1_{M}$ and $(I, I^{*}) ⟶ e$ .

Proof. It is easy.

Definition 4.18. An (L, M)-DFTML (L, η, η^*) is called T₂ if ∀ a, b ∈ c (L) with a ∧ b = 0_M (a and b are disjoint) implies $⋁_{u \lor v = 1_{L}} (R_{a}^{η} (u) \land R_{b}^{η} (v)) \neq 0_{M}$ and $⋀_{u \lor v = 1_{L}} (R_{a}^{*^{η^{*}}} (u) \lor R_{b}^{*^{η^{*}}} (v)) \neq 1_{M}$ .

Theorem 4.19. (L, η, η^*) is T₂ if and only if for each (L, M)-double fuzzy ideal $(I, I^{*})$ on L, $\lim (I, I^{*})$ contains no disjoint points.

Proof. Assume that $\lim (I, I^{*})$ contains two disjoint points a and b, i.e., $(I, I^{*}) ⟶ a$ and $(I, I^{*}) ⟶ b$ and a ∧ b = 0_M. Then, $\begin{matrix} ⋁_{u \lor v = 1_{L}} (R_{a}^{η} (u) \land R_{b}^{η} (v)) \leq ⋁_{u \lor v = 1_{L}} (I (u) \land I (v)) \\ \leq ⋁_{u \lor v = 1_{L}} I (u \lor v) = 0_{M}, \\ ⋀_{u \lor v = 1_{L}} (R_{a}^{*^{η^{*}}} (u) \lor R_{b}^{*^{η^{*}}} (v)) \\ \geq ⋀_{u \lor v = 1_{L}} (I^{*} (u) \lor I^{*} (v)) \\ \geq ⋀_{u \lor v = 1_{L}} I^{*} (u \lor v) = 1_{M} . \end{matrix}$ This is a contradicts with $\begin{matrix} ⋁_{u \lor v = 1_{L}} (R_{a}^{η} (u) \land R_{b}^{η} (v)) \neq 0_{M}, \\ ⋀_{u \lor v = 1_{L}} (R_{a}^{*^{η^{*}}} (u) \lor R_{b}^{*^{η^{*}}} (v)) \neq 1_{M} . \end{matrix}$ Conversely, Suppose that there exists a, b ∈ c (L) with a ∧ b = 0_M such that $⋁_{u \lor v = 1_{L}} (R_{a}^{η} (u) \land R_{b}^{η} (v)) = 0_{M}$ and $⋀_{u \lor v = 1_{L}} (R_{a}^{*^{η^{*}}} (u) \lor R_{b}^{*^{η^{*}}} (v)) = 1_{M}$ . Then we have, $(R_{b}^{η}, R_{b}^{*^{η^{*}}}) \propto a$ . From Theorem 4.13, there exists an (L, M)-double fuzzy ideal $(G, G^{*})$ such that $(R_{b}^{η}, R_{b}^{*^{η^{*}}}) \leq (G, G^{*})$ and $(G, G^{*}) ⟶ a$ . Then, $(G, G^{*}) ⟶ b$ and $(G, G^{*}) ⟶ a$ . Hence, $\lim (G, G^{*})$ contains two disjoint points. This contradicts the given condition.

Theorem 4.20. A GOH $f : (L_{1}, η_{1}, η_{1}^{*}) \to (L_{2}$ , $η_{2}, η_{2}^{*})$ is continuous if and only if $(f^{\to} (I), f^{\to} (I^{*})) ⟶ f (e)$ when $(I, I^{*}) ⟶ e$ , e ∈ c (L₁).

Proof. Let $(I, I^{*}) ⟶ e$ . Then, $R_{e}^{η_{1}} \leq I$ and $R_{e}^{*^{η_{1}^{*}}} \geq I^{*}$ . Therefore, ∀ u ∈ L₂, $\begin{matrix} R_{e}^{η_{1}} (f^{⊢} (u)) \leq I (f^{⊢} (u)) = f^{\to} (I) (u), \\ R_{e}^{*^{η_{1}^{*}}} (f^{⊢} (u)) \geq I^{*} (f^{⊢} (u)) = f^{\to} (I^{*}) (u) . \end{matrix}$ Since $f : (L_{1}, η_{1}, η_{1}^{*}) \to (L_{2}, η_{2}, η_{2}^{*})$ is continuous, we have, ∀ u ∈ L₂, $\begin{matrix} R_{f (e)}^{η_{2}} (u) \leq R_{e}^{η_{1}} (f^{⊢} (u)) \leq f^{\to} (I) (u), \\ R_{f (e)}^{*^{η_{2}^{*}}} (u) \geq R_{e}^{*^{η_{1}^{*}}} (f^{⊢} (u)) \geq f^{\to} (I^{*}) (u) . \end{matrix}$ Hence, $(f^{\to} (I), f^{\to} (I^{*})) ⟶ f (e)$ .

Conversely, we want to prove that: $\begin{matrix} R_{f (e)}^{η_{2}} (u) \leq R_{e}^{η_{1}} (f^{⊢} (u)), \\ R_{f (e)}^{*^{η_{2}^{*}}} (u) \geq R_{e}^{*^{η_{1}^{*}}} (f^{⊢} (u)), \forall u \in L_{2} . \end{matrix}$

Since $(R_{e}^{η_{1}}, R_{e}^{*^{η_{1}^{*}}}) ⟶ e$ , we have $(f^{\to} (R_{e}^{η_{1}}), f^{\to} (R_{e}^{*^{η_{1}^{*}}})) ⟶ f (e)$ In other words, $\begin{matrix} R_{f (e)}^{η_{2}} (u) \leq f^{\to} (R_{e}^{η_{1}}) (u) = R_{e}^{η_{1}} (f^{⊢} (u)), \\ R_{f (e)}^{*^{η_{2}^{*}}} (u) \geq f^{\to} (R_{e}^{*^{η_{1}^{*}}}) (u) = R_{e}^{*^{η_{1}^{*}}} (f^{⊢} (u)) . \end{matrix}$

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