Decompositions and representations of L -fuzzy natural numbers

Abstract

In this paper, decompositions of L-fuzzy natural numbers and cut sets of L-fuzzy sets on finite sets are presented. Then, based on some unified nature of L-fuzzy natural numbers, the notions of $ℕ_{β}^{L}$ -nested mappings and $ℕ_{α}^{L}$ -nested mappings are introduced and some representations of them are obtained. It is showed that if a $ℕ_{β}^{L}$ -nested mapping (resp. $ℕ_{α}^{L}$ -nested mapping) is generated by an L-fuzzy natural number, then its representation is exactly equal to itself. As for applications, the addition and the exponentiation of two $ℕ_{β}^{L}$ -nested mappings (resp. two $ℕ_{α}^{L}$ -nested mappings) are introduced. It is also showed that, given two $ℕ_{β}^{L}$ -nested mappings (resp. two $ℕ_{α}^{L}$ -nested mappings), cut sets that correspond to the representations of their addition (or, their exponentiation) are equal to cut sets of the addition (or, the exponentiations) of the given mappings.

Keywords

cut set addition,exponentiation

1 Introduction and preliminaries

The notion of fuzzy numbers was defined originally by Zadeh in [29]. Since then, this notion was study intensively. For example, in fuzzy set theories, it is well known that the fuzzy distance between two fuzzy sets is a fuzzy number, and that a fuzzy matric is defined upon non-negative fuzzy numbers [12, 16]. Later, this notion was redefined or extended by various approaches so as to satisfy various purposes [2 , 20]. Now, it have been applied into many fields, such as fuzzy statistics, fuzzy measures, fuzzy calculus, computer programming, engineering, decision-making, optimization, etc [1 , 25]. Among fuzzy numbers, fuzzy natural numbers are the most special ones. It has been introduced and applied by different means [3 , 5]. However, applications of both fuzzy numbers and fuzzy natural numbers are inevitably limited to fuzzy set theories. To break this limitation, Huang introduced the notion of L-fuzzy numbers in [15] and Shi introduced the notion of L-fuzzy natural numbers in [23]. It has been found that L-fuzzy natural numbers capture many features of L-fuzzifying matroids, (L, M)-fuzzy matroids and some related theories [14 , 26].

Decompositions and representations of fuzzy structures are vital importance in connecting classic theories with their corresponding fuzzy theories. Among all possible methods in obtaining decompositions and representations of fuzzy structures, cut sets are the most common and efficient. Lowen introduced cut sets of fuzzy sets and characterized fuzzy topologies [18]. Later, Wang extended those cut sets into L-fuzzy settings and obtained similar characterizations of L-fuzzy topologies [27]. By using properties of completely distributive lattices, Shi redefined cut sets of L-fuzzy sets and applied them in characterizing L-fuzzy algebras and L-fuzzy topologies [21]. Since then, decompositions and representations were widely used in various fuzzy theories [7 , 26].

Considering that L-fuzzy natural numbers are vital important in characterizing (L, M)-fuzzy matroids and its related theories, in this paper, we discuss decompositions and representations of L-fuzzy natural numbers. We present some decompositions of L-fuzzy natural numbers. Then, by generalizing properties of cut sets of them, we introduce the notions of $ℕ_{β}^{L}$ -nested mappings and $ℕ_{α}^{L}$ -nested mappings and observe some representation theorems of those mappings. We show that if a $ℕ_{β}^{L}$ -nested (resp. $ℕ_{α}^{L}$ -nested) mapping is generated by an L-fuzzy natural number, then its representation is exactly equal to itself. Also, we define the notions of additions and exponentiations of two $ℕ_{β}^{L}$ -nested (resp. two $ℕ_{α}^{L}$ -nested) mappings. We find that, given two $ℕ_{β}^{L}$ -nested (resp. two $ℕ_{α}^{L}$ -nested) mappings, cut sets that correspond to the representations of their addition (or, their exponentiation) are equal to cut sets of the addition (or, the exponentiations) of the given mappings.

Throughout this paper, L denotes a completely distributive lattice. The minimal element and the maximal element in L are denoted respectively by ⊥ and ⊤. If φ is a subset of L, then we denote ⋁φ = ⋁_b∈φb and ⋀φ = ⋀_b∈φb. In particular, we accept that ⋁∅ = ∅ and ⋀ ∅ = L. If $φ = {U_{k}}_{k \in K} \subseteq 2^{ℕ}$ where $ℕ$ is the natural set, we denote ⋃φ = ⋃_k∈KU_k and ⋂φ = ⋂_k∈KU_k. In particular, ⋃∅ = ∅ and $⋂ \emptyset = ℕ$ .

An element a ∈ L is called a prime element, if for all b, c ∈ L, b ∧ c ≤ a implies b ≤ a or c ≤ a. The set of all prime elements in $L \underline{{} ⊤}$ is denoted by P (L). Dually, an element a ∈ L is called a co-prime element, if for all b, c ∈ L, a ≤ b ∨ c implies a ≤ b or a ≤ c. The set of all co-prime elements in $L \underline{{} ⊥}$ is denoted by J (L). Wang showed that in a completely distributive lattice, each element is the supremum of some co-prime elements or the infimum of some prime elements [28].

A binary relation ≺ on L is defined by: for all a, b ∈ L, a ≺ b iff for all φsL, the relation b ≤ ⋁ φ always implies the existence of d ∈ φ such that a ≤ d. The opposite relation ≺^op of ≺ is defined by: for all a, b ∈ L, a ≺^opb iff b′ ≺ a′. The mapping β : Lr2^L, defined by β (a) = {b : b ≺ a} for all a ∈ L, is a ⋁-⋃ mapping (i.e., for all {a_k∈ L}_k∈KsL, β (⋁_k∈Ka_k) = ⋃_k∈Kβ (a_k)). The mapping α : Lr2^L, defined by α (a) = {b : b ≺^opa} for all a ∈ L, is a ⋀-⋃ mapping (i.e., for all {a_k∈ L}_k∈KsL, α (⋀_k∈Ka_k) = ⋃_k∈Kα (a_k)). Let β^*(a) = β (a) ∩ J (L) and α^*(a) = α (a) ∩ P (L). Then it is clear that a = ⋁ β (a) = ⋁ β^*(a) = ⋀ α (a) = ⋀ α^*(a) [27, 28].

L is called a β-lattice, denoted by L = L (β), if β (a ∧ b) = β (a) ∩ β (b) for all a, b ∈ L. Similarly, L is called an α-lattice, denoted by L = L (α), if α (a ∨ b) = α (a) ∩ α (b) for all a, b ∈ L.

An antitone mapping $λ : ℕ \to L$ is called an L-fuzzy natural number [22], if λ satisfies:

λ (0) =⊤;

$⋀_{n \in ℕ} λ (n) = ⊥$ .

The set of all L-fuzzy natural numbers is denoted by $ℕ (L)$ . For $λ, μ \in ℕ (L)$ , we denote λ ≤ μ, or μ ≤ λ, if λ (n) ≤ μ (n) for all $n \in ℕ$ .

Let $n \in ℕ$ . The mapping $\underline{n} : ℕ \to L$ is defined by $\underline{n} (t) = {\begin{matrix} ⊤, & t \leq n, \\ ⊥, & t > n, \end{matrix}$ for all $t \in ℕ$ is an L-fuzzy natural number [22].

Let L^X be the set of all L-fuzzy set on a finite set X. For any A ∈ L^X, the mapping $| A | : ℕ \to L$ , defined by $| A | (n) = ⋁ {a \in L : | A_{[a]} | \geq n}$ for all $n \in ℕ$ , is an L-fuzzy natural number [22].

Let ${λ_{k}}_{k \in K} \subseteq ℕ (L)$ . We define for all $n \in ℕ$ ,

(⋁_k∈Kλ_k) (n) = ⋁_k∈Kλ_k(n);

(⋀_k∈Kλ_k) (n) = ⋀_k∈Kλ_k(n).

Clearly, $⋀_{k \in K} λ_{k} \in ℕ (L)$ , and if K is finite, then $⋁_{k \in K} λ_{k} \in ℕ (L)$ . However, in general, $⋁_{k \in K} λ_{k} \notin ℕ (L)$ . In fact, $⋀_{n \in ℕ} ⋁_{m \in ℕ} \underline{m} (n) \geq ⋀_{n \in ℕ} \underline{n} (n) = ⊤$ , which implies $⋁_{m \in ℕ} \underline{m} \notin ℕ (L)$ .

For a ∈ L, cut sets of $λ \in ℕ (L)$ are as follows [22]:

$λ_{[a]} = {n \in ℕ : λ (n) \geq a}$ ;

$λ_{(a)} = {n \in ℕ : a \in β (λ (n))}$ ;

$λ^{[a]} = {n \in ℕ : a \notin α (λ (n))}$ ;

$λ^{(a)} = {n \in ℕ : λ (n) ≰ a}$ .

If a cut set of an L-fuzzy natural number is nonempty and finite, then there exists a one-to-one correspondence between the cut set and its maximal element. Thus, if we do not distinguish the cardinality of a cut set of an L-fuzzy natural number and its maximal element, then the cut set is a natural number [22, 23].

2 Decompositions of L-fuzzy natural numbers

Lemma 2.1. Let $λ \in ℕ (L)$ and a, b ∈ L. Then

a ∈ β (b) implies λ_[b]sλ_(a)sλ_[a];

a ∈ α (b) implies λ^[a]sλ^(b)sλ^[b].

Lemma 2.2. Let ${λ_{k}}_{k \in K} \subseteq ℕ (L)$ and a ∈ L. Then

(⋀_k∈Kλ_k)_[a]= ⋂_k∈K(λ_k)_[a];

(⋀_k∈Kλ_k)^[a]= ⋂_k∈K(λ_k)^[a].

If $⋁_{k \in K} λ_{k} \in ℕ (L)$ , then

(⋁_k∈Kλ_k)_(a)= ⋃_k∈K(λ_k)_(a);

(⋁_k∈Kλ_k)^(a)= ⋃_k∈K(λ_k)^(a).

In order to discuss decompositions of an L-fuzzy natural number, we introduce some notions as following.

Let $↓ ℕ = {\emptyset, ↓ n : n \in ℕ}$ , where $↓ n = {0, 1, \cdot, \cdot, \cdot, n}$ for all $n \in ℕ$ . For all $U \in ↓ ℕ$ and a ∈ L, two L-fuzzy sets $a \land U, a \lor U \in L^{↓ ℕ}$ are defined by $a \land U (t) = {\begin{matrix} a, & t \in U, \\ ⊥, & t \notin U . \end{matrix}; a \lor U (t) = {\begin{matrix} ⊤, & t \in U, \\ a, & t \notin U . \end{matrix}$ for all $t \in ℕ$ . Clearly, $⊤ \land U, ⊥ \lor U \in ℕ (L)$ iff U is a finite nonempty set.

Theorem 2.3. Let $λ \in ℕ (L)$ and a ∈ L. Then

λ = ⋁_a∈La ∧ λ_[a]= ⋁_a∈La ∧ λ_(a)

= ⋁_a∈J(L)a ∧ λ_[a]= ⋁_a∈J(L)a ∧ λ_(a);

λ = ⋀_a∈La ∨ λ^[a]= ⋀_a∈La ∨ λ^(a)

= ⋀_a∈P(L)a ∨ λ^[a]= ⋀_a∈P(L)a ∨ λ^(a);

λ_[a]= ⋂_b∈β(a)λ_[b]= ⋂_b∈β(a)λ_(b);

λ_(a)= ⋃_a∈β(b)λ_[b]= ⋃_a∈β(b)λ_(b);

λ^[a]= ⋂_a∈α(b)λ^[b]= ⋂_a∈α(b)λ^(b);

λ^(a)= ⋃_b∈α(a)λ^[b]= ⋃_b∈α(a)λ^(b).

Corollary 2.4. Let $λ \in ℕ (L)$ and $H : L \to ↓ ℕ$ be a mapping.

If λ_(a)sH (a) sλ_[a] for all a ∈ L, then λ = ⋁_a∈La ∧ H (a) = ⋁_a∈J(L)a ∧ H (a).

If λ^(a)sH (a) sλ^[a] for all a ∈ L, then λ = ⋀_a∈La ∨ H (a) = ⋀_a∈P(L)a ∨ H (a).

Corollary 2.5. Let X be a finite set, A ∈ L^X and $H : L \to ↓ ℕ$ be a mapping.

If |A|_(a)sH (a) s|A|_[a] for all a ∈ L, then |A| = ⋁_a∈La ∧ H (a) = ⋁_a∈J(L)a ∧ H (a).

If |A|^(a)sH (a) s|A|^[a] for all a ∈ L, then |A| = ⋀_a∈La ∨ H (a) = ⋀_a∈P(L)a ∨ H (a).

Lemma 2.6. Let X be a finite set and A ∈ L^X. Then

[23] |A|_[a]= |A_[a]| for all a ∈ J (L);

|A|^(a)= |A^(a)| for all a ∈ P (L);

|A|_(a)s|A_(a)|s|A_[a]|s|A|_[a] for all a ∈ L;

|A|^(a)s|A^(a)|s|A^[a]|s|A|^[a] for all a ∈ L.

Corollary 2.7. Let X be a finite set, A ∈ L^X and $H : L \to ↓ ℕ$ be a mapping.

|A| = ⋁_a∈La ∧ |A|_[a]= ⋁_a∈La ∧ |A|_(a)

= ⋁_a∈La ∧ |A_[a]| = ⋁_a∈La ∧ |A_(a)|;

|A| = ⋀_a∈La ∧ |A|^[a]= ⋀_a∈La ∧ |A|^(a)

= ⋀_a∈La ∧ |A^[a]| = ⋀_a∈La ∧ |A^(a)|;

|A|_[a]= ⋂_b∈β(a)|A|_(b)= ⋂_b∈β(a)|A_(b)|

= ⋂_b∈β(a)|A_[b]| = ⋂_b∈β(a)|A|_[b];

|A|_(a)= ⋃_a∈β(b)|A|_(b)= ⋃_a∈β(b)|A_(b)|

= ⋃_a∈β(b)|A_[b]| = ⋃_a∈β(b)|A|_[b];

|A|^[a]= ⋂_a∈α(b)|A|^(b)= ⋂_a∈α(b)|A^(b)|

= ⋂_a∈α(b)|A^[b]| = ⋂_a∈α(b)|A|^[b];

|A|^(a)= ⋃_b∈α(a)|A|^(b)= ⋂_b∈α(a)|A^(b)|

= ⋂_b∈α(a)|A^[b]| = ⋂_b∈α(a)|A|^[b].

3 Representations of L-fuzzy natural numbers

Let $ℕ^{L}$ be the set of all antitone mappings H : Lr $↓ ℕ$ satisfying:

${0} \subseteq H (⊥) \subseteq ℕ$ ;

for all $a \in L \underline{{} ⊥, ⊤}$ , there exists $m (a) \in ℕ$ such that ${0} \subseteq H (a) ↓ (a)$ .

Remark 3.1.

Since $H \in ℕ^{L}$ is antitone, there exists $m (⊤) \in ℕ$ such that $H (⊤) ↓ (⊤)$ ;

Let $λ \in ℕ (L)$ and $\tilde{H}, \hat{H}, \tilde{G}, \hat{G} : L \to ↓ ℕ$ be defined by: $\hat{H} (a) = λ_{[a]}; \hat{G} (a) = λ_{(a)}; \tilde{G} (a) = λ^{[a]}; \hat{G} (a) = λ^{(a)}$ for all a ∈ L. Then it follows from Lemma 2.1 that $\tilde{H}, \hat{H}, \tilde{G}, \hat{G} \in ℕ^{L}$ ;

If ${H_{k}}_{k \in K} \subseteq ℕ^{L}$ , we denote: for all a ∈ L,

(⋀_k∈KH_k) (a) = ⋂_k∈KH_k(a);

(⋁_k∈KH_k) (a) = ⋃_k∈KH_k(a).

It is easy to check that

⋀_{k \in K} H_{k} \in ℕ^{L}

, and if K is finite, then

⋁_{k \in K} H_{k} \in ℕ^{L}

. However, in general,

⋁_{k \in K} H_{k} \notin ℕ^{L}

Example 3.2. Let L = [0, 1], t ∈ L and $⌊ \frac{1}{t} ⌋ \in ℕ$ be the integer part of $\frac{1}{t}$ . Let $H_{t} : L \to ↓ ℕ$ be defined as: $H_{t} (a) = {\begin{matrix} ℕ, & a = ⊥, \\ ↓ (\frac{1}{t}) + 1, & a \in L \underline{{} ⊥} . \end{matrix}$

Then $H_{t} \in ℕ^{L}$ for all $t \in L \underline{{} 0}$ . For all $a \in L \underline{{} 0}$ , $\begin{matrix} (⋁_{t \in L \underline{{} ⊥}} H_{t}) (a) & \supseteq & ⋃_{n \in ℕ} H_{\frac{1}{n}} (a) \supseteq ⋃_{n \in ℕ} ↓ n = ℕ . \end{matrix}$

This shows $⋁_{k \in K} H_{k} \notin ℕ^{L}$ .

Definition 3.3. A mapping $H \in ℕ^{L}$ is called

a $ℕ_{β}^{L}$ -nested mapping, if a ∈ β (b) implies H (b) sH (a) for all a, b ∈ L;

a $ℕ_{α}^{L}$ -nested mapping, if a ∈ α (b) implies H (a) sH (b) for all a, b ∈ L.

The set of all $ℕ_{β}^{L}$ -nested (resp. $ℕ_{α}^{L}$ -nested) mappings is denoted by $𝒰 (ℕ_{β}^{L})$ (resp. $𝒰 (ℕ_{α}^{L})$ ). Clearly, $λ_{(a)}, λ_{[a]} \in 𝒰 (ℕ_{β}^{L})$ and $λ^{(a)}, λ^{[a]} \in 𝒰 (ℕ_{α}^{L})$ for all $λ \in ℕ (L)$ and a ∈ L.

Theorem 3.4. Let $λ \in ℕ (L)$ and $H \in ℕ^{L}$ .

If λ_(a)sH (a) sλ_[a] for all a ∈ L, then $H \in 𝒰 (ℕ_{β}^{L})$ .

If λ^(a)sH (a) sλ^[a] for all a ∈ L, then $H \in 𝒰 (ℕ_{α}^{L})$ .

Proof. (1) Let a ∈ β (b). By (1) of Lemma 2.2, λ_(a)sH (a) sλ_[a]sλ_(b)sH (b) sλ_[b]. Thus $H \in 𝒰 (ℕ_{β}^{L})$ .

(2) Let a ∈ α (b). By (2) of Lemma 2.2, λ^(b)sH (a) sλ^[b]sλ^(a)sH (b) sλ^[a]. Therefore $H \in 𝒰 (ℕ_{α}^{L})$ .

Theorem 3.5. Let ${H, H_{k}}_{k \in K} 𝒰 (ℕ_{β}^{L})$ and $f (H) = ⋁_{a \in L} a \land H (a) .$

Then

$f (H) \in ℕ (L)$ ;

for all a ∈ L, $f (H)_{(a)} \subseteq H (a) \subseteq f (H)_{[a]}$ ;

for all a ∈ L, $f (H)_{[a]} = ⋂_{b \in β (a)} H (b)$ ;

for all a ∈ L, $f (H)_{(a)} = ⋃_{a \in β (b)} H (b)$ ;

f is a surjective mapping from $𝒰 (ℕ_{β}^{L})$ to $ℕ (L)$ ;

$f (⋀_{k \in K} H_{k}) = ⋀_{k \in K} f (H_{k})$ ;

$f (⋁_{k \in K} H_{k}) = ⋁_{k \in K} f (H_{k})$ provided that $⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L})$ .

Proof. (1). (LN1). We have $\begin{matrix} f (H) (0) ⋁_{a \in L} a \land H (a) (0) \geq ⋁_{a \in L \underline{{} ⊤}} a \land H (a) (0) = ⊤ . \end{matrix}$

(LN2). Since $H \in ℕ^{L}$ , then for all $a \in L \underline{⊥}$ , there exists $m (a) \in ℕ$ such that $H (a) ↓ m (a)$ . Thus $\begin{matrix} ⋀_{n \in ℕ} f (H) (n) & = & ⋀_{n \in ℕ} (⋁_{a \in L} a \land H (a)) (n) \\ \leq & ⋁_{a \in L \underline{{} ⊥}} a \land H (a) (m (a) + 1) = ⊥ . \end{matrix}$

Therefore $f (H) \in ℕ (L)$ .

(2). Let a ∈ L. If $n \in f (H)_{(a)}$ , then $a \in β (f (H) (n)) = β (⋁_{b \in L} a \land H (b)) = ⋃_{b \in L} β (a \land H (b)) .$ Thus there exits b ∈ L such that a ∈ β (b ∧ H (b) (n)). Hence a ∈ β (b) and n ∈ H (b). Therefore n ∈ H (a) and $f (H)_{(a)} \subseteq H (a)$ .

If m ∈ H (a), then $f (H) (m) \geq a$ . Thus $n \in f (H)_{[a]}$ . Therefore $H (a) \subseteq f (H)_{[a]}$ .

(3). For all b ∈ β (a), by (2), we have $\begin{matrix} f (H)_{[a]} \subseteq f (H)_{(b)} \subseteq H (b) \subseteq f (H)_{[b]} . \end{matrix}$

Moreover, by (3) of Theorem 2.3, we have $\begin{matrix} f (H)_{[a]} \subseteq ⋂_{b \in β (a)} H (b) \subseteq ⋂_{b \in β (a)} f (H)_{[b]} = f (H)_{[a]} . \end{matrix}$

Therefore $f (H)_{[a]} = ⋂_{b \in β (a)} f (H)_{[b]}$ .

(4) For any a ∈ β (b), by (2), $f (H)_{(b)} \subseteq H (b) \subseteq f (H)_{[b]} \subseteq f (H)_{(a)} .$ Moreover, by (4) of Theorem 2.3, we have $\begin{matrix} f (H)_{(a)} = ⋃_{a \in β (b)} f (H)_{(b)} \subseteq ⋃_{a \in β (b)} H (b) \subseteq f (H)_{(a)} . \end{matrix}$

Therefore $f (H)_{(a)} = ⋃_{a \in β (b)} H (b)$ .

(5) Let $λ \in ℕ (L)$ . Then $\tilde{H} \in 𝒰 (ℕ_{β}^{L})$ by (1) of Theorem 3.4, and $λ = f (\tilde{H})$ by (1) of Corollary 2.4. Therefore $f$ is surjective.

(6) For all a ∈ L, by (1) of Lemma 2.2 and (3), $\begin{matrix} {(f (⋀_{k \in K} H_{k}))}_{[a]} \\ = & ⋂_{b \in β (a)} (⋀_{k \in K} H_{k}) (b) = ⋂_{b \in β (a)} ⋂_{k \in K} H_{k} (b) \\ = & ⋂_{k \in K} ⋂_{b \in β (a)} H_{k} (b) = ⋂_{k \in K} f (H_{k})_{[a]} \\ {(⋀_{k \in K} f (H_{k}))}_{[a]} . \end{matrix}$

Therefore $f (⋀_{k \in K} H_{k}) = ⋀_{k \in K} f (H_{k})$ .

(7). Let $⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L})$ . By (1), $f (⋁_{k \in K} H_{k}) \in ℕ (L)$ . For all a ∈ L, by (3) of Lemma 2.2 and (4), $\begin{matrix} {(f (⋁_{k \in K} H_{k}))}_{(a)} \\ = & ⋃_{a \in β (b)} (⋁_{k \in K} H_{k}) (b) = ⋃_{a \in β (b)} ⋃_{k \in K} H_{k} (b) \\ = & ⋃_{k \in K} ⋃_{a \in β (b)} H_{k} (b) = ⋃_{k \in K} f (H_{k})_{(a)} \\ = & {(⋁_{k \in K} f (H_{k}))}_{(a)} . \end{matrix}$

Therefore $f (⋁_{k \in K} H_{k}) = ⋁_{k \in K} f (H_{k})$ .

Theorem 3.6. Let ${G, G_{k}}_{k \in K} 𝒰 (ℕ_{α}^{L})$ and $g (G) = ⋀_{a \in L} a \lor G (a) .$

Then

$g (G) \in ℕ (L)$ ;

for all a ∈ L, $g (G)^{(a)} \subseteq G (a) \subseteq g (G)^{[a]}$ ;

for all a ∈ L, $g (G)^{[a]} = ⋂_{a \in α (b)} G (b)$ ;

for all a ∈ L, $g (G)^{(a)} = ⋃_{b \in α (a)} G (b)$ ;

g is a surjective mapping from $𝒰 (ℕ_{α}^{L})$ to $ℕ (L)$ ;

$g (⋀_{k \in K} G_{k}) = ⋀_{k \in K} g (G_{k})$ ;

$g (⋁_{k \in K} G_{k}) = ⋁_{k \in K} g (G_{k})$ provided that $⋁_{k \in K} G_{k} \in 𝒰 (ℕ_{α}^{L})$ .

Proof. (1). (LN1). We have $g (G) (0) \geq ⋀_{a \in L \underline{{} ⊤}} a \lor G (a) (0) = ⋀_{a \in L \underline{{} ⊤}} ⊤ = ⊤ .$ (LN2). Since $H \in ℕ^{L}$ , then for all $a \in L \underline{⊥}$ , there exists $m (a) \in ℕ$ such that $G (a) ↓ m (a)$ . Thus $\begin{matrix} ⋀_{n \in ℕ} g (G) (n) & = & ⋀_{n \in ℕ} (⋀_{a \in L} a \lor G (a)) (n) \\ \leq & ⋀_{a \in L \underline{{} ⊤}} a \lor G (a) (m (a) + 1) = ⊥ . \end{matrix}$

Therefore $g (G) \in ℕ (L)$ .

(2). Let a ∈ L and $n \in g (G)^{(a)}$ . Then $g (G) (n) ≰ a$ which implies the existence of b ∈ L such that b ≰ a and n ∈ G (b). Thus $g (G)^{(a)} \subseteq G (a)$ .

If $m \notin g (G)^{[a]}$ , then $a \in α (g (G) (m)) = ⋃_{b \in L} α (b \lor G (b) (m)) .$

Thus there exits b ∈ L such that a ∈ α (b ∨ G (b) (m)). Hence a ∈ α (b) and m ∉ G (b). Since G (a) sG (b), we have m ∉ G (a). Therefore $G (a) \subseteq g (G)^{[a]}$ .

(3). For all a ∈ α (b), by (2), we have $g (G)^{[a]} \subseteq g (G)^{(a)} \subseteq G (b) \subseteq g (G)^{[b]} .$

Thus, by (5) of Theorem 2.3, we have $g (G)^{[a]} \subseteq ⋂_{a \in α (b)} G (b) \subseteq ⋂_{a \in α (b)} g (G)^{[b]} = g (G)^{[a]} .$

Therefore $g (G)^{[a]} = ⋂_{a \in α (b)} G (b)$ .

(4). For all b ∈ α (a), by (2), $g (G)^{(b)} \subseteq G (b) \subseteq g (G)^{[b]} \subseteq g (G)^{(a)} .$

Thus, by (6) of Theorem 2.3, we have $g (G)^{(a)} = ⋃_{b \in α (a)} g (G)^{(b)} \subseteq ⋃_{b \in α (a)} G (b) \subseteq g (G)^{(a)} .$

Therefore $g (G)^{(a)} = ⋃_{b \in α (a)} G (b)$ .

(5). Let $λ \in ℕ (L)$ . Then, by (2) of Theorem 3.4, $\tilde{G} \in \subseteq (ℕ_{α}^{L})$ , and by (2) of Corollary 2.4, $g (\tilde{G}) = λ$ . Therefore $g$ is surjective.

(6). For all a ∈ L, by (3) and (2) of Lemma 2.2, $\begin{matrix} {(g (⋀_{k \in K} G_{k}))}^{[a]} \\ = & ⋂_{a \in α (b)} (⋀_{k \in K} G_{k}) (b) = ⋂_{a \in α (b)} ⋂_{k \in K} G_{k} (b) \\ = & ⋂_{k \in K} ⋂_{a \in α (b)} G_{k} (b) = ⋂_{k \in K} g (G_{k})^{[a]} \\ = & {(⋀_{k \in K} g (G_{k}))}^{[a]} . \end{matrix}$

Therefore $g (⋀_{k \in K} G_{k}) = ⋀_{k \in K} g (G_{k})$ .

(7). Let $⋁_{k \in K} G_{k} \in 𝒰 (ℕ_{α}^{L})$ . By (1), $g (⋁_{k \in K} G_{k}) \in ℕ (L)$ . For all a ∈ L, by (4) of Lemma 2.2 and (4), $\begin{matrix} {(g (⋁_{k \in K} G_{k}))}^{(a)} \\ = & ⋃_{b \in α (a)} (⋁_{k \in K} G_{k}) (b) = ⋃_{b \in α (a)} ⋃_{k \in K} G_{k} (b) \\ = & ⋃_{k \in K} ⋃_{b \in α (a)} G_{k} (b) = ⋃_{k \in K} g (G_{k})^{(a)} \\ = & {(⋁_{k \in K} g (G_{k}))}^{(a)} . \end{matrix}$

Therefore $g (⋁_{k \in K} G_{k}) = ⋁_{k \in K} g (G_{k})$ .

Theorem 3.7. Let $H \in 𝒰 (ℕ_{β}^{L})$ and $G \in 𝒰 (ℕ_{α}^{L})$ . Then

⋃_a∈β(b)H (b) = ⋃_a∈β(b)⋂_c∈β(b)H (c);

⋂_b∈β(a)H (b) = ⋂_b∈β(a)⋃_b∈β(c)H (c);

⋃_b∈α(a)G (b) = ⋃_b∈α(a)⋂_b∈α(c)G (c);

⋂_a∈α(b)G (b) = ⋂_a∈α(b)⋃_c∈α(b)G (c).

Proof. (1). By (4) of Theorem 2.3 and (3) and (4) of Theorem 3.5, we have $\begin{matrix} ⋃_{a \in β (b)} H (b) & = & f (H)_{(a)} = ⋃_{a \in β (b)} f (H)_{[b]} \\ = & ⋃_{a \in β (b)} ⋂_{c \in β (b)} H (c) . \end{matrix}$

Similarly, (2), (3) and (4) can be proved similarly.

Corollary 3.8. Let $λ \in ℕ (L)$ . Then

$\tilde{H}, \hat{H} \in 𝒰 (ℕ_{β}^{L})$ and $\tilde{G}, \hat{G} \in 𝒰 (ℕ_{α}^{L})$ ;

$f (\tilde{H}) = f (\hat{H}) = g (\tilde{G}) = g (\hat{G}) = λ$ .

Definition 3.9. Let $E, F \in ℕ^{L}$ .

E is called a meet $ℕ_{β}^{L}$ -nested mapping, if E (a) = ⋂_b∈β(a)E (b) for all a ∈ L;

E is called a meet $ℕ_{α}^{L}$ -nested mapping, if E (a) = ⋂_a∈α(b)E (b) for all a ∈ L;

F is called a join $ℕ_{β}^{L}$ -nested mapping, if F (a) = ⋃_a∈β(b)F (b) for all a ∈ L;

F is called a join $ℕ_{α}^{L}$ -nested mapping, if F (a) = ⋃_b∈α(a)F (b) for all a ∈ L.

The set of all meet $ℕ_{β}^{L}$ -nested (resp. meet $ℕ_{α}^{L}$ -nested) mappings is denoted by $Φ (ℕ_{β}^{L})$ (resp. $Φ (ℕ_{α}^{L})$ ). The set of all join $ℕ_{β}^{L}$ -nested (resp. join $ℕ_{α}^{L}$ -nested) mappings is denoted by $Ψ (ℕ_{β}^{L})$ (resp. $Ψ (ℕ_{α}^{L})$ ).

It is easy to check that $Φ (ℕ_{β}^{L}), Ψ (ℕ_{β}^{L}) \subseteq 𝒰 (ℕ_{β}^{L})$ and $Φ (ℕ_{α}^{L}), Ψ (ℕ_{α}^{L}) \subseteq 𝒰 (ℕ_{α}^{L})$ by Definition 3.9.

Theorem 3.10. Let $H \in 𝒰 (ℕ_{β}^{L})$ and $G \in 𝒰 (ℕ_{α}^{L})$ , and let $E_{H}, E_{G}, F_{H}, F_{G} \in ℕ^{L}$ be defined by: for all a ∈ L,

$E_{H} (a) = f (H)_{[a]}$ ;

$F_{H} (a) = f (H)_{(a)}$ ;

$E_{G} (a) = g (G)^{[a]}$ ;

$F_{G} (a) = g (G)^{(a)}$ .

Then

$E_{H} \in Φ (ℕ_{β}^{L}), E_{G} \in Φ (ℕ_{α}^{L}), F_{H} \in Ψ (ℕ_{β}^{L})$ and $F_{G} \in Ψ (ℕ_{α}^{L})$ ;

E_{E
_H}= E_{F
_H}= E_H and F_{F
_H}= F_{E
_H}= F_H;

E_{E
_G}= E_{F
_G}= E_G and F_{F
_G}= F_{E
_G}= F_G;

$f (E_{H}) = f (F_{H}) = f (H)$ ;

$g (E_{G}) = f (F_{G}) = f (G)$ .

Proof. (1). By (3) of Theorem 2.3, we have $⋂_{b \in β (a)} E_{H} (b) = ⋂_{b \in β (a)} f (H)_{[b]} = f (H)_{[a]} = E_{H} (a) .$

Thus $E_{H} \in Φ (ℕ_{β}^{L})$ . Other results are similar.

(2). By (3) of Theorem 3.5, we have $\begin{matrix} E_{E_{H}} (a) & = & f (E_{H})_{[a]} = ⋂_{b \in β (a)} E_{H} (b) = ⋂_{b \in β (a)} f (H)_{[b]} \\ = & ⋂_{b \in β (a)} ⋂_{c \in β (b)} H (c) = f (H)_{[a]} = E_{H} (a) \end{matrix}$ for all a ∈ L. Thus E_{E
_H}= E_H. By (3) of Theorem 3.5 and (2) of Theorem 3.7, we have $\begin{matrix} E_{F_{H}} (a) & = & f (F_{H})_{[a]} = ⋂_{b \in β (a)} F_{H} (b) = ⋂_{b \in β (a)} f (H)_{(b)} \\ = & ⋂_{b \in β (a)} ⋃_{b \in β (c)} H (c) = f (H)_{[a]} = E_{H} (a) . \end{matrix}$

Hence E_{F
_H}= E_H. Similarly, F_{F
_H}= F_{E
_H}= F_H.

(3) is similar to (2), and (4) and (5) are immediate.

Theorem 3.11. If ${H_{k}}_{k \in K} \subseteq 𝒰 (ℕ_{β}^{L})$ and ${G_{k}}_{k \in K} \subseteq 𝒰 (ℕ_{α}^{L})$ , then

E_{⋀_k∈KH_k}= ⋀_k∈KE_{H
_k} and F_{⋀_k∈KH_k}= ⋀_k∈KF_{H
_k};

E_{⋀_k∈KG_k}= ⋀_k∈KE_{G
_k} and F_{⋀_k∈KG_k}= ⋀_k∈KF_{G
_k};

E_{⋁_k∈KH_k}= ⋁_k∈KE_{H
_k} and F_{⋁_k∈KH_k}= ⋁_k∈KF_{H
_k} provided that $⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L})$ ;

E_{⋁_k∈KG_k}= ⋁_k∈KE_{G
_k} and F_{⋁_k∈KG_k}= ⋁_k∈KF_{G
_k} provided that $⋁_{k \in K} G_{k} \in 𝒰 (ℕ_{α}^{L})$ .

Proof. (1). By (6) of Theorem 3.5, (1) of Lemma 2.2 and (4) of Theorem 3.3, we have $\begin{matrix} E_{⋀_{k \in K} H_{k}} (a) & = & f {(⋀_{k \in K} H_{k})}_{[a]} = (⋀_{k \in K} f (H_{k}))_{[a]} \\ = & ⋂_{k \in K} f (H_{k})_{[a]} = ⋂_{k \in K} E_{H_{k}} (a) \\ = & (⋀_{k \in K} E_{H_{k}}) (a) \end{matrix}$ and $\begin{matrix} F_{⋀_{k \in K} H_{k}} (a) \\ = & f {(⋀_{k \in K} H_{k})}_{(a)} = {(⋀_{k \in K} f (H_{k}))}_{(a)} \\ = & ⋃_{a \in β (b)} {(⋀_{k \in K} f (H_{k}))}_{[b]} = ⋃_{a \in β (b)} ⋂_{k \in K} f (H_{k})_{[b]} \\ = & ⋂_{k \in K} ⋃_{a \in β (b)} f (H_{k})_{[b]} = ⋂_{k \in K} f (H_{k})_{(a)} \\ = & ⋂_{k \in K} F_{H_{k}} (a) = (⋀_{k \in K} F_{H_{k}}) (a) . \end{matrix}$

Therefore (1) holds.

(3). Let $⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L})$ . By (7) of Theorem 3.5 and (4) of Theorem 2.3, we have $\begin{matrix} E_{⋁_{k \in K} H_{k}} (a) \\ = & f {(⋁_{k \in K} H_{k})}_{[a]} = {(⋁_{k \in K} f (H_{k}))}_{[a]} \\ = & ⋂_{b \in β (a)} {(⋁_{k \in K} f (H_{k}))}_{(b)} = ⋂_{b \in β (a)} ⋃_{k \in K} f (H_{k})_{(b)} \\ = & ⋃_{k \in K} ⋂_{b \in β (a)} f (H_{k})_{(b)} = ⋃_{k \in K} f (H_{k})_{[a]} \\ = & ⋃_{k \in K} E_{H_{k}} (a) = (⋁_{k \in K} E_{H_{k}}) (a) . \end{matrix}$ and $\begin{matrix} F_{⋁_{k \in K} H_{k}} (a) & = & f {(⋁_{k \in K} H_{k})}_{(a)} = {(⋁_{k \in K} f (H_{k}))}_{(a)} \\ = & ⋃_{k \in K} f (H_{k})_{(a)} = ⋃_{k \in K} F_{H_{k}} (a) \\ = & (⋁_{k \in K} F_{H_{k}}) (a) . \end{matrix}$

Therefore (3) holds.

(2) and (4) are similar to (1) and (3).

Theorem 3.12. If $H \in 𝒰 (ℕ_{β}^{L})$ and $G \in 𝒰 (ℕ_{α}^{L})$ , then

$H \in Φ (ℕ_{β}^{L}) Ł H = E_{H} Ł H = E_{E_{H}} Ł H = E_{F_{H}}$ ;

$H \in Ψ (ℕ_{β}^{L}) Ł H = F_{H} Ł H = F_{F_{H}} Ł H = F_{E_{H}}$ ;

$G \in Φ (ℕ_{α}^{L}) Ł G = E_{G} Ł G = E_{E_{G}} Ł G = E_{F_{G}}$ ;

$G \in Ψ (ℕ_{α}^{L}) Ł G = F_{G} Ł H = F_{F_{G}} Ł G = F_{E_{G}}$ .

Proof. (1). By (3) of Theorem 3.5, $\begin{matrix} H \in Φ (ℕ_{β}^{L}) & Ł & \forall a \in L, H (a) = f (H)_{[a]} = E_{H} (a) \\ Ł & H = E_{H} . \end{matrix}$

Thus the first equivalence is valid. The second equivalence follows from (2) of Theorem 3.10.

(2), (3) and (4) can be proved similarly.

Corollary 3.13. Let $λ \in ℕ (L)$ . Then

$\tilde{H} = E_{\tilde{H}} = E_{E_{\tilde{H}}} = E_{F_{\tilde{H}}} \in Φ (ℕ_{β}^{L})$ ;

$\hat{H} = F_{\hat{H}} = F_{F_{\hat{H}}} = F_{E_{\hat{H}}} \in Ψ (ℕ_{β}^{L})$ ;

$\tilde{G} = E_{\tilde{G}} = E_{E_{\tilde{G}}} = E_{F_{\tilde{G}}} \in Φ (ℕ_{α}^{L})$ ;

$\hat{G} = F_{\hat{G}} = F_{F_{\hat{G}}} = F_{E_{\hat{G}}} \in Ψ (ℕ_{α}^{L})$ .

Proof. By (3) of Theorem 2.3 and (3) of Theorem 3.5, $\tilde{H} (a) = ⋂_{b \in β (a)} λ_{[b]} = ⋂_{b \in β (a)} \tilde{H} (b) = f (\tilde{H})_{[a]} = E_{\tilde{H}} (a)$ for any a ∈ L. Then $\tilde{H} = E_{\tilde{H}}$ and $\tilde{H} \in Φ (ℕ_{β}^{L})$ by (1) of Theorem 3.12. Other equalities follows from (2) of Theorem 3.10.

(2), (3) and (4) can be similarly proved.

Remark 3.14. If $H_{1}, H_{2} \in 𝒰 (ℕ_{β}^{L})$ and $G_{1}, G_{2} \in 𝒰 (ℕ_{α}^{L})$ , we denote

H₁∼_βH₂, if E_H
₁= E_H
₂;

G₁∼_αG₂, if E_G
₁= E_G
₂.

Theorem 3.15. Let $H_{1}, H_{2} \in 𝒰 (ℕ_{β}^{L})$ Then the followings are equivalent:

H₁∼_βH₂;

E_H
₁∼_βE_H
₂;

F_H
₁∼_βF_H
₂;

F_H
₁= F_H
₂;

$f (H_{1}) = f (H_{2})$ .

Proof. (1) R (2). If H₁∼_βH₂, then E_{H
₁}= E_{H
₂}. Thus, by (2) of Theorem 3.10, we have E_{E
_{H
₁}}= E_{H
₁}= E_{H
₂}= E_{E
_{H
₂}}. Hence E_{H
₁}∼_βE_{H
₂} by definition.

(2) R (3). If E_{H
₁}∼_βE_{H
₂}, then E_{F
_{H
₁}}= E_{E
_{H
₁}}= E_{E
_{H
₂}}= E_{F
_{H
₂}}. Thus F_{H
₁}∼_βF_{H
₂} by definition.

(3) R (4). Let F_{H
₁}∼_βF_{H
₂}. Then E_{F
_{H
₁}}= E_{F
_{H
₂}}. By (2) of Theorem 3.10, we have E_{H
₁}= E_{F
_{H
₁}}= E_{F
_{H
₂}}= E_{H
₂} and thus F_{H
₁}= F_{E
_{H
₁}}= F_{E
_{H
₂}}= F_{H
₂}. Therefore (4) is true.

(4) R (5). Let F_{H
₁}= F_{H
₂}. By (4) of Theorem 3.5, we have $f (H_{1})_{(a)} = F_{H_{1}} (a) = F_{H_{2}} (a) = f (H_{2})_{(a)}$ for all a ∈ L. Therefore $f (H_{1}) = f (H_{2})$ .

(5) R (1). Since $f (H_{1}) = f (H_{2})$ , we have E_{H
₁} $(a) = f (H_{1})_{[a]} = f (H_{2})_{[a]} = E_{H_{2}} (a)$ for all a ∈ L. Therefore H₁∼_βH₂.

Theorem 3.16. Let $G_{1}, G_{2} \in 𝒰 (ℕ_{α}^{L})$ . Then the followings are equivalent:

G₁∼_αG₂;

E_G
₁∼_αE_G
₂;

F_G
₁∼_αF_G
₂;

F_G
₁= F_G
₂;

$g (G_{1}) = g (G_{2})$ .

Proof. Similar to Theorem 3.14.

Theorem 3.17. If $H \in 𝒰 (ℕ_{β}^{L})$ and $G \in 𝒰 (ℕ_{α}^{L})$ , then

H ∼_βE_H∼_βF_H;

G ∼_αE_G∼_αF_G.

Proof. The proof follows from Theorem 3.10.

Theorem 3.18. Let ${H_{k}, T_{k}}_{k \in K} \subseteq 𝒰 (ℕ_{β}^{L})$ and ${G_{k}, R_{k}}_{k \in K} \subseteq 𝒰 (ℕ_{α}^{L})$ with H_k∼_βT_k and G_k∼_αR_k for any k ∈ K. Then

⋀_k∈KH_k∼_β⋀_k∈KT_k and ⋀_k∈KG_k∼_α⋀_k∈KR_k;

if $⋁_{k \in K} H_{k}, ⋁_{k \in K} T_{k} \in 𝒰 (ℕ_{β}^{L})$ , then ⋁_k∈KH_k∼_β⋁_k∈KT_k;

if $⋁_{k \in K} G_{k}, ⋁_{k \in K} R_{k} \in 𝒰 (ℕ_{α}^{L})$ , then ⋁_k∈KG_k∼_α⋁_k∈KR_k.

Proof. The proof follows from Theorem 3.11.

Remark 3.19. Let $H \in 𝒰 (ℕ_{β}^{L})$ and $G \in 𝒰 (ℕ_{α}^{L})$ and ∼_β, ∼_α be defined in Remark 3.14.

∼_β and ∼_α are equivalent relations on $𝒰 (ℕ_{β}^{L})$ and $𝒰 (ℕ_{α}^{L})$ . The equivalent classes of H and G are denoted by [H]_β and [G]_α respectively.

Let $𝒰 (ℕ_{β}^{L}) / \sim_{β} = {[H]_{β} : H \in 𝒰 (ℕ_{β}^{L})}$ and let ${[H_{k}]_{β}}_{k \in K} \subseteq 𝒰 (ℕ_{β}^{L}) / \sim_{β}$ . We define

⋀_k∈K[H_k]_β= [⋀_k∈KH_k]_β;

⋁_k∈K[H_k]_β= [⋁_k∈KH_k]_β provided that $⋁_{k \in K} H_{k} \in ℕ^{L}$ ;

Let $𝒰 (ℕ_{α}^{L}) / \sim_{α} = {[G]_{α} : G \in 𝒰 (ℕ_{α}^{L})}$ and let ${[G_{k}]_{α}}_{k \in K} \subseteq 𝒰 (ℕ_{α}^{L}) / \sim_{α}$ . We define

⋀_k∈K[G_k]_α= [⋀_k∈KG_k]_α;

⋁_k∈K[G_k]_α= [⋁_k∈KG_k]_α provided that $⋁_{k \in K} G_{k} \in ℕ^{L}$ ;

Consistencies of operations defined in (4) and (5) above can be seen in Theorem 3.18;

Let _LL1 and _LL2 be two lattices. A mapping φ :_LL1r_LL2 is called an isomorphism if it is bijective and homomorphic, i.e., {a_k}_k∈Ks_LL1,

φ (⋁_k∈Ka_k) = ⋁_k∈Kφ (a_k);

φ (⋀_k∈Ka_k) = ⋀_k∈Kφ (a_k).

Two lattices _LL1 and _LL2 is called isomorphic, denoted by _LL1≅_LL2, if there exists an isomorphism φ :_LL1r_LL2.

Theorem 3.20. Let $H \in 𝒰 (ℕ_{β}^{L})$ , $G \in 𝒰 (ℕ_{α}^{L})$ and $λ \in ℕ (L)$ . Then

[H]_β= [E_H]_β= [F_H]_β;

[G]_α= [E_G]_α= [F_G]_α;

$[\tilde{H}]_{β} = [\hat{H}]_{β} = {H \in ℕ^{L} : \forall a \in L, \hat{H} (a) \subseteq H (a) \subseteq \tilde{H} (a)}$ ;

$[\tilde{G}]_{α} = [\hat{G}]_{α} = {G \in ℕ^{L} : \forall a \in L, \hat{G} (a) \subseteq G (a) \subseteq \hat{G} (a)}$ .

Proof. (1) and (2) follows from Theorem 3.17.

(3). By (3) of Theorem 2.3 and (2) of Corollary 3.8, $\begin{matrix} E_{\tilde{H}} (a) & = & f (\tilde{H})_{[a]} = λ_{[a]} = ⋂_{b \in β (a)} λ_{(b)} = ⋂_{b \in β (a)} \hat{H} (b) \\ = & f (\hat{H})_{[a]} = E_{\hat{H}} (a) \end{matrix}$ for all a ∈ L. Thus $\tilde{H} \sim_{β} \hat{H}$ and $[\tilde{H}]_{β} = [\hat{H}]_{β}$ .

Let H be in the right side set. Then $E_{\tilde{H}} (a) \subseteq E_{H} (a) \subseteq E_{\hat{H}} (a)$ for all a ∈ L. Since $E_{\tilde{H}} = E_{\hat{H}}$ , $H \in [\tilde{H}]_{β}$ . Thus the right set is included by $[\tilde{H}]_{β}$ .

In order to prove the inverse inclusion, let $H \in [\tilde{H}]_{β}$ . Then $\tilde{H} \sim_{β} H \sim_{β} \hat{H}$ . Thus, by (1) of Remark 3.14 and (1) of Theorem 3.12, we have $E_{H} = E_{\tilde{H}} = \tilde{H}$ , and by (4) of Theorem 3.15 and (2) of Theorem 3.12, $\hat{H} = F_{\hat{H}} = F_{H}$ . Hence, by (2) of Theorem 3.5, we have $\begin{matrix} λ_{(a)} & = & \hat{H} (a) = F_{H} (a) = f (H)_{(a)} \subseteq H (a) \\ \subseteq & f (H)_{[a]} = E_{H} (a) = \tilde{H} (a) = λ_{[a]} \end{matrix}$ for all a ∈ L. Therefore H is in the right side set and $[\tilde{H}]_{β}$ is included by the right side set.

(4) is similar to (3).

Lemma 3.21. Let ${λ_{k}}_{k \in K} \subseteq ℕ (L)$ , ${H_{k}}_{k \in K} \subseteq 𝒰 (ℕ_{β}^{L})$ and ${G_{k}}_{k \in K} \subseteq 𝒰 (ℕ_{α}^{L})$ . Then

$⋁_{k \in K} λ_{k} \in ℕ (L) Ł ⋁_{k \in K} \tilde{H_{k}} \in Φ (ℕ_{β}^{L}) Ł ⋁_{k \in K} \hat{H_{k}} \in Ψ (ℕ_{β}^{L})$ ;

$⋁_{k \in K} λ_{k} \in ℕ (L) Ł ⋁_{k \in K} \tilde{G_{k}} \in Φ (ℕ_{α}^{L}) Ł ⋁_{k \in K} \hat{G_{k}} \in Ψ (ℕ_{α}^{L})$ ;

$⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L}) Ł ⋁_{k \in K} f (H_{k}) \in ℕ (L) Ł ⋁_{k \in K} E_{H_{k}} \in Φ (ℕ_{β}^{L}) Ł ⋁_{k \in K} F_{H_{k}} \in Ψ (ℕ_{β}^{L})$ ;

$⋁_{k \in K} G_{k} \in 𝒰 (ℕ_{α}^{L}) Ł ⋁_{k \in K} f (G_{k}) \in ℕ (L) Ł ⋁_{k \in K} E_{G_{k}} \in Φ (ℕ_{α}^{L}) Ł ⋁_{k \in K} F_{G_{k}} \in Ψ (ℕ_{α}^{L})$ .

Proof. (1). We only prove the first equivalence. The second equivalence is similar. Let $⋁_{k \in K} λ_{k} \in ℕ (L)$ and $\tilde{H}, \hat{H} : L \to ↓ ℕ$ be defined by: for all a ∈ L,

$\tilde{H} (a) = {(⋁_{k \in K} λ_{k})}_{[a]}$ ;

$\hat{H} (a) = {(⋁_{k \in K} λ_{k})}_{(a)}$ .

Then, for all a ∈ L, we have $\begin{matrix} \tilde{H} (a) & = & {(⋁_{k \in K} λ_{k})}_{[a]} = ⋂_{b \in β (a)} {(⋁_{k \in K} λ_{k})}_{(b)} \\ = & ⋂_{b \in β (a)} ⋃_{k \in K} (λ_{k})_{(b)} = ⋃_{k \in K} ⋂_{b \in β (a)} (λ_{k})_{(b)} \\ = & ⋃_{k \in K} (λ_{k})_{[a]} = ⋃_{k \in K} \tilde{H_{k}} (a) = (⋁_{k \in K} \tilde{H_{k}}) (a) \end{matrix}$ and $\begin{matrix} \hat{H} (a) & = & {(⋁_{k \in K} λ_{k})}_{(a)} = ⋃_{k \in K} \hat{H_{k}} (a) = (⋁_{k \in K} \hat{H_{k}}) (a) . \end{matrix}$

Thus $⋁_{k \in K} \tilde{H_{k}} = \tilde{H} \in Φ (ℕ_{β}^{L})$ and $⋁_{k \in K} \hat{H_{k}} = \hat{H} \in Ψ (ℕ_{β}^{L})$ by Corollary 3.13.

Conversely, let $⋁_{k \in K} \tilde{H_{k}} \in Φ (ℕ_{β}^{L})$ (resp. $⋁_{k \in K} \hat{H_{k}} \in Ψ (ℕ_{β}^{L})$ ). By Theorem 3.5 and Theorem 3.8, we have $⋁_{k \in K} λ_{k} = ⋁_{k \in K} f (\tilde{H_{k}}) = f (⋁_{k \in K} \tilde{H_{k}}) \in ℕ (L)$ and $⋁_{k \in K} λ_{k} = ⋁_{k \in K} f (\hat{H_{k}}) = f (⋁_{k \in K} \hat{H_{k}}) \in ℕ (L) .$

Therefore (1) holds.

(3). Let $⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L})$ . By (1) and (7) of Theorem 3.5, we have $⋁_{k \in K} f (H_{k}) = f (⋁_{k \in K} H_{k}) \in ℕ (L) .$

Conversely, let $⋁_{k \in K} f (H_{k}) \in ℕ (L)$ . By (1), we have $⋁_{k \in K} E_{H_{k}} \in Φ (ℕ_{β}^{L})$ . Moreover, by (2) of Theorem 3.5, $H_{k} (a) \subseteq f (H_{k})_{[a]} = E_{H_{k}} (a)$ for all a ∈ L and all k ∈ K. Thus H_k≤ E_{H
_k} for all k ∈ K. Hence ⋁_k∈KH_k≤ ⋁_k∈KE_{H
_k}, which implies $⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L})$ . Therefore the first equivalence is true.

Finally, let $λ_{k} = f (H_{k})$ for all k ∈ K. Then $\tilde{H_{k}^{'}} = E_{H_{k}}$ and $\hat{H_{k}^{'}} = F_{H_{k}}$ for all k ∈ K. The last two equivalences directly follow from (1). Therefore (3) holds.

(2) and (4) are similar to (1) and (3).

Theorem 3.22. We have

$ℕ (L) ≅ 𝒰 (ℕ_{β}^{L}) / \sim_{β} ≅ Φ (ℕ_{β}^{L}) / \sim_{β} ≅ Ψ (ℕ_{β}^{L}) / \sim_{β}$ ;

$ℕ (L) ≅ 𝒰 (ℕ_{α}^{L}) / \sim_{α} ≅ Φ (ℕ_{α}^{L}) / \sim_{α} ≅ Ψ (ℕ_{α}^{L}) / \sim_{α}$ .

Proof. (1). Let $φ : 𝒰 (ℕ_{β}^{L}) / \sim_{β} \to ℕ (L)$ be defined by $φ ([H]_{β}) = f (H)$ for all $[H]_{β} \in 𝒰 (ℕ_{β}^{L}) / \sim_{β}$ . We need to prove φ is an isomorphism.

(i) φ is bijective.

Let $λ \in ℕ (L)$ . By Corollary 3.8, we have $\tilde{H} \in 𝒰 (ℕ_{β}^{L})$ and $φ ([\tilde{H}]_{β}) = f (\tilde{H}) = λ$ . Thus φ is surjective.

Let $[H_{1}]_{β}, [H_{2}]_{β} \in 𝒰 (ℕ_{β}^{L}) / \sim_{β}$ with [H₁]_β≠[H₂]_β. Then H₁_notsimβH₂ and thus $f (H_{1}) \neq f (H_{2})$ by (5) of Theorem 3.15. Hence φ is injective.

(ii) φ is homomorphic.

Let ${[H_{k}]_{β}}_{k \in K} \subseteq 𝒰 (ℕ_{β}^{L}) / \sim_{β}$ . By (6) of Theorem 3.5, we have $\begin{matrix} φ (⋀_{k \in K} [H_{k}]_{β}) & = & φ ([⋀_{k \in K} H_{k}]_{β}) = f (⋀_{k \in K} H_{k}) \\ = & ⋀_{k \in K} f (H_{k}) = ⋀_{k \in K} φ ([H_{k}]_{β}) . \end{matrix}$ If $⋁_{k \in K} H_{k} \in 𝒰 (ℕ_{β}^{L})$ , then $⋁_{k \in K} f (H_{k}) \in ℕ (L)$ by (3) of Lemma 3.21. Finally, by (7) of Theorem 3.5, $\begin{matrix} φ (⋁_{k \in K} [H_{k}]_{β}) & = & φ ([⋁_{k \in K} H_{k}]_{β}) = f (⋁_{k \in K} H_{k}) \\ = & ⋁_{k \in K} f (H_{k}) = ⋁_{k \in K} φ ([H_{k}]_{β}) . \end{matrix}$

Therefore φ is isomorphic.

Similarly, let $φ : 𝒰 (ℕ_{β}^{L}) / \sim_{β} \to Φ (ℕ_{β}^{L}) / \sim_{β}$ and $ψ : 𝒰 (ℕ_{β}^{L}) / \sim_{β} \to Ψ (ℕ_{β}^{L}) / \sim_{β}$ be defined by φ ([H]_β) = [E_H]_β and ψ ([H]_β) = [F_H]_β for all $[H]_{β} \in 𝒰 (ℕ_{β}^{L}) / \sim_{β}$ .

Since $Φ (ℕ_{β}^{L}), Ψ (ℕ_{β}^{L}) \subseteq 𝒰 (ℕ_{β}^{L})$ , φ and ψ are surjective. By (1) of Theorem 3.19, φ and ψ are injective. Proofs that φ and ψ are homomorphic follow from (1) and (3) of Theorem Theorem 3.11 and (3) of Lemma 3.21. Therefore φ and ψ are isomorphisms.

(2) is similar to (1).

4 Decompositions and representations of additions and exponentiations of L-fuzzy natural numbers

Let $λ, μ \in ℕ (L)$ . The addition and multiplication of λ and μ are given respectively by: for all $n \in ℕ$ ,

(λ + μ) (n) = ⋁_k+l=nλ (k) ∧ μ (l);

(λ · μ) (n) = ⋁_k·l=nλ (k) ∧ μ (l).

Then $λ + μ, λ \cdot μ \in ℕ (L)$ [3,9, 3,9].

Let $H, G \in ℕ^{L}$ . The addition H + G and the multiplication H · G are defined respectively by: $\begin{matrix} (H + G) (a) & = & H (a) + G (a) \\ = & {n + m : n \in H (a), m \in G (a)} \end{matrix}$ and $\begin{matrix} (H \cdot G) (a) & = & H (a) \cdot G (a) \\ = & {n \cdot m : n \in H (a), m \in G (a)} \end{matrix}$

In particular, let (H + G) (a) = H (a) and (H · G) (a) = H (a) provided that G (a) =∅.

By definition of $ℕ^{L}$ , it is easy to check that $H + G, H \cdot G \in ℕ^{L}$ and that if $H, G \in 𝒰 (ℕ_{β}^{L})$ (resp. $𝒰 (ℕ_{α}^{L})$ ), then $H + G, H \cdot G \in 𝒰 (ℕ_{β}^{L})$ (resp. $𝒰 (ℕ_{α}^{L})$ ).

Since properties of exponentiations are quite similar to additions, we omit them.

Lemma 4.1. [22] Let $λ, μ, κ \in ℕ (L)$ and a ∈ L. Then

(λ + μ)_(a)sλ_(a)+ μ_(a)sλ_[a]+ μ_[a]

s (λ + μ)_[a];

(λ + μ)^(a)sλ^(a)+ μ^(a)sλ^[a]+ μ^[a]

s (λ + μ)^[a];

λ + μ = μ + λ;

(λ + μ) + κ = λ + (μ + κ).

Theorem 4.2. Let $λ, μ \in ℕ (L)$ and a ∈ L. Then

λ + μ = ⋁_a∈La ∧ (λ_[a]+ μ_[a])

= ⋁_a∈La ∧ (λ_(a)+ μ_(a))

= ⋁_a∈J(L)a ∧ (λ_[a]+ μ_[a])

= ⋁_a∈J(L)a ∧ (λ_(a)+ μ_(a));

λ + μ = ⋀_a∈La ∨ (λ^[a]+ μ^[a])

= ⋀_a∈La ∨ (λ^(a)+ μ^(a))

= ⋀_a∈P(L)a ∨ (λ^[a]+ μ^[a])

= ⋀_a∈P(L)a ∨ (λ^(a)+ μ^(a));

(λ + μ)_[a]= ⋂_b∈β(a)(λ_[b]+ μ_[b])

= ⋂_b∈β(a)(λ_(b)+ μ_(b));

(λ + μ)_(a)= ⋃_a∈β(b)(λ_[b]+ μ_[b])

= ⋃_a∈β(b)(λ_(b)+ μ_(b));

(λ + μ)^[a]= ⋂_a∈α(b)(λ^[b]+ μ^[b])

= ⋂_a∈α(b)(λ^(b)+ μ^(b));

(λ + μ)^(a)= ⋃_b∈α(a)(λ^[b]+ μ^[b])

= ⋃_b∈α(a)(λ^(b)+ μ^(b)).

Theorem 4.3. Let $H, G, T \in 𝒰 (ℕ_{β}^{L})$ (resp. $H, G, T \in 𝒰 (ℕ_{α}^{L})$ ). Then

H + G = G + H;

(H + G) + T = H + (G + T);

H · (G + T) = H · G + H · T.

Theorem 4.4. Let $H_{1}, H_{2}, H_{3} \in 𝒰 (ℕ_{β}^{L})$ and a ∈ L. Then

$(f (H_{1}) + f (H_{2}))_{(a)} \subseteq (H_{1} + H_{2}) (a)$

$\subseteq (f (H_{1}) + f (H_{2}))_{[a]}$ ;

$f (H_{1} + H_{2}) = f (H_{1}) + f (H_{2})$ .

Proof. The proof follows from Lemma 4.1 and Theorem 2.3.

Theorem 4.5. If $G_{1}, G_{2}, G_{3} \in 𝒰 (ℕ_{α}^{L})$ and a ∈ L, then

$(g (G_{1}) + g (G_{2}))^{(a)} \subseteq (G_{1} + G_{2}) (a)$ $\subseteq (g (G_{1}) + g (G_{2}))^{[a]}$ ;

$g (G_{1} + G_{2}) = g (G_{1}) + g (G_{2})$ .

Lemma 4.6. Let $λ, μ \in ℕ (L)$ and a ∈ L. Then

if a ∈ J (L), then (λ + μ)_[a]= λ_[a]+ μ_[a];

if a ∈ P (L), then (λ + μ)^(a)= λ^(a)+ μ^(a);

if L = L (α), then (λ + μ)^[a]= λ^[a]+ μ^[a];

if L = L (β), then (λ + μ)_(a)= λ_(a)+ μ_(a).

Proof. (1) and (2) are results Lemma 4.9 in [17] and Theorem 20 in [22] respectively.

(3). By (2) of Lemma 4.1, λ^[a]+ μ^[a]s (λ + μ)^[a]. Conversely, let n ∈ (λ + μ)^[a]. By assumption, $\begin{matrix} a \notin α ((λ + μ) (n)) & = & ⋂_{k + l = n} α (λ (k) \land μ (l)) . \end{matrix}$

Then there exist $k, l \in ℕ$ such that k + l = n and $a \notin α (λ (k) \land μ (l)) = α (λ (k)) \cup α (μ (l)) .$

Thus a ∉ α (λ (k)) and a ∉ α (μ (l)) which imply k ∈ λ^[a] and l ∈ μ^[a]. Hence n = k + l ∈ λ^[a]+ μ^[a]. Therefore (λ + μ)^[a]sλ^[a]+ μ^[a].

(4) is similar to (3).

Theorem 4.7. Let $H_{1}, H_{2} \in 𝒰 (ℕ_{β}^{L})$ , $G_{1}, G_{2} \in 𝒰 (ℕ_{α}^{L})$ and a ∈ L. Then

if $J (L) = L \underline{{} ⊥}$ , then E_H₁+H₂= E_H
₁+ E_H
₂;

if $P (L) = L \underline{{} ⊤}$ , then F_G₁+G₂= F_G
₁+ F_G
₂;

if L = L (β), then F_H₁+H₂= F_H
₁+ F_H
₂;

if L = L (α), then E_G₁+G₂= E_G
₁+ E_G
₂.

Proof. (1). By (3) of Theorem 4.4 and (1) of Theorem 4.6, for any a ∈ L = J (L), $\begin{matrix} E_{H_{1} + H_{2}} (a) & = & f (H_{1} + H_{2})_{[a]} = (f (H_{1}) + f (H_{2}))_{[a]} \\ = & f (H_{1})_{[a]} + f (H_{2})_{[a]} \\ = & E_{H_{1}} (a) + E_{H_{2}} (a) = (E_{H_{1}} + E_{H_{2}}) (a) . \end{matrix}$

Therefore E_H₁+H₂= E_{H
₁}+ E_{H
₂}.

(3). By (3) of Theorem 4.5, $f (H_{1} + H_{2}) = f (H_{1}) + f (H_{2}) .$

Thus by (4) of Lemma 4.6 and (2) of Theorem 3.10, $\begin{matrix} F_{H_{1} + H_{2}} (a) & = & f (H_{1} + H_{2})_{(a)} = (f (H_{1}) + f (H_{2}))_{(a)} \\ = & f (H_{1})_{(a)} + f (H_{2})_{(a)} \\ = & F_{H_{1}} (a) + F_{H_{2}} (a) \end{matrix}$ for any a ∈ L. Therefore F_H₁+H₂= F_{H
₁}+ F_{H
₂}.

(2) and (4) are similar to (1) and (3).

Remark 4.8.

Theorem 3.10 in [23] had proved the same result of (4) of Lemma 4.6 when a ∈ β (⊤);

The following corollary gives some conditions that enhance Theorem 4.4 and 4.5.

Corollary 4.9. Let $H_{1}, H_{2} \in Ψ (ℕ_{β}^{L})$ , $G_{1}, G_{2} \in Ψ (ℕ_{α}^{L})$ and a ∈ L. Then

if J (L) = L, then $f (H_{1} + H_{2})_{[a]} = (H_{1} + H_{2}) (a)$ ;

if P (L) = L, then $g (G_{1} + G_{2})^{(a)} = (G_{1} + G_{2}) (a)$ ;

if L = L (β), then $f (H_{1} + H_{2})_{(a)} = (H_{1} + H_{2}) (a)$ ;

if L = L (α), then $g (G_{1} + G_{2})^{[a]} = (G_{1} + G_{2}) (a)$ .

5 Conclusions

In this paper, we discuss decompositions and representations of L-fuzzy natural numbers. We focus heavily on representations because some of decompositions are similar to those of L-fuzzy numbers that has been discussed in [15]. Due to the limited space, apart from Corollary 2.5, Lemma 2.6 and Theorem 2.7, we omitted applications related to cardinalities of L-fuzzy sets.

Footnotes

Acknowledgments

The authors wish to thank Prof. Bijan Davvaz and the referees for their valuable suggestions.

This work is supported by the National Natural Science Foundation of China (No. 11471202); the Educational Commission of Human provice (No. 15C0586).

References

Aytar

and Pehlivan

, Statistically monotonic and statistically bounded sequences of fuzzy numbers, Inform Control176 (2006), 734–744.

Bodjanova

, Alpha-bounds of fuzzy numbers, Inform Sci8 (2003), 237–266.

Casasnovas

and Torrens

, An axiomatic approach to the fuzzy cardinality of finite fuzzy sets, Fuzzy Sets Syst133 (2003), 193–209.

Casasnovas

and Riera

J.V.

, Discrete fuzzy numbers defined on a subset of natural numbers, Adv Soft Comput42 (2007), 573–582.

Casasnovas

and Riera

J.V.

, Aggregation of bounded fuzzy natural number-valued multisets, Model Decis Artif Intell (2010), 67–78.

Cheng

C.B.

, Fuzzy process control: Construction of control charts with fuzzy numbers, Fuzzy Sets Syst154 (2005), 287–303.

El-Dardery

and Zhang

, On L-fuzzy proximity spaces, Int J Hybrid Intell Syst11 (2014), 137–144.

El-Dardery

, Ramadan

A.A.

and Kim

Y.C.

, L-Fuzzy topogenous orders and L-Fuzzy topologies, J Intell Fuzzy Syst24 (2013), 685–691.

Dubois

and Prade

, Operations on fuzzy numbers, Int J Syst Sci9 (1978), 613–626.

10.

Filev

D.P.

and Yager

R.R.

, Operations on fuzzy numbers via fuzzy reasoning, Fuzzy Sets Syst91 (1997), 137–142.

11.

Giachetti

R.E.

and Young

R.E.

, A parametric representation of fuzzy numbers and their arithmetic operators, Fuzzy Sets Syst91 (1997), 185–202.

12.

Grzegorzewski

, Metrics and orders in space of fuzzy numbers, Fuzzy Sets Syst97 (1998), 83–94.

13.

Grzegorzewski

, Nearest interval approximation of a fuzzy number, Fuzzy Sets Syst130 (2012), 321–330.

14.

Han

and Shi

F.G.

, Multiplication and exponentiation of L-fuzzy natural numbers, Fuzzy Syst Math1 (2010), 109–113.

15.

Huang

H.L.

and Shi

F.G.

, L-fuzzy numbers and there applications, Inform Sci178 (2008), 1141–1151.

16.

Janizade-Haji

, Zare

H.K.

, Eslamipoor

and Sepehriar

, A developed distance method for ranking generalized fuzzy numbers, Neural Comput Appl25 (2014), 727–731.

17.

Kaleva

and Swikkala

, On fuzzy metric spaces, Fuzzy Sets Syst12 (1984), 215–229.

18.

Lowen

, Fuzzy set theory2010–Springer-Verlag.

19.

Ranik

and Rinmanek

, Inequality between fuzzy numbers and its use in fuzzy potimization, Fuzzy Sets Syst16 (1985), 123–128.

20.

Rezvani

, A new approach ranking of exponential trapezoidal fuzzy numbers, J Phys Sci16 (2012), 45–57.

21.

Shi

F.G.

, Theory of L_β-nested mappings and L_α-nested mappings and applications, Fuzzy Syst Math4 (1995), 65–72. (Chinese).

22.

Shi

F.G.

, A new approach to the fuzzification of matroids, Fuzzy Sets Syst160 (2009), 696–675.

23.

Shi

F.G.

, (L, M)-fuzzy matroids, Fuzzy Sets Syst160 (2009), 2387–2400.

24.

Stefaninia

, Sorinia

and Guerra

M.L.

, Parametric representation of fuzzy numbers and application to fuzzy calculus, Fuzzy Sets Syst157 (2006), 2423–2455.

25.

Stupňanová

, A probabilistic approach to the arithmetics of fuzzy numbers, Fuzzy Sets Syst264 (2015), 64–75.

26.

Xin

and Shi

F.G.

, Rank functions for closed and perfect [0,1]-matroids, Hacettetepe J Math Statist39 (2010), 31–39.

27.

Wang

G.J.

, Theory of L-fuzzy topological spaces, Xi’an: Shanxi Normal University Press (1988). (Chinese).

28.

Wang

G.J.

, Theory of topological molecular lattice, Fuzzy Sets Syst47 (1992), 351–376.

29.

Zadeh

L.A.

, Fuzzy sets, Inform Control8 (1965), 338–353.