Roughness in m -semilattices 1

Abstract

In this paper, we consider roughness in m-semilattices, which combine the ∨-semilattice structure and the operation · of a semigroup. Firstly, we propose some kinds of equivalence relations on m-semilattices and investigate the properties of Pawlak rough sets w.r.t. them. Secondly, by replacing equivalence classes in Pawlak rough approximations with neighborhoods, we introduce and study minimal neighborhood approximation operators on m-semilattices, and obtain some order properties of the set of all fixed points of minimal neighborhood approximation operators. Finally, we present the definition of fuzzy rough sets based on fuzzy coverings of m-semilattices, discuss some fundamental properties of fuzzy rough sets and investigate relations between fuzzy rough sets and fuzzy coverings.

Keywords

Pawlak rough set minimal neighborhood approximation operator fuzzy covering fuzzy rough set

1 Introduction

Rough set theory, a new mathematical approach to deal with inexact and uncertain knowledge in information systems, was originally introduced by Pawlak [16]. It has turned out to be fundamentally important in fields such as knowledge acquisition, decision analysis and pattern recognition. Rough set theory is an extension of set theory, in which a subset of a universe is described by a pair of ordinary sets called the lower and upper approximations. Inspired by the construction of Pawlak rough set algebras and the investigation in algebraic properties of rough sets [10 , 27], many researchers have applied notions and methods of rough set theory to various algebraic structures [1 , 25] and lattice structures [3 , 28]. In Pawlak’s original proposal, indiscernibility is modeled by an equivalence relation. Since then, many generalizations of rough set theory have been proposed, one of which is to replace the partition obtained by an equivalence relation with a covering [21 , 33]. Unlike in classical rough set theory, there is no unique way to define lower and upper approximation operators in covering based rough set theory. For example, Y.Y. Yao and B.X. Yao [29] considered twenty pairs of covering based lower and upper approximation operators. Based on the operators discussed in [23 , 29], M. Restrepo et al. [20] introduced a partial order “is finer than” among the pairs of approximation operators, studied this order relation and concluded that $((H_{5}^{ℂ})^{\partial}, H_{5}^{ℂ})$ and $({\underline{apr}}_{N_{1}}, {\bar{apr}}_{N_{1}})$ are not comparable w.r.t. this partial order, but they produce the finest approximations. In [7], Dubois and Prade combined fuzzy sets and rough sets in a fruitful way by defining rough fuzzy sets and fuzzy rough sets. By now, fuzzy rough sets have received wide attention both on practical applications [11, 31] and on theory as well [14 , 24]. For example, Deng et al. [5] fuzzified the general rough approximation operators, presented two kinds of approximation operators based on fuzzy coverings and studied the link between them. However, in the theoretical study of fuzzy rough approximations, much attentions have been paid to set approximations induced by fuzzy relations, while little work has been done on extensions to the universe. Hence, it is of interest to extend the universe to much wider classes of mathematical objects. Then in [13, 14], the properties of (ϑ, T)-fuzzy rough approximation operators on a semigroup and a ring were respectively studied.

Motivated by the studies of roughness in algebraic systems and partially ordered sets such as semigroups, rings, lattices and quantales, we shall consider roughness in m-semilattices, and the research methods mentioned above can also be used in the field of m-semilattices. In fact, there are abundant contents in the structure of m-semilattices, and m-semilattices can be regarded as generalizations of residuated lattices, frames, quantales and lattice-ordered semigroups. Moreover, Rosenthal pointed out in [22] that each coherent quantale is isomorphic to a quantale consisting of all ∨-semilattice ideals of an m-semilattice with a top element. In this paper, we investigate some properties of Pawlak rough set in m-semilattices by proposing some kinds equivalence relations. Compared with $((H_{5}^{ℂ})^{\partial}, H_{5}^{ℂ})$ , we think $({\underline{apr}}_{N_{1}}, {\bar{apr}}_{N_{1}})$ is much more general, so we will investigate properties of $({\underline{apr}}_{N_{1}}, {\bar{apr}}_{N_{1}})$ on m-semilattices. In particular, the properties of fixed points of ${\bar{apr}}_{N_{1}}$ provide more ways to obtain algebraic lattices. Then we introduce fuzzy rough sets based on fuzzy coverings and study properties of fuzzy rough sets within the framework of m-semilattices.

This paper is organized as follows. In Section 2, we recall some basic notions and results which will be used in the paper. In Section 3, some kinds of equivalence relations are introduced and some relative properties of Pawlak rough sets induced by them are discussed. The minimal neighborhood approximation operators are studied on m-semilattices in Section 4. In Section 5, the properties of fuzzy rough sets based on fuzzy coverings are investigated on m-semilattices.

2 Preliminaries

An m-semilattice is a ∨-semilattice (S, ∨) equipped with an associative binary operation “·” such that the operation “·” distributes over ∨ on the left and the right.

An m-semilattice S is called negative if a · b ≤ a and a · b ≤ b for every a, b ∈ S. In the dual case, we call it positive. In the sequel, unless otherwise stated, S always denotes an m-semilattice and a · b is simply written as ab for convenience.

An equivalence θ on S is called an m-semilattice congruence (or congruence for brevity) on S if for all a, b, c, d ∈ S, (a, b) ∈ θ and (c, d) ∈ θ imply (ac, bd) ∈ θ and (a ∨ c, b ∨ d) ∈ θ. For x ∈ S, let [x] _θ denote the equivalence (congruence) class of x. Con(S) denotes the set of all congruences on S. Then Con(S) is a complete lattice with respect to set inclusion.

Definition 2.1. Let I be a subset of S. Consider the following conditions: (i) a, b ∈ I ⇒ a ∨ b ∈ I for all a, b ∈ S; (ii) b ≤ a ∈ I ⇒ b ∈ I for all a, b ∈ S; (iii) b ∈ I ⇒ ab ∈ I and ba ∈ I for all a, b ∈ S, I is called a ∨-semilattice ideal of S if it satisfies (i) and (ii). If I satisfies (i), (ii) and (iii), then I is called an m-ideal (or ideal for brevity) of S. An ideal P of S is called prime if P ≠ S and for all a, b ∈ S, ab ∈ P implies a ∈ P or b ∈ P.

By Definition 2.1, we know that the empty set ∅ is a prime ideal of S, and if {I_λ} _λ∈Λ is a family of ideals, then ⋂_λ∈ΛI_λ is an ideal of S. We denote by Idl(S) the set of all ideals of S.

Definition 2.2. Let S₁ and S₂ be two m-semilattices and f : S₁ → S₂ be a map. If ∀a, b ∈ S, f (a ∨ b) = f (a) ∨ f (b) and f (ab) = f (a) f (b), then f is called an m-semilattice homomorphism.

In this paper, we will assume that U is a non-empty set, and $P$ (U) represents the powerset of U. A pair (U, θ) is called an approximation space, where θ is an equivalence on U. For a subset A ⊆ U, the θ-upper and θ-lower Pawlak rough approximation of A are respectively defined as $\bar{θ} (A) = {x \in U : [x]_{θ} \cap A \neq \emptyset}$ and $\underline{θ} (A) = {x \in U : [x]_{θ} \subseteq A}$ . It is easy to verify that for all $A \subseteq U, \underline{θ} (A) \subseteq A \subseteq \bar{θ} (A)$ .

Definition 2.3. (Zhu [32]). Let $ℂ$ = {K_i} be a family of nonempty subsets of U. $ℂ$ is called a covering of U if ∪K_i = U. The pair $(U, ℂ)$ is called a covering approximation space.

It is clear that a partition generated by an equivalence on U is a special case of a covering of U, so the concept of a covering is an extension of a partition.

Definition 2.4. (Yao and Yao [29]). If $ℂ$ is a covering of U, a neighborhood system for x ∈ U, 𝒞 $(ℂ, x)$ is defined by 𝒞 $(ℂ, x) = {K \in ℂ : x \in K}$ . Clearly, 𝒞 $(ℂ, x)$ is a subset of $ℂ$ .

Definition 2.5. (Yao and Yao [29]). A mapping $N : U \to P (U)$ is called a neighborhood operator. If x ∈ N (x) for all x ∈ U, then N is called a reflexive neighborhood operator.

Definition 2.6. (Bonikowski and Brynarski [2]). Let $(U, ℂ)$ be a covering approximation space and x ∈ U. The set $md (ℂ, x) = {K \in$ 𝒞 $(ℂ, x) : (\forall S \in$ 𝒞 $(ℂ, x)) S \subseteq K \Rightarrow K = S}$ is called the minimal description of the object x.

Yao and Yao [29] proposed four different neighborhood operators: N_i (i = 1, 2, 3, 4), in which $N_{1} (x) = \cap {K : K \in md (ℂ, x)}$ and the set N₁ (x) is called the minimal neighborhood of x. [29] pointed out that by replacing the equivalence class [x] _θ in Pawlak rough approximations with N₁ (x), a pair of approximation operators is defined as follows: ${\underline{apr}}_{N_{1}} (A) = {x \in U : N_{1} (x) \subseteq A}, {\bar{apr}}_{N_{1}} (A) = {x \in U : N_{1} (x) \cap A \neq \emptyset} .$ In this paper, N₁ is simply written as N for convenience, and we call the pair $({\underline{apr}}_{N}, {\bar{apr}}_{N})$ minimal neighborhood approximation operator. From [29] we know that if N is a reflexive neighborhood operator, then $\forall A \in P$ (U), ${\underline{apr}}_{N} (A) \subseteq A \subseteq {\bar{apr}}_{N} (A)$ .

Throughout this paper, L always denotes a complete lattice with its bottom element 0 and top element 1, and a fuzzy subset of U is defined as a map from U to L, which is a generalization of the notion of Zadeh’s fuzzy sets [30]. Let L^U = {f : f : U → L}. For f, g ∈ L^U, by f ⊆ g we mean that f (x) ≤ g (x) (∀ x ∈ U), the union and intersection of f and g are defined as fuzzy subsets of U by (f ∪ g) (x) = f (x) ∨ g (x) , (f ∩ g) (x) = f (x) ∧ g (x) for all x ∈ U. Then L^U with the order ⊆ forms a complete lattice.

Definition 2.7. (Deng et al. [5]). A binary logical operator ∗ : L × L → L is called a fuzzy conjunction if it is non-decreasing in both arguments and satisfies 0 ∗ 1 =1 ∗ 0 =0 and 1 ∗ 1 =1. A fuzzy implication → is a binary logical operator from L × L to L that is non-increasing in its first argument, non-decreasing in its second, and satisfies 0 → 0 =1 → 1 =1 and 1 → 0 =0. The pair (∗ , →) is called an adjunction if ∀a, b, c ∈ L, a ∗ b ≤ c ⇔ b ≤ a → c.

A t-norm ∗ is a commutative and associative fuzzy conjunction satisfying condition (P):1 ∗ l = l for all l ∈ L.

Deng and Heijmans [6] pointed out that if (∗ , →) is an adjunction on L, then ∀a ∈ L, {l_i} _i∈I ⊆ L, a ∗ ⋁ _i∈Il_i = ⋁ _i∈I (a ∗ l_i) and a → ⋀ _i∈Il_i = ⋀ _i∈I (a → l_i). Clearly, if (∗ , →) is an adjunction and the condition (P) holds, then ∀a, b ∈ L, a ≤ b ⇔ a → b = 1.

Throughout this paper, we assume that every conjunction means a fuzzy conjunction and every implication means a fuzzy implication when no confusions can arise. Clearly, if ∗ is a conjunction and → is an implication on L, then ∀l ∈ L, l ∗ 0 =0 ∗ l = 0 and 0 → l = l → 1 =1.

Definition 2.8. (Deng et al. [5]). Let ∗ be a conjunction on L. A fuzzy relation R on U is a fuzzy subset in L^U×U. If ∀x ∈ U, R (x, x) =1, then R is called reflexive; if ∀x, y ∈ U, R (x, y) = R (y, x), then R is called symmetric; if ∀x, y, z ∈ U, R (x, y) ∗ R (y, z) ≤ R (x, z), then R is called ∗-transitive. If R is reflexive, symmetric and ∗-transitive, then R is called a fuzzy ∗-similarity relation.

Definition 2.9. (Deng et al. [5]). A fuzzy covering of U is a collection of fuzzy subsets Φ ⊆ L^U satisfying: (1) every fuzzy subset f ∈ Φ is non-null, i.e., f≠ ∅; (2) ∀x ∈ U, ⋁ _f∈Φf (x) >0.

Given a reflexive fuzzy relation R, let [x] _R (y) = R (x, y), then the collection Φ = {[x] _R : x ∈ U} is a fuzzy covering of U. Conversely, given a fuzzy covering Φ = {f_i} _i∈I, let R (x, y) = ⋁ _i∈I (f_i (x) ∗ f_i (y)), then R is a fuzzy relations derived from Φ.

Let Φ be a fuzzy covering of U and R be a fuzzy relation derived from Φ. The pair (L^U, Φ) or (L^U, R) is called a covering fuzzy rough universe.

For definitions not given in this paper, please refer to [9, 10].

3 Pawlak rough sets in m-semilattices

Definition 3.1. Let θ be an equivalence on S.

(i) If θ satisfies {ab : a ∈ [x] _θ, b ∈ [y] _θ} ⊆ [xy] _θ for all x, y ∈ S, then θ is called compatible with ·;

(ii) If θ satisfies {a ∨ b : a ∈ [x] _θ, b ∈ [y] _θ} ⊆ [x ∨ y] _θ for all x, y ∈ S, then θ is called compatiblewith ∨;

(iii) If θ satisfies (x, ab) ∈ θ ⇒ ∃ x_a, x_b ∈ S such that x = x_ax_b, (x_a, a) ∈ θ and (x_b, b) ∈ θ for all x, a, b ∈ S, then θ is called ·-decomposed;

(iv) If θ satisfies (x, a ∨ b) ∈ θ ⇒ ∃ x_a, x_b ∈ S such that x = x_a ∨ x_b, (x_a, a) ∈ θ and (x_b, b) ∈ θ for all x, a, b ∈ S, then θ is called ∨-decomposed;

(v) If θ is both ·-decomposed and ∨-decomposed, then θ is called complete decomposed;

(vi) If θ satisfies [x ∨ y] _θ = {a ∨ b : a ∈ [x] _θ, b ∈ [y] _θ} for all x, y ∈ S, then θ is called ∨- complete;

(vii) If θ satisfies [xy] _θ = {ab : a ∈ [x] _θ, b ∈ [y] _θ} for all x, y ∈ S, then θ is called ·-complete.

(viii) If θ is both ∨-complete and ·-complete, then θ is called a complete equivalence.

We can see that if θ is an equivalence on S, then θ is a congruence on S if and only if θ is both compatible with · and compatible with ∨.

Proposition 3.2. Let θ be an equivalence on S. Then we have:

θ is ·-decomposed if and only if [xy] _θ ⊆ {ab : a ∈ [x] _θ, b ∈ [y] _θ} for all x, y ∈ S;

θ is ∨-decomposed if and only if [x ∨ y] _θ ⊆ {a ∨ b : a ∈ [x] _θ, b ∈ [y] _θ} for all x, y ∈ S;

θ is a complete equivalence if and only if θ is a complete decomposed congruence;

θ is ·-complete if and only if θ is both compatible with · and ·-decomposed;

θ is ∨-complete if and only if θ is both compatible with ∨ and ∨-decomposed.

Define two binary operations · and ∨ on the powerset $P$ (S) as follows: ( $\forall A, B \in P$ (S)) A · B = {ab : a ∈ A, b ∈ B} and A ∨ B = {a ∨ b : a ∈ A, b ∈ B}. And we setting ↓A = {x ∈ S : ∃ a ∈ A such that x ≤ a}, ↑A = {x ∈ S : ∃ a ∈ A such that x ≥ a}, whereA ⊆ S.

Proposition 3.3. Let θ be an equivalence on S. Then the following statements are true:

if θ is ·-decomposed, then $\forall A, B \in P$ (S), $\underline{θ} (A) \cdot \underline{θ} (B) \subseteq \underline{θ} (A \cdot B)$ ;

if θ is ∨-decomposed, then $\forall A, B \in P$ (S), $\underline{θ} (A) \lor \underline{θ} (B) \subseteq \underline{θ} (A \lor B)$ ;

$\forall A, B \in P$ (S), if A = ↑ A and B = ↑ B, then $\underline{θ} (A \lor B) \subseteq \underline{θ} (A) \lor \underline{θ} (B)$ ;

θ is ·-complete if and only if $\bar{θ} (A \cdot B) = \bar{θ} (A) \cdot \bar{θ} (B)$ for all $A, B \in P$ (S);

θ is ∨-complete if and only if $\bar{θ} (A \lor B) = \bar{θ} (A) \lor \bar{θ} (B)$ for all $A, B \in P$ (S).

Corollary 3.4. Let θ be an equivalence on S. Then we have

$(P$ (S) , · , ∪) is an m-semilattice;

θ is ·-complete if and only if $\bar{θ} : P$ $(S) \to P$ (S) is an m-semilattice homomorphism from $P$ (S) into $P$ (S).

Let θ₁ and θ₂ be two binary relations on S. The composition θ₁ ∘ θ₂ is defined as follows: θ₁ ∘ θ₂ = {(x, y) ∈ S × S : (x, z) ∈ θ₁ and (z, y) ∈ θ₂ forsome z ∈ S} . If θ₁ and θ₂ are two equivalences, then we know that θ₁ ∘ θ₂ = θ₂ ∘ θ₁ if and only if θ₁ ∘ θ₂ is an equivalence on S.

Let A be a subset of S. If A satisfies (i) in Definition 2.1, then A is called a directed subset of S. A is called to satisfy absorption law if it satisfies (iii) in Definition 2.1.

Proposition 3.5. Let θ₁ and θ₂ be two equivalences on S such that θ₁ ∘ θ₂ = θ₂ ∘ θ₁, and let A be a subset of S satisfying absorption law. Then

if θ₁ and θ₂ are both compatible with ·, then $\bar{θ_{1}} (A) \cdot \bar{θ_{2}} (A) \subseteq \bar{θ_{1} \circ θ_{2}} (A)$ ;

if θ₁ and θ₂ are comparable and are both ·-decomposed, then $\underline{θ_{1}} (A) \cdot \underline{θ_{2}} (A) \subseteq \underline{θ_{1} \circ θ_{2}} (A)$ .

Now, we give an example to show that the conclusion of Proposition 3.5 (2) may not hold without the condition “θ₁ and θ₂ are both ·-decomposed and A satisfies absorption law”.

Example 3.6. Consider the bounded lattice S = {⊥ , a, b, c, ⊤} with the partial order ≤ defined by: ⊥≤ a ≤ b, c ≤ ⊤, whose multiplication · is defined to be ∧. Then it is straightforward to verify that (S, · , ∨) is an m-semilattice. Let θ₁ be an equivalence on S with the following blocks: [⊥] _{θ
₁} = {⊥ , a} ; [b] _{θ
₁} = {b, ⊤} ; [c] _{θ
₁} = {c}, and let θ₂ be an equivalence on S with the following blocks: [⊥] _{θ
₂} = {⊥ , a} ; [b] _{θ
₂} = {b, c, ⊤}. We can see that θ₁ ⊆ θ₂ and θ₁ ∘ θ₂ = θ₂ ∘ θ₁. Since [bc] _{θ
₁} = {⊥ , a} ⊈ {a, c} = {mn : m ∈ [b] _{θ
₁}, n ∈ [c] _{θ
₁}} and [bc] _{θ
₂} = {⊥ , a} ⊈ {a, b, c, ⊤} = {pq : p ∈ [b] _{θ
₂}, q ∈ [c] _{θ
₂}}, by Proposition 3.2, we have that neither θ₁ nor θ₂ is ·-decomposed. Let A = {a, b, c, ⊤}. Then A does not satisfy absorption law and $\underline{θ_{1}} (A) \cdot \underline{θ_{2}} (A) = {a, b, c, ⊤} ⊈ {b, c, ⊤} = \underline{θ_{1} \circ θ_{2}} (A)$ .

Proposition 3.7. Let θ₁ and θ₂ be two equivalences on S such that θ₁ ∘ θ₂ = θ₂ ∘ θ₁, and let A be a directed subset of S. If θ₁ and θ₂ are both compatible with ∨, then $\bar{θ_{1}} (A) \lor \bar{θ_{2}} (A) \subseteq \bar{θ_{1} \circ θ_{2}} (A)$ .

Proposition 3.8. Let θ be an equivalence on a positive (resp. negative) m-semilattice S and let A be a subset of S that satisfies absorption law. Then the following statements hold:

if θ is compatible with ·, then $\bar{θ} (↓ A) = ↓ \bar{θ} (A) ($ resp. $\bar{θ} (↑ A) = ↑ \bar{θ} (A))$ ;

if θ is ·-complete and $\underline{θ} (A) \neq \emptyset$ , then $\underline{θ} (↓ A) = ↓ \underline{θ} (A) ($ resp. $\underline{θ} (↑ A) = ↑ \underline{θ} (A))$ .

Proposition 3.9. Let θ be an equivalence on a positive m-semilattice S and let A and B be two subsets of S that satisfy absorption law. Then the following statements hold:

if θ is compatible with ∨, then $\bar{θ} (A \cdot B) \subseteq \bar{θ} (A) \cap \bar{θ} (B) \subseteq \bar{θ} (↓ (A \cdot B)) = ↓ \bar{θ} (A \cdot B)$ ;

$\underline{θ} (A \cdot B) \subseteq \underline{θ} (A \cap B) \subseteq \underline{θ} (↓ (A \cdot B))$ .

Proposition 3.10. Let θ be an equivalence on S and I be an ideal of S. Then we have

if θ is a ∨-complete congruence, then $\bar{θ} (I)$ is an ideal of S;

if S is negative and θ is ∨-complete, then $\underline{θ} (I)$ is an ideal of S;

if θ is ∨-complete and ·-decomposed, then $\underline{θ} (I)$ is an ideal of S.

Proposition 3.11. Let θ be a complete equivalence on S and I be a prime ideal of S. If $\bar{θ} (I) \neq S$ , then $\bar{θ} (I)$ and $\underline{θ} (I)$ are both prime ideals of S.

4 Minimal neighborhood approximation operators on m-semilattices

Let A ⊆ S. A is called a down-set of S if A = ↓ A, and A is called an up-set if A = ↑ A. We denote the set of all down-sets (resp. up-sets) of S by D (S) (resp. U (S)). Then we can see D (S) and U (S) are both coverings of S.

Lemma 4.1. Let $ℂ$ be a covering of S. For every x ∈ S,

if $ℂ = D (S)$ , then N (x) = ↓ x;

if $ℂ = U (S)$ , then N (x) = ↑ x.

Proposition 4.2. Let $ℂ$ be a covering of S and A ⊆ S. Then we have:

if $ℂ = D (S)$ , then ${\bar{apr}}_{N} (A) = ↑ A$ and ${\bar{apr}}_{N} (↑ A) = ↑ {\bar{apr}}_{N} (A)$ ;

if $ℂ = U (S)$ , then ${\bar{apr}}_{N} (A) = ↓ A$ and ${\bar{apr}}_{N} (↓ A) = ↓ {\bar{apr}}_{N} (A)$ .

Definition 4.3. Let F be a subset of S. If F is an up-set that satisfies absorption law, then F is called a filter of S.

Lemma 4.4. Let A ⊆ S. Then the following statements are true:

if S is negative (resp. positive), then ↓A (resp.↑A) satisfies absorption law;

if A satisfies absorption law, then ↓A and ↑A both satisfy absorption law.

Proposition 4.5. Let $ℂ$ be a covering of S and A ⊆ S. Then we have

if S is positive and $ℂ = D (S)$ , then ${\bar{apr}}_{N} (A)$ is a filter of S;

if A satisfies absorption law and $ℂ = D (S)$ , then ${\bar{apr}}_{N} (A)$ is a filter of S;

if S is negative, $ℂ = U (S)$ and A is directive, then ${\bar{apr}}_{N} (A)$ is an ideal of S;

if $ℂ = U (S)$ , A is directive and satisfies absorption law, then ${\bar{apr}}_{N} (A)$ is an ideal of S;

if S is positive, $ℂ = U (S)$ and A is an up-set, then ${\underline{apr}}_{N} (A)$ is a filter of S;

if $ℂ = D (S)$ and A is an ideal, then ${\underline{apr}}_{N} (A)$ is an ideal of S.

Proposition 4.6. Let $ℂ$ be a covering of S andA, B ⊆ S. Then we have

if $ℂ = D (S)$ or U (S), then ${\bar{apr}}_{N} (A) \cdot {\bar{apr}}_{N} (B) \subseteq {\bar{apr}}_{N} (A \cdot B)$ ;

if $ℂ = D (S)$ , then ${\bar{apr}}_{N} (A \lor B) = {\bar{apr}}_{N} (A) \lor {\bar{apr}}_{N} (B)$ ;

if S is negative and satisfies ∀a ∈ S, a² = a and $ℂ = D (S)$ , then ${\underline{apr}}_{N} (A) \cdot {\underline{apr}}_{N} (B) \subseteq {\underline{apr}}_{N} (A \cdot B)$ ;

if S satisfies ∀a ∈ S, a² = a and A and B both satisfy absorption law, then ${\underline{apr}}_{N} (A \cdot B) \subseteq {\underline{apr}}_{N} (A) \cdot {\underline{apr}}_{N} (B)$ ;

if $ℂ = U (S)$ and A, B ∈ U (S), then ${\underline{apr}}_{N} (A \lor B) = {\underline{apr}}_{N} (A) \lor {\underline{apr}}_{N} (B)$ .

Recall that c ∈ L is called compact if c ≤ ⋁ _λ∈Λx_λ implies that c ≤ ⋁ _λ∈Λ₁x_λ for a finite subset Λ₁ of Λ. L is called algebraic if each x ∈ L is a supremum of compact elements. The reader can refer to [9] for unknown definitions. Then we have: (1) (Idl(S) , ⊆) is a complete lattice, in particular, ∀ {I_λ} _λ∈Λ⊆Idl(S), $⋀_{λ \in Λ}^{Idl (S)} I_{λ} = ⋂_{λ \in Λ} I_{λ}$ and $⋁_{λ \in Λ}^{Idl (S)} I_{λ} = ⋃ {↓ (x_{λ_{1}} \lor \dots \lor x_{λ_{n}}) : x_{λ_{i}} \in I_{λ_{i}}, λ_{1}, \dots, λ_{n} \in Λ}$ ; (2) if S is negative, then for every a ∈ S, ↓ a is a compact element of Idl(S).

Let $ℂ$ be a covering of S. An ideal I of S is called a fixed point of ${\bar{apr}}_{N}$ on Idl(S), if ${\bar{apr}}_{N} (I) = I$ . Let ${Fix}_{{\bar{apr}}_{N}} (S)$ denote the family of all fixed points of ${\bar{apr}}_{N}$ on Idl(S).

Note that if $ℂ = U (S)$ or D (S), then N is a reflexive neighborhood operator, and so $\forall A \in P$ (S), $A \subseteq {\bar{apr}}_{N} (A)$ .

Lemma 4.7. Let $ℂ$ be a covering of S and A ⊆ S. If $ℂ = U (S)$ , then we have:

${\bar{apr}}_{N} ({\bar{apr}}_{N} (A)) = {\bar{apr}}_{N} (A)$ ;

( ${Fix}_{{\bar{apr}}_{N}} (S), \subseteq$ ) is a complete lattice. In particular, ∀ {I_λ} _λ∈Λ⊆ ${Fix}_{{\bar{apr}}_{N}} (S), ⋀_{λ \in Λ}^{{Fix}_{{\bar{apr}}_{N}} (S)} I_{λ}$ = ⋂ _λ∈ΛI_λ and $⋁_{λ \in Λ}^{{Fix}_{{\bar{apr}}_{N}} (S)} I_{λ} = {\bar{apr}}_{N} (⋁_{λ \in Λ}^{Idl (S)} I_{λ})$ ;

if S is negative, then for every J∈Idl(S), ${\bar{apr}}_{N} (J) = ⋁_{a \in J}^{Idl (S)} {\bar{apr}}_{N} (↓ a)$ .

Theorem 4.8. Let S be a negative m-semilattice and $ℂ$ be a covering of S. If $ℂ = U (S)$ , then ( ${Fix}_{{\bar{apr}}_{N}} (S), \subseteq$ ) is an algebraic lattice and the compact elements are exactly the ones of the form ${\bar{apr}}_{N} (↓ x)$ for all x ∈ S.

5 Fuzzy rough sets based on a fuzzy covering

Firstly, we present a kind of approximation operators, which have been given in [5] without formal names.

Definition 5.1. Let (∗ , →) be an adjunction on L, (L^U, Φ) be a covering fuzzy rough universe and R be a fuzzy relation derived form Φ. Define the following two mappings $\underline{R}$ and $\bar{R} : L^{U} \to L^{U}$ , called Φ-lower fuzzy rough approximation operator and Φ-upper fuzzy rough approximation operator respectively, as follows: ∀x ∈ U, f ∈ L^U, $\underline{R} (f) (x) = ⋀_{y \in U} (R (y, x) \to f (y)) and \bar{R} (f) (x) = ⋁_{y \in U} (R (x, y) * f (y)) .$ The pair $(\underline{R} (f), \bar{R} (f))$ models a fuzzy rough set based on Φ of f if $\underline{R} (f) \neq \bar{R} (f)$ .

From [5], we know that: (i) $\underline{R}$ and $\bar{R}$ are monotone; (ii) if ∗ is a conjunction on L with the condition (P), (∗ , →) is an adjunction and R is reflexive, then $\forall f \in L^{U}, \underline{R} (f) \subseteq f \subseteq \bar{R} (f)$ ; (iii) in the case of ∗ being min, → being the standard ν-implication of ∗, and R being a fuzzy similarity relation, the pair $(\underline{R}, \bar{R})$ models the Dubois-Prade fuzzy rough set in [7]; when L is a residuated lattice, ∗ is a t-norm on L, → is the ∗-residuated implication and R is a fuzzy ∗-similarity relation, the pair also reduces to the fuzzy rough sets presented in [13, 14].

Proposition 5.2. Let (L^U, R) be a covering fuzzy rough universe, ∗ be an associative conjunction on L with the condition (P) and (∗ , →) be an adjunction. If R is reflexive and ∗-transitive, then $\underline{R}$ and $\bar{R}$ are idempotent.

Definition 5.3. Let f ∈ L^S. If f satisfies for all x, y ∈ S: (i) f (x) ∗ f (y) ≤ f (x ∨ y); (ii) x ≤ y ⇒ f (y) ≤ f (x); (iii) f (x) ∨ f (y) ≤ f (xy), then f is called an ∗-fuzzy ideal of S. We use ∗-FIdl(S) to denote the set consisting of all ∗-fuzzy ideals of S.

Lemma 5.4. Let ∗ be a commutative and associative conjunction on L and (∗ , →) be an adjunction on L, then ∀a, b, c, d ∈ L, (a → b) ∗ (c → d) ≤ (a ∗ c) → (b ∗ d).

Proposition 5.5. Let (L^S, R) be a covering fuzzy rough universe of a negative m-semilattice S, ∗ be a commutative and associative conjunction on L and (∗ , →) be an adjunction. Then

if R satisfies ∀a, b, c, d ∈ S, R (_-, a) preserves inverse order and R (a, b) ∗ R (c, d) ≤ R (a ∨ c, b ∨ d), then ∀f∈∗-FIdl(S), $\bar{R} (f)$ is an ∗-fuzzy ideal of S;

if ∗ is a t-norm, R satisfies ∀a, b, c ∈ S, R (a, b) ∗ R (a, c) = R (a, b ∨ c) and R (a, _-) preserves order, then ∀f ∈ L^S, $\underline{R} (f)$ is an ∗-fuzzy ideal of S.

Now we give two examples of fuzzy relations satisfying the conditions in Proposition 5.5.

Example 5.6. Let Φ = {f_i} _i∈I be a fuzzy covering of S. (1) Let (∗ , →) be an adjunction on L. Define R ∈ L^S×S by ∀x, y ∈ S, R (x, y) = ⋀ _i∈I (f_i (x) ∗ f_i (y)). If Φ = {f_i} _i∈I ⊆ ∗-FIdl(S) and ∗ is commutative and associative, then R satisfies ∀a, b, c, d ∈ S, R (a, b) ∗ R (c, d) ≤ R (a ∨ c, b ∨ d) and R (_-, a) preserves inverse order. (2) Let L be a frame and (∧ , →) be an adjunction on L. Define W ∈ L^S×S by ∀x, y ∈ S, W (x, y) = ⋀ _i∈If_i (x). Then W satisfies ∀a, b, c ∈ S, W (a, b) ∧ W (a, c) = W (a, b ∨ c) and W (a, _-) preserves order.

Let ∗ be a conjunction on L. ∀f, g ∈ L^S, the product f ∗ g is defined by (∀ x ∈ S) (f ∗ g) (x) = ⋁ _x≤yz (f (y) ∗ g (z)). If L is a frame and (∧ , →) is an adjunction on L, then (f ∗ g) (x) = ⋁ _x≤yz (f (y) ∧ g (z)).

Note that L is called ∗-idempotent, if ∀l ∈ L, l ∗ l = l, where ∗ is a conjunction on L.

Proposition 5.7. Let (L^S, R) be a covering fuzzy rough universe, ∗ be a commutative and associative conjunction on L and (∗ , →) be an adjunction. Then ∀f, g ∈ L^S,

if R satisfies that ∀a, b, c, d ∈ S, R (a, b) ∗ R (c, d) ≤ R (ac, bd) and R (_-, a) preserves inverse order, then $\bar{R} (f) * \bar{R} (g) \subseteq \bar{R} (f * g)$ ;

if S is negative, L is ∗-idempotent and ∀a ∈ S, R (a, _-) and R (_-, a) both preserve order, then $\bar{R} (f * g) \subseteq \bar{R} (f) * \bar{R} (g)$ .

Note that if (∗ , →) is an adjunction on L, then (L, ∗ , ∨) is also an m-semilattice. Now we give two examples of fuzzy relations satisfying the conditions proposed in Proposition 5.7.

Example 5.8. Let ∗ be a commutative and associative conjunction on L, (∗ , →) be an adjunction on L, and Φ = {f_i} _i∈I be a fuzzy covering of S, where ∀i ∈ I, f_i : S → L is an m-semilattice homomorphism. (1) Define R ∈ L^S×S by ∀x, y ∈ S, R (x, y) = ⋀ _i∈I (f_i (x) → f_i (y)). Then R satisfies ∀a, b, c, d ∈ S, R (a, b) ∗ R (c, d) ≤ R (ac, bd) and R (_-, a) preserves inverse order. (2) Define W ∈ L^S×S by ∀x, y ∈ S, W (x, y) = ⋁ _i∈I (f_i (x) ∗ f_i (y)). Then W satisfies ∀a ∈ S, W (a, _-) and W (_-, a) both preserve order.

Lemma 5.9. Let S be a positive m-semilattice. Then

if ∗ is a conjunction on L with the condition (P), then ∀f, g ∈ ∗-FIdl(S), f ∗ g ⊆ f ∩ g;

if L is a frame and (∧ , →) is an adjunction, then ∀f, g ∈ L^S, f ∩ g ⊆ f ∗ g.

Proposition 5.10. Let (L^S, R) be a covering fuzzy rough universe of a positive m-semilattice S and f, g ∈ L^S. Then

if ∗ is a commutative and associative conjunction on L, (∗ , →) is an adjunction, R satisfies ∀a, b, c ∈ S, R (a, b) ∗ R (a, c) ≥ R (a, bc) and R (a, _-) preserves order, then $\underline{R} (f) * \underline{R} (g) \subseteq \underline{R} (f * g)$ ;

if L is a frame, (∧ , →) is an adjunction and f, g ∈ ∗-FIdl(S), then $\underline{R} (f * g) \subseteq \underline{R} (f) * \underline{R} (g)$ .

We denote the set of all fuzzy ∗-ideals of S with 1 in their images by ∗-FIdl(S) ¹.

Lemma 5.11.Let ∗ be a t-norm on L. Then ∀f₁, f₂∈∗-FIdl(S) ¹, the map f ∈ L^S defined by ∀x ∈ S, f (x) = ⋁ _{x≤y₁∨y₂} (f₁ (y₁) ∗ f₂ (y₂)) is the supremum of f₁ and f₂ in ∗-FIdl(S) ¹, and we use f = f₁ ∨ f₂ to denote this situation.

Proposition 5.12. Let (L^S, R) be a covering fuzzy rough universe, ∗ be a t-norm on L and (∗ , →)be an adjunction and let f, g∈∗-FIdl(S) ¹ with $\bar{R} (f), \bar{R} (g) \in$ ∗-FIdl(S) ¹. Then

if R satisfies ∀a, b, c, d ∈ S, R (_-, a) preserves inverse order, and R (a, b) ∗ R (c, d) ≤ R (a ∨ c, b ∨ d), then $\bar{R} (f) \lor \bar{R} (g) \subseteq \bar{R} (f \lor g)$ ;

if R satisfies ∀a, b, c ∈ S, R (a, _-) preserves order, and R (a, b) ∗ R (a, c) ≥ R (a, b ∨ c), then $\bar{R} (f \lor g) \subseteq \bar{R} (f) \lor \bar{R} (g)$ .

Proposition 5.13. Let (L^S, R) be a covering fuzzy rough universe, ∗ be a t-norm on L and (∗ , →) be an adjunction, and let f, g∈∗-FIdl(S) ¹ with $\underline{R} (f), \underline{R} (g) \in$ ∗-FIdl(S) ¹. If R satisfies ∀a, b, c ∈ S, R (a, _-) preserves order, and R (a, b) ∗ R (a, c) ≥ R (a, b ∨ c), then $\underline{R} (f) \lor \underline{R} (g) \subseteq \underline{R} (f \lor g)$ .

Footnotes

Acknowledgements

The authors would like to thank the referees and Professor Susana Montes, associate editor, for providing very helpful comments and suggestions.

References

Biswas

and Nanda

, Rough groups and rough subgroups, Bulletin of the Polish Academy of Sciences Mathematics42 (1994), 251–254.

Bonikowski

and Brynarski

, Extensions and intensions in rough set theory, Information Sciences107 (1998), 149–167.

Chen

D.G.

, Zhang

W.X.

, Yeung

and Tsang

E.C.C.

, Rough approximations on a complete completely distributive lattice with applications to generalized rough sets, Information Sciences176 (2006), 1829–1848.

Davvaz

, Roughness in rings, Information Sciences164 (2004), 147–163.

Deng

T.Q.

, Chen

Y.M.

, Xu

W.L.

and Dai

Q.H.

, A novel approach to fuzzy rough sets based on a fuzzy covering, Information Sciences177 (2007), 2308–2326.

Deng

T.Q.

and Heijmans

H.J.A.M.

, Grey-scale morphology based on fuzzy logic, Journal of Mathematical Imaging and Vision16 (2002), 155–171.

Dubois

and Prade

, Rough fuzzy sets and fuzzy rough sets, International Journal of General Systems17(23) (1990), 191–209.

Estaji

A.A.

, Hooshmandasl

M.R.

and Davvaz

, Rough set theory applied to lattice theory, Information Sciences200 (2012), 108–122.

Gierz

, Hofmann

K.H.

, Keimel

, Lawson

J.D.

, Mislove

and Scott

D.S.

, Continuous Lattices and Domains, Cambridge University Press, 2003.

10.

Järvinen

, Lattice Theory for Rough Sets, Transactions on Rough Sets VI, LNCS 4374, Springer-Verlag, Berlin, Heidelberg, 2007, pp. 400–498.

11.

Jensen

and Shen

, Fuzzy-rough data reduction with ant colony optimization, Fuzzy Sets and Systems149 (2005), 5–20.

12.

Kuroki

, Rough ideals in semigroups, Information Sciences100 (1997), 139–163.

13.

and Yin

Y.Q.

, The ϑ-lower and T-upper fuzzy rough approximation operators on a semigroup, Information Sciences195 (2012), 241–255.

14.

, Yin

and Lu

, (ϑ, T)-fuzzy rough approximation operators and TL-fuzzy rough ideals on a ring, Information Sciences177 (2007), 4711–4726.

15.

Liu

G.L.

and Zhu

, The algebraic structures of generalized rough set theory, Information Sciences178 (2008), 4105–4113.

16.

Pawlak

, Rough sets, International Journal of Computer and Information Sciences11 (1982), 341–356.

17.

and Liu

, Rough operations on Boolean algebras, Information Sciences173 (2005), 49–63.

18.

Rasouli

and Davvaz

, Roughness in MV-algebras, Information Sciences180 (2010), 737–747.

19.

Radzikowska

A.M.

and Kerre

E.E.

, A comparative study of fuzzy rough sets, Fuzzy Sets and Systems126 (2002), 137–155.

20.

Restrepo

, Cornelis

and Gómez

, Partial order relation for approximation operators in covering based rough sets, Information Sciences284 (2014), 44–59.

21.

Restrepo

, Cornelis

and Gómez

, Duality, conjugacy and adjointness of approximation operators in covering-based rough sets, International Journal of Approximate Reasoning55 (2014), 469–485.

22.

Rosenthal

K.I.

, Quantales and Their Applications, New York: Longman Scientific and Technical, 1990.

23.

Wang

L.J.

, Yang

X.B.

, Yang

J.Y.

and Wu

, Relationships among generalized rough sets in six coverings and pure reexive neighborhood system, Information Sciences207 (2012), 66–78.

24.

W.Z.

, Leung

and Mi

J.S.

, On characterizations of(ℐ, 𝒥)-fuzzy rough approximation operators, Fuzzy Sets and Systems154 (2005), 76–102.

25.

Yamak

, Kazanci

and Davvaz

, Generalized lower and upper approximations in a ring, Information Sciences180 (2010), 1759–1768.

26.

Yang

and Li

Q.G.

, Reduction about approximation spaces of covering generalized rough sets, International Journal of Approximate Reasoning51 (2010), 335–345.

27.

Yang

L.Y.

and Xu

L.S.

, Algebraic aspects of generalized approximation spaces, International Journal of Approximate Reasoning51 (2009), 151–161.

28.

Yang

L.Y.

and Xu

L.S.

, Roughness in quantales, Information Sciences220 (2013), 568–579.

29.

Yao

Y.Y.

and Yao

B.X.

, Covering based rough set approximations, Information Sciences200 (2012), 91–107.

30.

Zadeh

L.A.

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

31.

Zhang

W.X.

, Liang

and Wu

W.Z.

, Information System and Knowledge Discovery, Science Publishing Press, Beijing, 2005.

32.

Zhu

, Properties of the first type of covering-based rough sets, in: Proceedings of Sixth IEEE International Conference on Data Mining-Workshops, 2006, pp. 407–411.

33.

Zhu

and Wang

F.Y.

, On three types of covering based rough sets, IEEE Transactions on Knowledge and Data Engineering19(8) (2007), 1131–1144.