T -similarity of fuzzy relations on a complete residuated lattice and its algebraic structures

Abstract

This paper studies T-similarity of L-fuzzy relations and its algebraic structures where L is a complete residuated lattice. L-fuzzy topologies induced by L-fuzzy rough approximation operators are investigated and T-similarity of L-fuzzy relations is first introduced. Next, a characteristic condition for L-fuzzy relations to be T-transitive is established. Finally, algebraic structures on T-similarity of L-fuzzy relations are obtained and an example is given in order to illustrate the practical significance and possible applications of the proposed concepts.

Keywords

Complete residuated lattice

1 Introduction

Rough set theory, proposed by Pawlak [25], is a new mathematical tool for data reasoning. It has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, image processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [26 , 30–32].

The basic structure of rough set theory is an approximation space. Lower and upper approximations can be built based on approximation space. Using these rough approximations, knowledge hidden in information systems may be revealed and expressed in the form of decision rules [26 , 31].

Pawlak’s rough set model are build at equivalence relations. The equivalence relation is too restrictive for many practical applications. For example, in an incomplete information system (U, A), we define a binary relation on U by

SIM (A) = {(x, y) ∈ U × U : ∀ a ∈ A, a (x) = a (y) or a (x) = * or a (y) = *} ,

where * is a missing value. Clearly, SIM (A) is a tolerance relation (i.e., reflexive and symmetric relation) on U. But SIM (A) is not an equivalence relation on U. To address this issue, many interesting and meaningful extensions of Pawlak’s rough sets have been presented. Equivalence relations can be replaced by tolerance relations [37], similarity relations [39], binary relations [19 , 54] and so on.

Over the past few decades, various fuzzy generalizations of rough approximations have been proposed [5 , 45]. The most common fuzzy rough sets are obtained by replacing the crisp relations with fuzzy relations on the universe and crisp subsets with fuzzy sets. Dubois et al. [5] first proposed the concept of rough fuzzy sets and fuzzy rough sets and pointed out that a rough fuzzy set is a special case of a fuzzy rough set. Now, fuzzy rough sets have been used to solve practical problems such as data mining [24], approximate reasoning [32], medical time series and case generation [28].

Two approaches of fuzzy rough set theory are developed as in classical cases. One is the constructive approach in which lower and upper approximation operators are constructed from fuzzy relations. The other one is the axiomatic approach in which the lower and upper approximation operators satisfy a set of axioms are the same as those constructed ones. Morsi et al. [21] provided the axioms for fuzzy rough set model. Wu et al. [45] presented a general framework for the study of fuzzy rough sets in which both constructive and axiomatic approaches were used. Yeung et al. [49] investigated the generalization of fuzzy rough sets and established a connection between fuzzy preorders and saturated Kuratowski fuzzy closure operators satisfying an additional condition.

Topology is a branch of mathematics, whose concepts exist not only in almost all branches of mathematics, but also in many real life applications. Topological structure is an important base for knowledge extraction and processing. Therefore, an interesting and natural research topic in rough set theory is to study the relationship between rough sets and topologies. Many authors studied topological properties of rough sets [11 , 50]. It is known that the pair of lower and upper approximation operators induced by a reflexive and transitive relation is exactly the pair of interior and closure operators of a topology [11, 48]. In the study of topological properties of fuzzy rough sets and fuzzy approximation spaces, Qin et al. [33] investigated the topological properties of fuzzy rough sets and pointed out that there exists a one-to-one correspondence between the set of all preorder fuzzy relations and the set of all fuzzy topologies satisfying (TC) axiom; Hao et al. [8] discussed the relationship between L-fuzzy rough set and L-topology and pointed out that there exists a one-to-one correspondence between the set of all preorder L-fuzzy relations and the set of all Alexandrov L-fuzzy topologies on U. She et al. [38] applied (C₁) and (C₂) axioms to describe the lower and upper approximation operators. Zhou et al. [55], Wu et al. [44] and Zhang et al. [57] studied intuitionistic fuzzy rough sets and intuitionistic fuzzy topologies.

For researching inherent property of fuzzy relations, Li et al. [16] proposed similarity of fuzzy relations based on the fuzzy topologies. This similarity is the mutual relationship between fuzzy relations that can induce the same fuzzy topology. They proved that ever fuzzy relation uniquely similar to some fuzzy preorder relation and obtained algebraic structures on similarity of fuzzy relations. Afterwards, Li et al. [17] generalized these results to T-similarity of fuzzy relations.

The purpose of this paper is to generalize the above results to T-similarity of fuzzy relations on a complete residuated lattice.

The remaining part of this paper is organized as follows. In Section 2, complete residuated lattices and L-fuzzy topology are recalled. In Section 3, L-fuzzy rough approximation operators is investigated. In Section 4, relationships between L-fuzzy relations and L-fuzzy topologies are studied. In Section 5, T-similarity of L-fuzzy relations is introduced and its properties are given. In Section 6, algebraic structures on T-similarity of L-fuzzy relations are obtained.

2 Preliminaries

In this section, we recall some basic concepts about complete residuated lattices and L-fuzzy topology.

Throughout this paper, X denotes a non-empty set which may be infinite, L denotes a complete residuated lattice (L, ∗ , → , ∨ , ∧ , 0, 1) where the partial order of L is ⪯, 2^X denotes the family of all subsets of X and L^X denotes the family of all L-fuzzy sets in X.

2.1 Complete residuated lattices

Definition 2.1. [56] Let ∗ : L × L → L be a mapping. If ∗ satisfies the following conditions:

a ∗ b = b ∗ a,

(a ∗ b) ∗ c = a ∗ (b ∗ c),

a ≤ c, b ≤ d ⇒ a ∗ b ≤ c ∗ d,

a ∗ 1 = a for each a ∈ I,then ∗ is called a triangular norm (for short, t-norm) on L.

Definition 2.2. [56] Let → : L × L → L be a mapping and ∗ a t-norm in L. If for any a, b, c ∈ L, $a \leq b \to c \Leftrightarrow a * b \leq c,$ then → is called residuum of ∗.

Definition 2.3. [43] Let (L, ∨ , ∧ , 0, 1) be a bounded lattice with the greatest element 1 and the smallest element 0, ∗ a t-norm on L and → the residuum of ∗. Then (L, ∗ , → , ∨ , ∧ , 0, 1) is called a residuated lattice.

A residuated lattice is said to be complete if the underlying lattice is complete.

Proposition 2.4. [1, 41] (1) a → b = ∨ {w ∈ L : a ∗ w ≤ b};

(2) b ≤ a → b, 1 → a = a;

(3) a ≤ b implies a → b = 1, b → c ≤ a → c and c → a ≤ c → b;

(4) a → (b → c) = b → (a → c) = (a ∗ b) → c;

(5) a ∗ (⋁ _i∈Γb_i) = ⋁ _i∈Γ (a ∗ b_i);

(6) a → (⋀ _i∈Γb_i) = ⋀ _i∈Γ (a → b_i),

(⋁ _i∈Γa_i) → b = ⋀ _i∈Γ (a → b_i).

2.2 L-fuzzy topology

Definition 2.5. [7] A is called an L-fuzzy set in X, if A is a mapping given by A : X → L.

For a ∈ L, $\bar{a}$ denotes the constant L-fuzzy set on X.

Some relations and operations of L-fuzzy sets are defined as follows [7]: for any A, B ∈ L^X and λ ∈ L,

A = B ⇔ A (x) = B (x) for each x ∈ X;

A ⊆ B ⇔ A (x) ≤ B (x) for each x ∈ X;

(A ∗ B) (x) = A (x) ∗ B (x) for each x ∈ X;

(A → B) (x) = A (x) → B (x) for each x ∈ X;

(A ∩ B) (x)=A (x) ∧ B (x) for each x ∈ X;

(A ∪ B) (x)=A (x) ∨ B (x) for each x ∈ X.

(¬ A) (x) = A (x) →0 for each x ∈ X.

$λ A = \bar{λ} * A (x)$ for each x ∈ X.

If ¬¬ a = a holds for all a ∈ L, then L is called involutive.

An L-fuzzy set is called an L-fuzzy point in U, if it takes the value 0 for each y ∈ U except one, say, x ∈ U. If its value at x is λ (0 < λ ≤ 1), we denote this L-fuzzy point by x_λ, where the point x is called its support and λ is called its height (see [15, 27]). Denote $P (U) = {x_{λ} : x \in U, λ \in L} .$ $Specially, x_{1} (y) = {\begin{matrix} 1, & y = x, \\ 0, & y \neq x . \end{matrix}$

For x_λ and A ∈ L^U, x_λ ∈ A ⇔ x_λ ⊆ A.

Remark 2.6. [1] a → b = ∨ {w ∈ L : a ∗ w ≤ b};

Proposition 2.7. [1, 41] (1) b ≤ a → b, 1 → a = a;

(2) a ≤ b implies a → b = 1;

(3) a → ¬ b = ¬ (a ∗ b);

(4) a ≤ b implies b → c ≤ a → c and c → a ≤ c → b.

Remark 2.8. A = ⋃ _x∈U (A (x) x₁) (A ∈ L^U).

Definition 2.9. [42] τ ⊆ L^X is called an L-fuzzy topology on X, if it satisfies the following conditions:

$\bar{0}, \bar{1} \in τ$ ;

A, B ∈ τ ⇒ A ∩ B ∈ τ;

{A_j : j ∈ J} ⊆ τ ⇒ ⋃ _j∈JA_j ∈ τ.

An L-fuzzy topology τ is called strong [4], if it further satisfies the following conditions:

$\bar{a} * A \in τ$ for all A ∈ τ, a ∈ L;

$\bar{a} \to A \in τ$ for all A ∈ τ, a ∈ L;

An L-fuzzy strong topology τ is called stratified [51], if it further satisfies the following condition:

(1′) $\bar{a} \in τ$ for all a ∈ L.

A stratified L-fuzzy topology τ is called Alexandrov [10], if it also satisfies the following condition:

(2′) {A_j : j ∈ J} ⊆ τ ⇒ ⋂ {A_J : j ∈ J} ∈ τ.

3 L-fuzzy rough approximation operators

Recall that R is an L-fuzzy relation on X if R ∈ L^X×X.

Definition 3.1. [20, 42] Let R be an L-fuzzy relation on X. Then R is called

reflexive, if R (x, x) =1 for any x ∈ X;

symmetric, if R (x, y) = R (y, x) for any x, y ∈ X;

T-transitive, if R (x, z) ≥ R (x, y) ∗ R (y, z) for any x, y, z ∈ X.

T-preorder, if R is reflexive and T-transitive.

Definition 3.2. [35, 38] Let R be an L-fuzzy relation on X. Then (X, R) is called an L-fuzzy approximation space. Based on (X, R), $\underline{R} (A)$ and $\bar{R} (A)$ are defined as follows: $\begin{matrix} \underline{R} (A) (x) & = & ⋀_{y \in X} (R (x, y) \to A (y)) (x \in X), \\ \bar{R} (A) (x) & = & ⋁_{y \in X} (R (x, y) * A (y)) (x \in X) . \end{matrix}$

Then the pair $(\underline{R} (A), \bar{R} (A))$ is called the L-fuzzy rough set of A with respect to (U, R).

In this case, $\underline{R} : L^{X} \to L^{X}$ and $\bar{R} : L^{X} \to L^{X}$ are called the L-fuzzy lower approximation operator and the L-fuzzy upper approximation operator, respectively. In general, we refer to $\underline{R}$ and $\bar{R}$ as the L-fuzzy rough approximation operators.

Remark 3.3. [38] $\bar{R} (x_{1}) (y) = R (y, x), \underline{R} ((x_{1})^{c}) (y) = 1 - R (y, x) (x, y \in U)$ .

Proposition 3.4. [35, 38] Let (X, R) be an L-fuzzy approximation space. Then for any A, B ∈ L^X, {A_j : j ∈ J} ⊆ L^X and a ∈ L,

$\underline{R} (\bar{1}) = \bar{1}$ , $\bar{R} (\bar{0}) = \bar{0}$ .

$A \subseteq B \Rightarrow \underline{R} (A) \subseteq \underline{R} (B)$ , $\bar{R} (A) \subseteq \bar{R} (B)$ .

$\underline{R} (A) \subseteq \neg \bar{R} (\neg A), \bar{R} (A) \subseteq \neg \underline{R} (\neg A), \neg \bar{R} (A) = \underline{R} (\neg A)$ .

$\underline{R} (⋂_{j \in J} A_{j}) = ⋂_{j \in J} \underline{R} (A_{j}), \bar{R} (⋃_{j \in J} A_{j}) = ⋃_{j \in J} \bar{R} (A_{j})$ .

$\underline{R} (\bar{a} \to A) = \bar{a} \to \underline{R} (A), \bar{R} (\bar{a} * A) = \bar{a} * \bar{R} (A)$ .

$\bar{R} (\bar{a}) \subseteq \bar{a} \subseteq \underline{R} (\bar{a})$ .

$\underline{R} (y_{1} \to \bar{a}) = R (x, y) \to a$ , $\bar{R} (y_{1} * \bar{a}) (x) = R (x, y) * a$ .

By Proposition 3.4(3), the lower and upper approximation operators have the certain duality. Thus, we may only consider the lower approximation operator, which is also based on the custom.

Proposition 3.5. Let {R_α : α ∈ Γ} be a family of L-fuzzy relations on X. Then for any A ∈ L^X, we have

$\underline{⋃_{α \in Γ} R_{α}} (A) = ⋂_{α \in Γ} \underline{R_{α}} (A)$ ,

$\bar{⋃_{α \in Γ} R_{α}} (A) = ⋃_{α \in Γ} \bar{R_{α}} (A)$ .

Proof. (1) Let x ∈ X. By Proposition 2.4, $\begin{matrix} (\underline{⋃_{α \in Γ} R_{α}}) (A) (x) & = & ⋀_{y \in U} ((⋃_{α \in Γ} R_{α}) (x, y) \to A (y)) \\ = & ⋀_{y \in U} ((⋁_{α \in Γ} R_{α} (x, y)) \to A (y)) \\ = & ⋀_{y \in U} (⋀_{α \in Γ} (R_{α} (x, y) \to A (y))) \\ = & ⋀_{α \in Γ} (⋀_{y \in U} (R_{α} (x, y) * A (y))) \\ = & ⋀_{α \in Γ} \underline{R_{α}} (A) (x) \\ = & (⋂_{α \in Γ} \underline{R_{α}} (A)) (x) . \end{matrix}$

Hence $\underline{⋃_{α \in Γ} R_{α}} (A) = ⋂_{α \in Γ} \underline{R_{α}} (A) .$

(2) Let x ∈ X. By Proposition 2.4, $\begin{matrix} (\bar{⋃_{α \in Γ} R_{α}}) (A) (x) & = & ⋁_{y \in U} ((⋃_{α \in Γ} R_{α}) (x, y) * A (y)) \\ = & ⋁_{y \in U} ((⋁_{α \in Γ} R_{α} (x, y)) * A (y)) \\ = & ⋁_{y \in U} (⋁_{α \in Γ} (R_{α} (x, y) * A (y))) \\ = & ⋁_{α \in Γ} (⋁_{y \in U} (R_{α} (x, y) * A (y))) \\ = & ⋁_{α \in Γ} \bar{R_{α}} (A) (x) \\ = & (⋃_{α \in Γ} \bar{R_{α}} (A)) (x), \end{matrix}$

Hence $\bar{⋃_{α \in Γ} R_{α}} (A) = ⋃_{α \in Γ} \bar{R_{α}} (A) . □$

Proposition 3.6. Let (X, R) be an L-fuzzy approximation space. Then for any A ∈ L^X,

(1) $\underline{R^{r}} (A) = A$ ⇔ (2) $\underline{R} (A) \supseteq A$ ⇔ (3) $\underline{R^{0}} (A) \supseteq A$ .

Proof. (1) ⇒ (2) and (2) ⇒ (3) are obvious.

(3) ⇒ (1). Let x ∈ X. By Proposition 2.4, we have R⁰ (x, x) → A (x) =0 → A (x) =1 and R^r (x, x) → A (x) =1 → A (x) = A (x). Then $\begin{matrix} ⋀_{y \neq x} (R^{0} (x, y) \to A (y)) & = & ⋀_{y \in X} (R^{0} (x, y) \to A (y)) \\ = & (\underline{R^{0}} (A)) (x) \geq A (x) . \end{matrix}$

Hence $\begin{matrix} \underline{R^{r}} (A) (x) \\ = ⋀_{y \in X} (R^{r} (x, y) \to A (y)) \\ = (R^{r} (x, x) \to A (x)) \land (⋀_{y \neq x} (R^{0} (x, y) \to A (y))) \\ = A (x) \land (⋀_{y \neq x} (R^{0} (x, y) \to A (y))) = A (x) . \end{matrix}$

Thus $\underline{R^{r}} (A) = A$ . □

4 Relationships between L-fuzzy relations and L-fuzzy topologies

In this section, we investigate relationships between L-fuzzy relations and L-fuzzy topologies.

4.1 L-fuzzy topologies induced by L-fuzzy relations

Let R be an L-fuzzy relation on X, we denote $τ_{R} = {A \in L^{X} : A \subseteq \underline{R} (A)} .$

Theorem 4.1. Let R be an L-fuzzy relation on X. Then

τ_R is an Alexandrov L-fuzzy topology on X;

∀ A ∈ L^X, ${int}_{τ_{R}} (A) \subseteq \underline{R} (A)$ ;

If R is reflexive, then $τ_{R} = {A \in L^{X} : \underline{R} (A) = A}$ ;

τ_{R
⁰} = τ_R = τ_{R
^r}.

Proof. (1) By Proposition 3.4, $\bar{a} \in τ_{R}$ for each a ∈ I.

Let {A_α : α ∈ Γ} ⊆ τ_R. Then $A_{α} \subseteq \underline{R} (A_{α})$ for each α ∈ Γ.

By Proposition 3.4,

$⋂_{α \in Γ} A_{α} \subseteq ⋂_{α \in Γ} \underline{R} (A_{α}) = \underline{R} (⋂_{α \in Γ} A_{α})$ ,

$⋃_{α \in Γ} A_{α} \subseteq ⋃_{α \in Γ} \underline{R} (A_{α}) \subseteq \underline{R} (⋃_{α \in Γ} A_{α}) .$

So ⋂_α∈ΓA_α, ⋃ _α∈ΓA_α ∈ τ_R.

Hence τ_R is Alexandrov.

(2) By Proposition 3.4, $\begin{matrix} {int}_{τ_{R}} (A) & = & ⋃ {B \in τ_{R} : B \subseteq A} \\ \subseteq & ⋃ {B \in τ_{R} : \underline{R} (B) \subseteq \underline{R} (A)} \\ = & ⋃ {B \in L^{X} : B \subseteq \underline{R} (B) \subseteq \underline{R} (A)} \\ \subseteq & \underline{R} (A) . \end{matrix}$

(3) This holds by Proposition 3.4.

(4) This holds by Proposition 3.6 and Theorem 4.1(3). □

Definition 4.2. Let R be a reflexive L-fuzzy relation on U. Then τ_R is called the L-fuzzy topology induced by R on U.

Theorem 4.3. Let ρ and R be two L-fuzzy relations on X. Then $ρ \subseteq R \Rightarrow τ_{ρ} \supseteq τ_{R} .$

Proof. Let A ∈ τ_R and x ∈ X, $A \subseteq \underline{R} (A)$ .

By ρ ⊆ R and Proposition 3.4, $\begin{matrix} \underline{R} (A) (x) & = & ⋀_{y \in X} (R (x, y) \to A (y)) \\ \leq & ⋀_{y \in X} (ρ (x, y) \to A (y)) \\ = & \underline{ρ} (A) (x) . \end{matrix}$

Then $\underline{R} (A) \subseteq \underline{ρ} (A)$ . So $A \subseteq \underline{ρ} (A)$ .

This implies A ∈ τ_ρ.

Thus τ_R ⊆ τ_ρ. □

Proposition 4.4. Let {R_α : α ∈ Γ} be a family of L-fuzzy relations on X. Then $τ_{_{\underset{α \in Γ}{\cup} R_{α}}} = \underset{α \in Γ}{\cap} τ_{R_{α}} .$

Proof. By Theorem 4.3, τ_{_{_⋃α∈Γ
R
_α}} ⊆ ⋂ _α∈Γτ_{R
_α}.

Let A ∈ ⋂ _α∈Γτ_{R
_α}. We have $A \subseteq \underline{R_{α}} (A)$ for each α ∈ Γ. By Proposition 3.5, $A \subseteq ⋂_{α \in Γ} \underline{R_{α}} (A) = \underline{⋃_{α \in Γ} R_{α}} (A) .$

So A ∈ τ_{_{_⋃α∈Γ
R
_α}}. Thus τ_{_{_⋃α∈Γ
R
_α}} ⊇ ⋂ _α∈Γτ_{R
_α}.

Hence τ_{_{_⋃α∈Γ
R
_α}} = ⋂ _α∈Γτ_{R
_α}. □

4.2 L-fuzzy relations induced by L-fuzzy topologies

Definition 4.5. Let τ be an L-fuzzy topology on X. Define an L-fuzzy relation R_τ on X by $R_{τ} (x, y) = {cl}_{τ} (y_{1}) (x) (x, y \in X) .$

Then R_τ is called the L-fuzzy relation induced by τ on X.

Theorem 4.6. Let L be involutive, let τ be an L-fuzzy topology on X and let R_τ be the L-fuzzy relation induced by τ on X.

R_τ is reflexive.

If τ is Alexandrov, then $\begin{matrix} \underline{R_{τ}} (A) & \subseteq & {int}_{τ} (A) \subseteq A \subseteq {cl}_{τ} (A) \\ \subseteq & \bar{R_{τ}} (A) (A \in L^{X}) . \end{matrix}$

Proof. (1) For each x ∈ X, $R_{τ} (x, x) = {cl}_{τ} (x_{1}) (x) \geq (x_{1}) (x) = 1 .$

Then R_τ is reflexive.

(2) Let A ∈ L^X. Note that τ is Alexandrov. Then, by Remark 2.8 and Proposition 3.2(2), $\begin{matrix} {cl}_{τ} (A) & = & {cl}_{τ} (⋃_{y \in U} (A (y) y_{1})) = ⋃_{y \in X} {cl}_{τ} (A (y) y_{1}) \\ \subseteq & ⋃_{y \in X} ({cl}_{τ} (\bar{A (y)}) * {cl}_{τ} (y_{1})) \\ = & ⋃_{y \in X} (\bar{A (y)} * {cl}_{τ} (y_{1})) . \end{matrix}$

So for each x ∈ X, $\begin{matrix} {cl}_{τ} (A) (x) & \leq & ⋁_{y \in U} (\bar{A (y)} (x) * {cl}_{τ} (y_{1}) (x)) \\ = & ⋁_{y \in U} (A (y) * R_{τ} (x, y)) \\ = & \bar{R_{τ}} (A) (x) . \end{matrix}$

This implies ${cl}_{τ} (A) \subseteq \bar{R_{τ}} (A)$ .

Note that L is involutive. Then $\begin{matrix} {int}_{τ} (A) & = & \neg {cl}_{τ} (\neg A) \supseteq \neg \bar{R_{τ}} (\neg A) \\ = & \underline{R_{τ}} (\neg \neg A) = \underline{R_{τ}} (A) . \end{matrix}$

Hence

$\underline{R_{τ}} (A) \subseteq {int}_{τ} (A) \subseteq A \subseteq {cl}_{τ} (A) \subseteq \bar{R_{τ}} (A) . □$

Theorem 4.7. Let L be involutive and let R be a preorder L-fuzzy relation on X. Let τ_R be the L-fuzzy topology induced by R on X and let R_{τ
_R} be the L-fuzzy relation induced by τ_R on X. Then R_{τ
_R} = R.

Proof. For each (x, y) ∈ X × U, by Remark 3.3, Proposition 4.1 and Theorem 4.3(3), $\begin{matrix} R (x, y) & = & \bar{R} (y_{1}) (x) = {cl}_{θ_{R}} (y_{1}) (x) \\ = & {cl}_{τ_{R}} (y_{1}) (x) = R_{τ_{R}} (x, y) . \end{matrix}$

Then R_{τ
_R} = R. □

4.3 (C₁) and (C₂) axioms

The following conditions for an L-fuzzy topology τ on X are respectively called (C₁) and (C₂) axioms ([38]): $\begin{matrix} {cl}_{τ} (λ A) = λ {cl}_{τ} (A) (λ \in L, A \in L^{X}), \\ {cl}_{τ} (⋃_{α \in Γ} A_{α}) = ⋃_{α \in Γ} {cl}_{τ} (A_{α}) (A_{α} : α \in Γ} \subseteq L^{X}) . \end{matrix}$

Proposition 4.8. Let τ be an L-fuzzy topology on X. If τ satisfies (C₁) and (C₂) axioms, then R_τ is perorder.

Proof. By Theorem 4.8(1), R_τ is reflexive. For any x, y, z ∈ X, put cl_τ (z₁) (y) = λ. By Remark 2.8, $\begin{matrix} λ {cl}_{τ} (y_{1}) & = & {cl}_{τ} (λ y_{1}) = {cl}_{τ} ({cl}_{τ} (z_{1}) (y) y_{1}) \\ \subseteq & {cl}_{τ} (⋃_{t \in X} ({cl}_{τ} (z_{1}) (t) t_{1})) \\ = & {cl}_{τ} ({cl}_{τ} (z_{1})) = {cl}_{τ} (z_{1}) . \end{matrix}$

Then $\begin{matrix} R_{τ} (x, y) * R_{τ} (y, z) & = & {cl}_{τ} (y_{1}) (x) * {cl}_{τ} (z_{1}) (y) \\ = & {cl}_{τ} (y_{1}) (x) * λ \\ = & (λ {cl}_{τ} (y_{1})) (x) \\ \leq & {cl}_{τ} (z_{1}) (x) = R_{τ} (x, z) . \end{matrix}$

This implies R_τ is transitive.

Thus R_τ is perorder. □

Proposition 4.9. Let τ be an L-fuzzy topology on X. τ satisfies (C₁) and (C₂) axioms if and only if $\bar{R_{τ}}$ is the ∗-closure operator of τ.

Proof. Necessity. Let A ∈ L^X. By Remark 2.8, $\begin{matrix} {cl}_{τ} (A) & = & {cl}_{τ} (⋃_{y \in U} (A (y) y_{1})) \\ = & ⋃_{y \in X} {cl}_{τ} (A (y) y_{1}) \\ = & ⋃_{y \in X} (A (y) {cl}_{τ} (y_{1})) . \end{matrix}$

Then for each x ∈ X, $\begin{matrix} {cl}_{τ} (A) (x) & = & ⋁_{y \in U} (A (y) * {cl}_{τ} (y_{1}) (x)) \\ = & ⋁_{y \in X} (A (y) * R_{τ} (x, y)) \\ = & \bar{R_{τ}} (A) (x) . \end{matrix}$

Thus $\bar{R_{τ}} (A) = {cl}_{τ} (A)$ . So $\bar{R_{τ}}$ is the ∗-closure operator of τ.

Sufficiency. This holds by Proposition 3.2. □

Corollary 4.10. Let L be a complete residuated lattice and τ be an L-fuzzy topology on X, then there exists an L-fuzzy reflexive and transitive relation R on R such that $\bar{R} = {ci}_{τ}^{*}$ if and only if τ satisfies (C₁) and (C₂) axioms.

Proof. This holds by Propositions 4.10 and 4.11. □

Corollary 4.12 is Theorem 6.2 in [38]. But this theorem does not imply Propositions 4.10 and 4.11.

Proposition 4.11. Let L be involutive and let R be a preorder L-fuzzy relation on X. Then τ_R satisfies (C₁) and (C₂) axioms.

Proof. For any λ ∈ L and A ∈ L^X, by Proposition 3.2(5), Proposition 4.1 and Theorem 4.3(3), $\begin{matrix} {cl}_{τ_{R}} (λ A) & = & {cl}_{θ_{R}} (λ A) = \bar{R} (λ A) = λ \bar{R} (A) \\ = & λ {cl}_{θ_{R}} (A) = λ {cl}_{τ_{R}} (A) . \end{matrix}$

For any {A_α : α ∈ Γ} ⊆ L^X, by Proposition 3.2(4), Proposition 4.1 and Theorem 4.3(3), $\begin{matrix} {cl}_{τ_{R}} (⋃_{α \in Γ} A_{α}) & = & \bar{R} (⋃_{α \in Γ} A_{α}) = ⋃_{α \in Γ} \bar{R} (A_{α}) \\ = & ⋃_{α \in Γ} {cl}_{τ_{R}} (A_{α}) . \end{matrix}$

Thus τ_R satisfies (C₁) and (C₂) axioms. □

Theorem 4.12. Let τ be an L-fuzzy topology on X, let R_τ be the L-fuzzy relation induced by τ on X and let τ_{R
_τ} be the L-fuzzy topology induced by R_τ on X. If L is involutive, then

τ_{R
_τ} = τ if and only if τ satisfies (C₁) and (C₂) axiom.

Proof. Necessity. By Theorem 4.8(1), R_τ is reflexive. By Theorem 4.4, τ_{R
_τ} is Alexandrov. Note that τ_{R
_τ} = τ. Then τ is Alexandrov.

For each A ∈ L^X, by Theorems 4.6 and 4.8(2), $\bar{R_{τ}} (A) \subseteq {cl}_{τ_{R_{τ}}} (A) = {cl}_{τ} (A) \subseteq \bar{R_{τ}} (A) .$

Then ${cl}_{τ} = \bar{R_{τ}}$ . By Theorem 3.7(1), R_τ is preorder. By Proposition 4.13 and τ = τ_{R
_τ}, τ satisfies (C₁) and (C₂) axioms.

Sufficiency. By Proposition 4.10, R_τ is perorder. For each A ∈ L^X, by Proposition 4.1, Theorem 4.3(3) and Proposition 4.11,

${cl}_{τ_{R_{τ}}} (A) = {cl}_{θ_{R_{τ}}} (A) = \bar{R_{τ}} (A) = {cl}_{τ} (A) .$

Then τ_{R
_τ} = τ. □

Theorem 4.13. Let $Σ = {R : R is a preorder L - fuzzy relation on U}$ and $\begin{matrix} Γ & = & {τ : τ is an L - fuzzy topology on U \\ satisfying (C_{1}) and (C_{2}) axioms} . \end{matrix}$

If L is involutive, then there exists a one-to-one correspondence between Σ and Γ.

Proof. Two mappings f : Σ → Γ and g : Γ → Σ are defined as follows: $\begin{matrix} f (R) & = & τ_{R} (R \in Σ), \\ g (τ) & = & R_{τ} (τ \in Γ) . \end{matrix}$

By Theorem 4.9, $g \circ f = i_{Σ},$ where g ∘ f is the composition of f and g, and i_Σ is the identity mapping on Γ.

By Theorem 4.14, $f \circ g = i_{Γ},$ where f ∘ g is the composition of g and f, and i_Γ is the identity mapping on Σ.

Hence f and g are two one-to-one correspondences. This prove that there exists a one-to-one correspondence between Σ and Γ. □

5 T-similarity of L-fuzzy relations

Let ρ and R be two L-fuzzy relations on X. Denote $(ρ ⊙ R) (x, y) = ⋁_{t \in X} (ρ (x, t) * R (t, y)) (x, y \in X) .$

5.1 Some results of T-similarity of L-fuzzy relations

Definition 5.1. Let ρ and R be two L-fuzzy relations on X. ∗ is a t-norm on L. If τ_ρ = τ_R, then ρ is called T-similar to R. We write ρ ∼ _{_T}R.

Remark 5.2. ∼_{_T} is an equivalence relation.

Let R be an L-fuzzy relation on X. Denote $[R] = {ρ \in I^{X \times X} : ρ \sim_{T} R},$

Theorem 5.3. Let R be an L-fuzzy relation on X. Then

R^o, R^r ∈ [R].

If {ρ_α : α ∈ Γ} ⊆ [R], then ⋃_α∈Γρ_α ∈ [R].

If ρ, λ ∈ [R] and ρ ⊆ δ ⊆ λ, then δ ∈ [R].

Proof. (1) This holds by Theorem 4.1.

(2) This holds by Proposition 4.4.

(3) By ρ ⊆ δ ⊆ λ and Theorem 4.3, $τ_{R} = τ_{λ} \subseteq τ_{δ} \subseteq τ_{ρ} = τ_{R} .$

Then τ_δ = τ_R and so δ ∈ [R]. □

Let R be an L-fuzzy relation on X. For x, y, u, v, w ∈ X, denote $\begin{matrix} R^{uvw} (x, y) \\ = {\begin{matrix} R (u, v) \lor (R (u, w) * R (w, v)), & x = u, y = v \\ R (x, y), & x \neq u or y \neq v; \end{matrix} \end{matrix}$ R^uv = ⋃ _w∈XR^uvw.

Proposition 5.4. Let R be an L-fuzzy relation on X. Then for any u, v . w ∈ X, R^uvw, R^uv ∈ [R].

Proof. (1) 1) If u = v, we have 0 ≤ R^uvw (u, v) ≤1. Then $R^{0} \subseteq R^{uvw} \subseteq R^{r} .$

By Theorem 5.3, R^uvw ∈ [R].

2) If u ≠ v, we have R^uvw (u, v) = R (u, w) ∗ R (w, v) and R ⊆ R^uvw. Let A ∈ τ_R. Then $A \subseteq \underline{R} (A)$ . We have $\begin{matrix} R (w, v) \to A (v) \\ \geq ⋀_{y \in X} (R (w, y) \to A (y)) = \underline{R} (A) (w) \geq A (w), \\ R (u, w) \to A (w) \\ \geq ⋀_{y \in X} (R (u, y) \to A (y)) = \underline{R} (A) (u) \geq A (u), \\ ⋀_{y \neq v} (R (u, y) \to A (y)) \\ \geq ⋀_{y \in X} (R (u, y) \to A (y)) = \underline{R} (A) (u) \geq A (u) . \end{matrix}$

By Proposition 2.4, $\begin{matrix} R^{uvw} (u, v) \to A (v) \\ = (R (u, w) * R (w, v)) \to A (v) \\ = R (u, w) \to (R (w, v) \to A (v)) \\ \geq R (u, w) \to A (w) \geq A (u) . \end{matrix}$

So $\underline{R^{uvw}} (A) (u) = ⋀_{y \in X} (R (u, y) \to A (y)) \geq A (u) .$

Since $\underline{R^{uvw}} (A) (x) = \underline{R} (A) (x) \geq A (x)$ for any x ∈ X - {u}, we have $A \subseteq \underline{R^{uvw}} (A)$ .

Then A ∈ τ_{R
^uvw}.

Thus τ_R ⊆ τ_{R
^uvw}.

By R ⊆ R^uvw and Theorem 4.3, τ_R ⊇ τ_{R
^uvw}.

Therefore τ_R = τ_{R
^uvw}.

This shows R^uvw ∈ [R].

(2) By (1) and Theorem 5.3, R^uv ∈ [R]. □

Denote R₀ = R. R_n (n ∈ ω) are defined as follows: $R_{n + 1} = R_{n} ⊙ (R_{n})^{r} .$

Obviously, R_n+1 = ⋃ _u,v∈X (R_n) ^uv where $\begin{matrix} (R_{n})^{uv} (x, y) & = & {\begin{matrix} (R_{n} ⊙ R_{n}) (u, v), & x = u, y = v, \\ R_{n} (x, y), & x \neq u or y \neq v; \end{matrix} \\ R_{n} (x, y) & = & R_{n - 1} (x, y) \lor ((R_{n - 1} ⊙ R_{n - 1}) (x, y)) . \end{matrix}$

Theorem 5.5. Let R be an L-fuzzy relation on X. Then $R is T - transitive \Leftrightarrow R = R_{1} .$

Proof. Necessity. Obviously.

Sufficiency. Suppose R = R₁. Let x, y ∈ X. Then $R (x, y) = R_{1} (x, y) = ⋁_{t \in X - {x, y}} (R (x, t) * R (t, y)) .$

Note that R (x, y) ≥ R (x, x) ∗ R (x, y) and R (x, y) ≥ R (x, y) ∗ R (y, y). Then $R (x, y) \geq ⋁_{t \in X} (R (x, t) * R (t, y)) .$

Thus, R is T-transitive. □

Remark 5.6. (1) {R_n}↑.

(2) ∀ n ∈ N, R_n ∈ [R].

(3) ρ ⊆ R ⇒ ρ_n ⊆ R_n, n ∈ ω.

(4) If R is T-transitive, then for any n ∈ ω, R_n = R.

Proposition 5.7. Let R be an L-fuzzy relation on X. If ρ, λ ∈ [R], then $ρ ⊙ λ^{r}, ρ^{r} ⊙ λ, ρ^{r} ⊙ λ^{r} \in [R] .$

Proof. By Theorem 5.3 and Remark 5.6, (ρ ∪ λ) ₁ ∈ [R]. Note that ρ ⊆ ρ ⊙ λ^r ⊆ (ρ ∪ λ) ₁. Then ρ ⊙ λ^r ∈ [R].

Similarly, ρ^r ⊙ λ ∈ [R]. Note that ρ^r, λ ∈ [R]. Then ρ^r ⊙ λ^r ∈ [R]. □

5.2 The preorder expression and the transmitting expression of R

[R] ^t = {ρ ∈ [R] : ρ ⊇ R and ρ is T-transitive } .

Definition 5.8. Let R be an L-fuzzy relation on X. Then $R^{t} = ⋃_{n = 0}^{\infty} R_{n}$ is called the transmitting expression of R and $R^{p} = ⋃_{n = 0}^{\infty} (R^{r})_{n}$ is called the T-preorder expression of R.

Theorem 5.9. Let R be an L-fuzzy relation on X. Then

R ⊆ R^t ⊆ R^p.

R^p = (R^r) ^t = (R^t) ^r = R^t ∪ R^r.

R^t is T-transitive; R^p is T-preorder.

R^t, R^p ∈ [R] ^t.

R^t = ⋂ [R] ^t.

Proof. (1) This holds by Remark 5.6.

(2) This proof is obvious.

(3) By Remark 5.6, {R_n}↑. Then $\begin{matrix} R^{t} (x, y) & = & ⋃_{n = 1}^{\infty} R_{n} (x, y) \\ = & ⋁_{n = 1}^{\infty} (⋁_{t \in X - {x, y}} (R_{n - 1} (x, t) * R_{n - 1} (t, y))) \\ = & ⋁_{t \in X - {x, y}} (⋁_{n = 1}^{\infty} (R_{n - 1} (x, t) * R_{n - 1} (t, y))) \\ = & ⋁_{t \in X - {x, y}} (R^{t} (x, t) * R^{t} (t, y)) . \end{matrix}$

Note that R^t (x, y) ≥ R^t (x, x) ∗ R^t (x, y) and R^t (x, y) ≥ R^t (x, y) ∗ R^t (y, y). Then R^t (x, y) ≥ ⋁ _t∈X (R^t (x, t) ∗ R^t (t, y)).

Thus, R^t is T-transitive. Note that R^p = (R^r) ^t and R are reflexive. Then R^p is T-preorder.

(4) This holds by Theorem 5.9(3).

(5) This is obvious. □

Proposition 5.10. Let R be an L-fuzzy relation on X. Then $\begin{matrix} R^{t} (x, y) \\ = {\begin{matrix} R^{p} (x, y), & x \neq y, \\ R (x, y) \lor (⋁_{z \neq x} (R^{p} (x, z)) * R^{p} (z, y)), & x = y . \end{matrix} \end{matrix}$

Proof. Pick $\begin{matrix} λ (x, y) \\ = {\begin{matrix} R^{p} (x, y), & x \neq y, \\ R (x, y) \lor ⋁_{z \neq x} T (R^{p} (x, z), R^{p} (z, y)), & x = y . \end{matrix} \end{matrix}$

Obviously, (R^p) ⁰ ⊆ λ ⊆ R^p, λ is T-transitive.

By Theorem 5.3(3), λ ∈ [R].

Note that λ ⊇ R. Then λ ∈ [R] ^t.

Let x, y ∈ X. If x ≠ y, by (1), (R^p) ⁰ (x, y) ≤ ρ (x, y) ≤ R^p (x, y) for any ρ ∈ [R] ^t.

Then

ρ (x, y) = R^p (x, y),

$\begin{matrix} R^{t} (x, y) & = & ⋀_{ρ \in [R]^{t}} ρ (x, y) = ⋀_{ρ \in [R]^{t}} R^{p} (x, y) \\ = & R^{p} (x, y) = λ (x, y) . \end{matrix}$

If x = y, ∀ ρ ∈ [R] ^t and z ≠ x, by (1), we have $\begin{matrix} (R^{p})^{0} (x, z) \subseteq ρ (x, z) \subseteq R^{p} (x, z), \\ (R^{p})^{0} (z, y) \subseteq ρ (z, y) \subseteq R^{p} (z, y) . \end{matrix}$

Then ρ (x, z) = R^p (x, z) and ρ (z, y) = R^p (z, y).

Note that ρ is T-transitive. Then $\begin{matrix} ρ (x, y) & \geq & ⋁_{z \neq x} T (ρ (x, z), ρ (z, y)) \\ = & ⋁_{z \neq x} T (R^{p} (x, z), R^{p} (z, y)) \\ = & λ (x, y) . \end{matrix}$

So R^t (x, y) = ⋀ _ρ∈[R]^tρ (x, y) ≥ λ (x, y).

Since λ ∈ [R] ^t, we have R^t (x, y) = ρ (x, y).

Thus R^t (x, y) = λ (x, y).

This implies that R^t = λ. □

5.3 The first and second kinds of transformation for R

Definition 5.11. Let ρ and R be two L-fuzzy relations on X.

ρ is called the first kind of transformation for R, if R⁰ ⊆ ρ ⊆ R^r.

ρ is called the second kind of transformation for R, if ρ ∈ [R] and ρ (x, x) = R (x, x) (x ∈ X).

Remark 5.12. (1) R^r is the first kind of transformation for R.

(2) If ρ is the first kind of transformation for R, then ρ ∈ [R].

(3) If u ≠ v, then R^uvw, R^uv are the second kind of transformation for R.

Denote

Remark 5.13. (1) R⁰, R^r ∈ [R] ¹.

(2) If ρ ∈ [R] ¹, then ρ ∈ [R].

(3) If k ≠ h, then R^khl, R^kh ∈ [R] ².

(4) R ∈ [R] ¹, R ∈ [R] ².

(5) If {ρ_α : α ∈ Γ} ⊆ [R] ¹, then ⋃_α∈Γρ_α ∈ [R] ¹.

(6) If {ρ_α : α ∈ Γ} ⊆ [R] ², then ⋃_α∈Γρ_α ∈ [R] ².

6 Algebra structures on T-similarity of L-fuzzy relations

Let R be an L-fuzzy relation on X. Denote $\begin{matrix} [R]^{r} & = & {ρ^{r} : ρ \in [R]}, [R]^{0} = {ρ^{0} : ρ \in [R]}, \\ [R]^{1} & = & {ρ : ρ is the first kind of \\ transformation for R}, \\ [R]^{2} & = & {ρ : ρ is the second kind of \\ transformation for R}, \\ [R]^{t} & = & {ρ \in [R] : ρ is T - transitive} . \end{matrix}$

Theorem 6.1. (1) ([R] , ∪) is a commutative semigroup.

(2) ([R] ⁰, ∪) and ([R] ^r, ∪) are two sub-semigroups of ([R] , ∪).

Proof. This holds by Proposition 4.4. □

Corollary 6.2. Let R be an L-fuzzy relation on X. Then ([R] ^r, ⊙) is a commutative semigroup.

Proof. This holds by Proposition 5.7. □

Theorem 6.3. Let R be an L-fuzzy relation on X. Then ([R] ¹, ∪) and ([R] ², ∪) are two sub-semigroups of ([R] , ∪).

Proof. This holds by Remark 5.13. □

Theorem 6.4. Let R be an L-fuzzy relation on X. Then

([R] , ∪) is a commutative semigroup.

([R] ⁰, ∪), ([R] ^r, ∪), ([R] ¹, ∪) and ([R] ², ∪) are four sub-semigroups of ([R] , ∪).

Proof. This holds by Theorem 5.3. □

Theorem 6.5. Let R be an L-fuzzy relation on X. Then ([R] ^r, ⊙) is a commutative semigroup.

Proof. This holds by Proposition 5.7. □

In order to illustrate the practical significance and possible applications of the proposed concepts, we give the following example.

Example 6.6. Let X = {x₁, x₂, x₃} be a universe consisting of three persons as possible communication. We consider their communication circumstances. For convenience, we take i class emails to express all emails that written by x_i (i = 1, 2, 3). Let T (a, b) = a × b.

(1) p_ij denotes the probability that x_i sends the emails to x_j (i, j = {1, 2, 3}). It can be expressed as an L-fuzzy relation on X and be denoted by R. Then $R = (p_{ij})_{3 \times 3} = (\begin{matrix} 0 & 0 & 0.3 \\ 0.6 & 0 & 1 \\ 0 & 0.9 & 0.1 \end{matrix})$ and so $R (x_{i}, x_{j}) = p_{ij}, i, j \in {1, 2, 3} .$

Since $R^{uv} = {\begin{matrix} (\begin{matrix} 0 & 0.27 & 0.3 \\ 0.6 & 0 & 1 \\ 0 & 0.9 & 0.1 \end{matrix}), u = 1, v = 2 \\ (\begin{matrix} 0 & 0 & 0.3 \\ 0.6 & 0.9 & 1 \\ 0 & 0.9 & 0.1 \end{matrix}), u = 2, v = 2 \\ (\begin{matrix} 0 & 0 & 0.3 \\ 0.6 & 0 & 1 \\ 0.54 & 0.9 & 0.1 \end{matrix}), u = 3, v = 1 \\ (\begin{matrix} 0 & 0 & 0.3 \\ 0.6 & 0 & 1 \\ 0 & 0.9 & 0.9 \end{matrix}), u = 3, v = 3 \\ R, other \end{matrix},$ we have $\begin{matrix} R_{1} = ⋃_{u, v \in X} R^{uv} = (\begin{matrix} 0 & 0.27 & 0.3 \\ 0.6 & 0.9 & 1 \\ 0.54 & 0.9 & 0.9 \end{matrix}) . \\ (R_{1})^{uv} = {\begin{matrix} (\begin{matrix} 0.162 & 0.27 & 0.3 \\ 0.6 & 0.9 & 1 \\ 0.54 & 0.9 & 0.9 \end{matrix}), u = 1, v = 1 \\ R_{1}, other \end{matrix}, \end{matrix}$ we have $R_{2} = ⋃_{u, v \in X} (R_{1})^{uv} = (\begin{matrix} 0.162 & 0.27 & 0.3 \\ 0.6 & 0.9 & 1 \\ 0.54 & 0.9 & 0.9 \end{matrix}) .$

Note that (R₂) ^uv = R₂ (u, v ∈ {1, 2, 3}). Then R₃ = ⋃ _u,v∈X (R₂) ^uv = R₂. So R₂ is T-preorder. Hence $\begin{matrix} R^{p} = R_{2} \cup R^{r} = (\begin{matrix} 1 & 0.27 & 0.3 \\ 0.6 & 1 & 1 \\ 0.54 & 0.9 & 1 \end{matrix}), \\ R^{t} = (\begin{matrix} 0.162 & 0.27 & 0.3 \\ 0.6 & 0.9 & 1 \\ 0.54 & 0.9 & 0.9 \end{matrix}) . \end{matrix}$

(2) (i) Let l_ij be the maximum probability that x_j receives the emails of i class by only a way (i, j = {1, 2, 3}). Then l_ij = R₃ (x_i, x_j) and so $(l_{ij})_{3 \times 3} = R_{2} .$

(ii) Let $l_{ij}^{'}$ be the minimum probability that x_j receives the emails of i class by any way (i, j = 1, 2, 3. Then $\begin{matrix} l_{ij}^{'} & = & 1 - (1 - l_{ij}) \times (\prod_{t \in M - {i, j}} (1 - l_{it} \times l_{tj})) \end{matrix}$ and so $(l_{ij}^{'})_{3 \times 3} = (\begin{matrix} 0.4115 & 0.4671 & 0.4890 \\ 0.8160 & 0.9916 & 1 \\ 0.7884 & 0.9146 & 0.9916 \end{matrix}) .$

(3) (i) Let q_ij be the maximum probability that x_j can see the emails of i class by only a way (i, j = {1, 2, 3}). Then q_ij = R^p (x_i, x_j) and so $(q_{ij})_{3 \times 3} = R^{p} .$

(ii) Let $q_{ij}^{'}$ be the minimum probability that x_j can see the emails of i class by any way (i, j = 1, 2, 3). Then $\begin{matrix} q_{ij}^{'} & = & 1 - (1 - q_{ij}) \times (\prod_{t \in M - {i, j}} (1 - q_{it} \times q_{tj})) \end{matrix}$ and so $(q_{ij}^{'})_{3 \times 3} = (\begin{matrix} 1 & 0.4671 & 0.4890 \\ 0.8160 & 1 & 1 \\ 0.7884 & 0.9146 & 1 \end{matrix}) .$

7 Conclusions

In this paper, T-similarity of L-fuzzy relations based on L-fuzzy topologies induced by L-fuzzy relations is proposed. Variations of the same L-fuzzy relation are investigated. A characteristic condition for L-fuzzy relations to be T-transitive is established. Algebraic structures on T-similarity of L-fuzzy relations are given. We believe that T-similarity of L-fuzzy relations will be helpful in studying L-fuzzy relations. In future work, we will consider concrete applications of our results.

Footnotes

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their valuable comments and suggestions which have helped immensely in improving the quality of the paper. This work is supported by Guangxi University Science and Technology Research Project (KY2015YB266), Given Point on Master of Applied Statistics in Guangxi University of Finance and Economics (2016TJYB03), Quantitative Economics Key Laboratory Program of Guangxi University of Finance and Economics (2014SYS11) and Guangxi Province Universities and Colleges Excellence Scholar and Innovation Team Funded Scheme.

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T -similarity of fuzzy relations on a complete residuated lattice and its algebraic structures

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 Complete residuated lattices

2.2 L-fuzzy topology

3 L-fuzzy rough approximation operators

4 Relationships between L-fuzzy relations and L-fuzzy topologies

4.1 L-fuzzy topologies induced by L-fuzzy relations

4.2 L-fuzzy relations induced by L-fuzzy topologies

4.3 (C1) and (C2) axioms

5 T-similarity of L-fuzzy relations

5.1 Some results of T-similarity of L-fuzzy relations

5.2 The preorder expression and the transmitting expression of R

5.3 The first and second kinds of transformation for R

6 Algebra structures on T-similarity of L-fuzzy relations

7 Conclusions

Footnotes

Acknowledgments

References

4.3 (C₁) and (C₂) axioms