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Keywords:

  • Fuzzy set;
  • L-ordered set;
  • Fuzzy closure system;
  • Fuzzy closure L-system;
  • MSC (2010) 03E72, 06A15

Abstract

  1. Top of page
  2. Abstract
  3. Introduction
  4. L—ordered sets and fuzzy closure operators
  5. Fuzzy closure systems on L—ordered sets
  6. Fuzzy closure L—systems on L—ordered sets
  7. References

In this paper, notions of fuzzy closure system and fuzzy closure L—system on L—ordered sets are introduced from the fuzzy point of view. We first explore the fundamental properties of fuzzy closure systems. Then the correspondence between fuzzy closure systems (fuzzy closure L—systems) and fuzzy closure operators is established. Finally, we study the connections between fuzzy closure systems and fuzzy Galois connections. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim


Introduction

  1. Top of page
  2. Abstract
  3. Introduction
  4. L—ordered sets and fuzzy closure operators
  5. Fuzzy closure systems on L—ordered sets
  6. Fuzzy closure L—systems on L—ordered sets
  7. References

Closure systems and closure operators have been studied in many mathematical branches such as topology, algebra and logic. In the classical order theory, they are particularly related to theoretical computer science and have a close relation with Galois connections 9, 15. Recently, the notions of fuzzy closure system and fuzzy closure operator have been investigated on the L—powerset LX, i.e., all the L—sets in a given universe X, where L is an appropriate truth value structure. In 2, the one-to-one correspondence between fuzzy closure systems and fuzzy closure operators was established within a general fuzzy framework in which complete residuated lattice serves as the truth value structure. Later in 12, fuzzy closure systems and fuzzy closure operators were studied within the non-commutative fuzzy framework, where the conjunction operator in the truth value structure is not necessarily commutative. Additionally, fuzzy interior operators and fuzzy interior systems on LX were investigated in 7, 8, 13. It was shown that the correspondence between fuzzy interior operators and fuzzy interior systems can be established analogously.

On the other hand, the generalization of the crisp partial order has been studied from the fuzzy point of view 6, 10, 17, 25. One of them is the L—ordered set which was originally introduced in 6 in order to fuzzify the fundamental theorem of concept lattices. The basic classical notions, such as infimum and supremum, were also fuzzified on L—ordered set. In addition, the notions of fuzzy closure (respectively, interior) operator and Galois connection have been developed directly on L—ordered sets 4, 22, 23. They can be viewed as the generalization of the notions defined on crisp partial order sets and L—powersets 1, 2, 14. From this point of view, the need for development of fuzzy closure (interior) systems on L—ordered sets is apparent.

The aim of this paper is to propose the notion of fuzzy closure system on L—ordered sets and study its relationship with fuzzy closure operators and fuzzy Galois connections. The paper is organized as follows: In Section 2, the notions of L—ordered set and fuzzy closure operator are briefly reviewed. Section 3 focuses on the new notion of fuzzy closure systems. It is shown that there exists a bijective correspondence between fuzzy closure systems and fuzzy closure operators on a common L—ordered set. In the last section, we develop another type of fuzzy closure system, namely fuzzy closure L—system, and study its relation with fuzzy closure system and fuzzy closure operator.

L—ordered sets and fuzzy closure operators

  1. Top of page
  2. Abstract
  3. Introduction
  4. L—ordered sets and fuzzy closure operators
  5. Fuzzy closure systems on L—ordered sets
  6. Fuzzy closure L—systems on L—ordered sets
  7. References

Truth value structure, formally, a set of truth values equipped with some special structure, plays an important role in fuzzy set theory and fuzzy logic. A traditional and common choice is the real interval [0, 1] with an appropriate structure 24. A more general one is the complete residuated lattice, which is significant in fuzzy logic in narrow sense 16, 18.

Definition 2.1 (5, 16) A complete residuated lattice is a structure L = (L, ∧ , ∨ , ⊗ , → , 0, 1) such that

  • (i)
    (L, ∧ , ∨ , 0, 1) is a complete lattice with the least element 0 and the greatest element 1;
  • (ii)
    (L, ⊗ , 1) is a commutative monoid, i.e., ⊗ is commutative, associative, and x ⊗ 1 = x holds for any xL;
  • (iii)
    ⊗ and → form an adjoint pair, i.e., for any x, y, zL,
    • equation image

Let L be a complete residuated lattice. Given x, y, zL, (xi)iIL, where I is an index set, the following properties will be needed in this paper.

  • (1)
    0 → x = 1,
  • (2)
    1 → x = x,
  • (3)
    xyxzyz,
  • (4)
    xyzxzy,
  • (5)
    xyxzyz,
  • (6)
    xy iff xy = 1,
  • (7)
    x ⊗ (xy) ≤ y,
  • (8)
    x ≤ (xy) → y,
  • (9)
    x → (yz) = y → (xz),
  • (10)
    equation image,
  • (11)
    equation image.

More properties about complete residuated lattice can be found in 5.

Definition 2.2 (6) Let X be a non-empty set. A binary L—relation ≈ on X is called an L—equality if it satisfies that for any x, y, zX, (xx) = 1 (reflexivity); (xy) = (yx) (symmetry); (xy) ⊗ (yz) ≤ (xz) (transitivity); and (xy) = 1 implies x=y.

Let X and Y be two sets, ≈X and ≈Y the L—equalities on X and Y respectively. A binary L—relation R: X × YL is said to be compatible w.r.t. ≈X and ≈Y if for any x1, x2X and y1, y2Y

  • equation image

For the particular situation of an L—relation e on X with an L—equality ≈, we say e is compatible w.r.t. ≈ for simplicity.

Definition 2.3 (6) Let X be a non-empty set and ≈ an L—equality on X. An L−order on X is a binary L—relation e which is compatible w.r.t. ≈ and satisfies

  • (1)
    for any xX, e(x, x)=1;
  • (2)
    for any x, yX, e(x, y) ∧ e(y, x) ≤ (xy);
  • (3)
    for any x, y, zX, e(x, y) ⊗ e(y, z) ≤ e(x, z).

We call ((X, ≈), e) an L—ordered set.

Example 2.4 (1) Given an L—ordered set ((X, ≈), e), e−1: X × XL is defined by e−1(x, y)=e(y, x) for any x, yX. Then ((X, ≈), e−1) is an L—ordered set as well and it is called the dual L—ordered set of ((X, ≈), e).

(2) Suppose ((X, ≈), e) is an L—ordered set. For any nonempty subset MX, ((M, ≈), e) is an L—ordered set, where e and ≈ are inherited from ((X, ≈), e).

(3) Let (X, ≤) be a crisp partial order set. Choose the truth value structure L as the Boolean algebra 2 = {0, 1}. If we define the L—equality ≈ on X by (xy) = 1 iff x=y, and e: X × XL by e (x, y) = 1 iff xy, then ((X, ≈), e is an L—ordered set.

(4) An L—set A in a given universe X is a mapping A: XL. Given A, BLX, the subsethood degree S(A, B) of A in B is defined by equation image. The equality degree E(A, B) between A and B is defined by equation image. Then ((LX, E), S) is an L—ordered set, where LX is the collection of all L—sets on X.

It is noteworthy that an L—order defined in Definition Definition 2.3 is equivalent to a fuzzy partial order given by Fan and Zhang (see 10, and also 22, 27). Based on the introduction of L—ordered set, the fundamental notions, such as join, meet and complete lattice, can be established as an approach to generalize the classical order theory. In the following, we only recall some basic notions and give a fundamental proposition. One can refer to 6, 27 for further details.

Given an L—ordered set ((X, ≈), e), a pair of operaters equation image on LX are defined by

  • equation image

For some ALX, we denote equation image by Au and equation image by Al, and use the notation Alu to replace (Al)u, etc. In addition, AinfLX is defined by

  • equation image

and AsupLX is defined by

  • equation image

A is called an L—singleton if there exists x0X such that A(x) = (x0x) for any xX. Furthermore, we have the following proposition.

Proposition 2.56, 21, 27 Let ((X, ≈), e) be an L—ordered set and ALX. The following are equivalent:

  • (1)
    Ainf (respectively, Asup) is an L—singleton.
  • (2)
    There exists (unique) x0X such that Ainf(x0)=1 (respectively, Asup(x0)=1).
  • (3)
    There exists (unique) x0X such that A(x) ≤ e(x0, x) (respectively, A(x) ≤ e(x, x0)) for any xX, and equation image (respectively, equation image) for any yX.

Remark 2.6 Notice that Proposition 2.5 provides different ways to define the join and meet of L— sets. As a matter of fact, this has been shown in 21 where L is a frame as the truth value structure (note: every frame can be viewed as a complete residuated lattice), and the notion of L—fuzzy complete lattice introduced there is consistent with Bělohlávek’s completely lattice L—ordered set (see 6).

Let equation image and equation image? be two L—ordered sets. A mapping f: XY is said to be fuzzy order-preserving if for any x1, x2X, equation image. If f is fuzzy order-preserving and ff = f, then f is called a fuzzy projection.

Definition 2.7 (23) Let ((X, ≈), e) be an L—ordered set. An order-preserving mapping f: XX is said to be a fuzzy closure (respectively, interior) operator if for any xX,

  • equation image

Example 2.8 Let X be a non-empty set and L a complete residuated lattice. Recall a mapping C : LXLX is an L—closure operator if for any A, A1, A2LX, (i) S(A, C(A)) = 1; (ii) S(A1, A2) ≤ S(C(A1), C(A2)); (iii) C(A)=C(C(A)). Apparently, C is a fuzzy closure operator on ((LX, E), S). In fact, L—closure operators are special cases of LK—closure operators when K = L. One can refer to 2 for more details about LK—closure operators.

Remark 2.9 (1) The notion of LK—closure operator on L—ordered sets was introduced in 4. It can be seen as a generalization of that in 2. It is clear that fuzzy closure operators defined in Definition 2.7 are exactly extreme cases of LK—closure operators in 4 when K = L.

(2) A fuzzy closure operator on an L—ordered set is an extension of a classical closure operator on a crisp partial order set. Given an L—ordered set ((X, ≈), e), define ≤eX × X by xey: ⇔ e(x, y) = 1 for any x, yX. Then f: XX is a fuzzy closure operator iff f is fuzzy order-preserving and a classical closure operator on (X, ≤ e).

(3) Every fuzzy closure (respectively, interior) operator on ((X, ≈), e) is a fuzzy interior (respectively, closure) operator on the dual L—ordered set ((X, ≈), e−1).

As a special case of the fuzzy system closed under SK—intersections which is introduced initially in 2, we recall that a system ℘ = {AiLX|iI} is said to be closed under S—intersections if for any ALX, it holds that equation image, where for any xX,

  • equation image

The following proposition provides not only equivalent characterizations of this notion, but also a basic reference to generalize the notion of fuzzy closure system onto L—ordered sets in next section.

Proposition 2.10 Let X be a non-empty set, L a complete residuated lattice and ℘ = {AiLX|iI} a subset of LX. Then the following are equivalent:

  • (1)
    ℘ is closed under S−intersections.
  • (2)
    For any ALX, there exists A0 ∈ ℘ such that S(A, A0) = 1 and equation image for any jI.
  • (3)
    For any ALX, there exists A0 ∈ ℘ such that S(A, A0)=1 and S(Aj, A0) ⊗ S(A, Ai) ≤ S(Aj, Ai) for any i, jI.

Proof. (1) ⇒ (2). For any ALX let equation image.

Firstly, we have

  • equation image

Secondly, we show that S(A, Ai) ≤ S(A0, Ai) for any Ai ∈ ℘, i.e., equation image for any Ai ∈ ℘, which is equivalent to S(A, Ai) ≤ A0(x) → Ai(x) for any xX iff S(A, Ai) ⊗ A0(x) ≤ Ai(x), i.e., equation image for any xX. Indeed, the last inequality is true:

  • equation image

Therefore, for any i, jI, S(Aj, A0) ⊗ S(A, Ai) ≤ S(Aj, A0) ⊗ S(A0, Ai) ≤ S(Aj, Ai) which implies that S(Aj, A0) ≤ S(A, Ai) → S(Aj, Ai). It follows that equation image for any jI. In addition, for any Aj ∈ ℘, since A0 ∈ ℘, we have

  • equation image

Thus equation image for any jI.

(2) ⇒ (3). Since equation image for any jI, we have for any i, jI, S(Aj, A0) ≤ S(A, Ai) → S(Aj, Ai) which is equivalent to S(Aj, A0) ⊗ S(A, Ai) ≤ S(Aj, Ai).

(3) ⇒ (1). Suppose A0 satisfies the conditions in (3), we want to show equation image, i.e., for any xX, equation image On one hand, since A0 ∈ ℘,

  • equation image

On the other hand, we need to prove equation image for any xX iff A0(x) ≤ S(A, Ai) → Ai(x) for any iI iff S(A, Ai) ≤ A0(x) → Ai(x) which is true. Indeed, since S(Aj, A0) ⊗ S(A, Ai) ≤ S(Aj, Ai) for any i, jI and A0 ∈ ℘,

  • equation image

which implies S(A, Ai) ≤ A0(x) → Ai(x) for any xX.

Fuzzy closure systems on L—ordered sets

  1. Top of page
  2. Abstract
  3. Introduction
  4. L—ordered sets and fuzzy closure operators
  5. Fuzzy closure systems on L—ordered sets
  6. Fuzzy closure L—systems on L—ordered sets
  7. References

In this section, we are going to develop the notion of fuzzy closure system on L—ordered sets and investigate the relationship between fuzzy closure systems and closure operators. It is shown that the notion of fuzzy closure system provides an alternative way to think about fuzzy closure operators.

Definition 3.1 Let ((X, ≈), e) be an L—ordered set. A non-empty subset MX is called a fuzzy closure (respectively, interior) system on X if for any xX, there exists mxM such that

  • (1)
    e(x, mx)=1 (respectively, e(mx, x)=1), and
  • (2)
    e(n, mx) ⊗ e(x, m) ≤ e(n, m) (respectively, e(m, x) ⊗ e(mx, n) ≤ e(m, n)) for any m, nM.

We call mx the least upper bound (respectively, the greatest lower bound) of x in M.

Remark 3.2 (1) Some basic properties of fuzzy closure system are obvious. Given a fuzzy closure system M on L—ordered set ((X, ≈), e), the least upper bound of x in M is unique. Every fuzzy closure system on ((X, ≈), e) is a fuzzy interior system on the dual L—ordered set ((X, ≈), e−1). Moreover, the least upper bound of x in M (as a fuzzy closure system) w.r.t. ((X, ≈), e) is the greatest lower bound of x in M (as a fuzzy interior system) w.r.t. ((X, ≈), e−1). Similar properties hold for fuzzy interior systems.

(2) A fuzzy closure system on an L—ordered set is an extension of a classical closure system on a crisp partial order set. Let ((X, ≈), e) be an L—ordered set and M a non-empty subset X. For some xX, denote the principal ideal (respectively, filter) generated by x w.r.t. ≤e by equation image (respectively, equation image), i.e., equation image (respectively, equation image). Then M is a fuzzy closure system iff M is a classical closure system on (X, ≤ e) and e(x, m) ≤ e(x0, m) for any xX and mM, where x0 is the minimum upper bound of x in M w.r.t. ≤e, i.e., equation image. Dually, NX is a fuzzy interior system iff N is a classical interior system on (X, ≤ e) and e(n, x) ≤ e(n, x0) for any xX and nN, where x0 is the maximum lower bound of x in N w.r.t. ≤e, i.e., equation image.

Example 3.3 Let X be a non-empty set, L a complete residuated lattice, and ℘ = (Ai)iI a subset of LX. By Proposition 2.10, ℘ is a fuzzy closure system on ((LX, E), S) iff ℘ is closedunder S−intersections.

Proposition 3.4 Let ((X, ≈), e) be an L—ordered set and M a non-empty subset of X. The following are equivalent:

  • (1)
    M is a fuzzy closure system.
  • (2)
    For any xX, there exists (unique) m0M such that e(x, m0)=1, and equation image for any nM.

Proof. Suppose M is a fuzzy closure system. For any xX, let mx be the least upper bound of x in M. It is clear that e(x, mx)=1. Furthermore, for any nM,

  • equation image

In addition, for any mM, since e(n, mx) ⊗ e(x, m) ≤ e(n, m), we have e(n, mx) ≤ e(x, m) → e(n, m). This implies that equation image. Hence, equation image for any nM.

Conversely, we show m0 is exactly the least upper bound of x in M: Apparently, e(x, m0)=1. Moreover, for any nM, since equation image, we have e(n, m0) ≤ e(x, m) → e(n, m) for any mM. It follows that e(n, m0) ⊗ e(x, m) ≤ e(n, m) for any m, nM.

Dually, we have

Proposition 3.5 Let ((X, ≈), e) be an L—ordered set and M a non-empty subset of X. The following are equivalent:

  • (1)
    M is a fuzzy interior system.
  • (2)
    For any xX, there exists (unique) m0X such that e(m0, x)=1, and equation image for any nM.

Proposition 3.6 Let ((X, ≈), e) be an L—ordered set, and M a fuzzy closure (respectively, interior) system on X. Then there exists (unique) mxM for any xX such that:

  • (1)
    e(x, mx)=1 (respectively, e(mx, x)=1),
  • (2)
    e(x, m) ≤ e(mx, m) (respectively, e(m, x) ≤ e(m, mx)) for any mM, and
  • (3)
    equation image (respectively, equation image) for any nM.

Moreover, given xX, if there exists yX such that

  • (1′)
    e(x, y)=1 (respectively, e(y, x)=1),
  • (2′)
    e(x, m) ≤ e(y, m) (respectively, e(m, x) ≤ e(m, y)) for any mM, and
  • (3′)
    equation image (respectively, equation image) for any nM,

then y is exactly the least upper (respectively, greatest lower) bound of x in M.

Proof. Trivial.

Proposition 3.7 Let ((X, ≈), e) be an L—ordered set, MX a fuzzy closure (respectively, interior) system, and x, yX. Then the following are equivalent:

  • (1)
    x and y have the same least upper (respectively, greatest lower) bound in M.
  • (2)
    e(x, m)=e(y, m) (respectively, e(m, x)=e(m, y)) for any mM.

Proof. Trivial.

Now we investigate the connection between fuzzy closure (respectively, interior) systems and fuzzy closure (respectively, interior) operators on a common L—ordered set.

Theorem 3.8 Let ((X, ≈), e) be an L—ordered set, f a fuzzy closure (respectively, interior) operator and MX a fuzzy closure (respectively, interior) system. Then Mf:f(X) is a fuzzy closure (respectively, interior) system on X and fM is a fuzzy closure (respectively, interior) operator on X, where fM is defined by associating each xX with the least upper (respectively, greatest lower) bound of x in M. Moreover, equation image and equation image, i.e., mappings equation image and equation image are mutually inverse.

Proof. We only prove the statements related to fuzzy closure systems. The proof for fuzzy interior systems can be obtained analogously.

Suppose f is a fuzzy closure operator on X and xX. It is obvious that f(x) ∈ Mf. By Definition 2.7, e(x, f(x))=1. For any m, nMf, since f is a fuzzy closure operator, f(m)=m and f(n)=n. Then e(n, f(x)) ⊗ e(x, m) ≤ e(n, f(x)) ⊗ e(f(x), m) ≤ e(n, m). By Definition 3.1, f(x) is the least upper bound of x in Mf and thus Mf is a fuzzy closure system on X.

On the other hand, suppose MX is a fuzzy closure system. From Remark 3.2 (1), equation image is well-defined. For any xX, it is clear that equation image by Definition 3.1. Suppose x1, x2X, then equation image. Since equation image, by Proposition 3.6, equation image, which implies that equation image. This means equation image is fuzzy order-preserving. Moreover, because equation image and equation image, we have equation image. Therefore, equation image is a fuzzy closure operator on X.

Finally, from the above proof, it is clear that mappings equation image and equation image are mutually inverse.

Corollary 3.9 Let ((X, ≈), e) be an L—ordered set and f a mapping on X. Then f is a fuzzy closure (respectively, interior) operator iff f(X) is a fuzzy closure (respectively, interior) system.

In the classical setting, Galois connections have a close relation with closure operators and closure systems 9, 15. In formal concept analysis, formal concept lattice is extracted from a formal context based on Galois connection 11, 20, 26. In the fuzzy setting, the notion of fuzzy Galois connection have been introduced in different frameworks and they serve as important tools in fuzzy concept analysis 1, 3, 13, and also 19.

In the following, we are going to investigate the relationship between fuzzy Galois connections and fuzzy closure (interior) systems. The results can be seen as a generalization of those presented by Bělohlávek in 2.

Definition 3.10 (23) Let equation image and equation image be two L—ordered sets and equation image and equation image two mappings. (φ, ψ) is called a fuzzy Galois connection between X and Y if both φ and ψ are fuzzy order-preserving and equation image for any xX, yY. In this case, φ is called the left adjoint of ψ and dually ψ the right adjoint of φ.

Proposition 3.11 Let equation image and equation image be two L—ordered sets and equation image and equation image two mappings. (φ, ψ) is a fuzzy Galois connection if and only if for all x, x1, x2X and y, y1, y2Y,

  • (1)
    equation image
  • (2)
    equation image
  • (3)
    equation image
  • (4)
    equation image

Proof. Straightforward.

Proposition 3.12 Let equation image and equation image be two L—ordered sets. For any fuzzy Galois connection (φ, ψ) between X and Y,

  • equation image

Proof. Suppose xX. Since φ is fuzzy order-preserving, it holds that equation image. By Proposition 3.11, we have equation image. On the other hand, by Proposition 3.11 again, it is obvious that equation image. Therefore, φψφ(x) = φ(x). This means φψφ = φ.

The second equation can be proved similarly.

Remark 3.13 Given a fuzzy Galois connection, the left (respectively, right) adjoint uniquely determine the right (respectively, left) adjoint. Indeed, suppose both (φ, ψ1) and (φ, ψ2) are fuzzy Galois connections between X and Y. By Definition 3.10 and Proposition 3.11, equation image for any yY. Dually, it holds that equation image for any yY. We thus have ψ1 = ψ2.

Given a fuzzy closure system MX on equation image and a fuzzy interior system MY on equation image. We call MX and MY are e−isomorphic iff there is a bijective mapping equation image such that equation image for any x1, x2MX. We call ω the e−isomorphism between MX and MY. In addition, the inverse of ω, denoted by ω−1, is also a bijective mapping and equation image for any y1, y2MY.

Theorem 3.14 Let (φ, ψ) be a fuzzy Galois connection between X and Y, f a fuzzy closure operator on X and g a fuzzy interior operator on Y such that f(X) and g(Y) are e−isomorphic with ω being the e−isomorphism. Put f(φ,ψ) = ψφ and g(φ,ψ) = φψ. Let φ(f,g)(x) = ω(f(x)) and ψ(f,g)(y) = ω−1(g(y)) for any xX, yY. Then

  • (1)
    f(φ,ψ) is a fuzzy closure operator on X and g(φ,ψ) is a fuzzy interior operator on Y; f(φ,ψ)(X) and g(φ,ψ)(Y) are e−isomorphic;
  • (2)
    (f,g), ψ(f,g)) is a fuzzy Galois connection;
  • (3)
    the correspondences defined by equation image and equation image are mutually inverse mappings.

Proof. (1) Suppose x1, x2X. Since both ψ and φ are fuzzy order-preserving, equation image ψφ(x2)), which means f(φ,ψ) is fuzzy order-preserving. By Proposition 3.11, equation image =1. By Proposition 3.12, equation image. Therefore, f(φ,ψ) is a fuzzy closure operator on X. One can similarly prove g(φ,ψ) is a fuzzy interior operator on Y.

By Theorem 3.8, f(φ,ψ)(X) is a fuzzy closure system on X and g(φ,ψ)(Y) is a fuzzy interior system on Y. Define equation image by ω(φ,ψ)(x) = φ(x) for any xf(φ,ψ)(X). It is routine to verify that ω(φ,ψ) is the e−isomorphism between f(φ,ψ)(X) and g(φ,ψ)(Y).

(2) It is obvious that both φ(f,g) and ψ(f,g) are fuzzy order-preserving.

Suppose xX, yY. On one hand,

  • equation image

On the other hand,

  • equation image

Therefore, equation image holds for any xX, yY. By Definition 3.10, (φ(f,g), ψ(f,g)) is a fuzzy Galois connection.

(3) By Remark 3.13, in order to show equation image, we only need to prove that equation image. For any xX,

  • equation image

In addition, for any xX,

  • equation image

Therefore, equation image. The equation equation image can be proved similarly.

Theorem 3.15 Let ((X, ≈), e) be an L—ordered set. Then MX is a fuzzy closure (respectively, interior) system if and only if there is a fuzzy Galois connection (g, h) (respectively, (h, g)) between ((X, ≈), e) and some fuzzy ordered set ((Y, ≈ ′), e′) (respectively, between some fuzzy ordered set ((Y, ≈ ′), e′) and ((X, ≈), e)) such that M=hg(X).

Proof. We only give the proof for the fuzzy closure systems. The statements about the fuzzy interior systems can be proved analogously.

Suppose M is a fuzzy closure system on X. Then ((M, ≈), e) is an L—ordered set. Define equation image as for any xX, g(x)=mx which is the smallest upper bound of x in M. Denote equation image as the restriction of idX on M, i.e., for any mM, h(m)=m. It is obvious that both g and h are fuzzy order-preserving. Moreover, it is easy to verify that e(x, m)=e(mx, m) for any xX and mM. This implies that (g, h) is a fuzzy Galois connection between ((X, ≈), e) and ((M, ≈), e). Finally, M=hg(X) is obvious.

The converse immediately follows from Theorem 3.14.

Fuzzy closure L—systems on L—ordered sets

  1. Top of page
  2. Abstract
  3. Introduction
  4. L—ordered sets and fuzzy closure operators
  5. Fuzzy closure systems on L—ordered sets
  6. Fuzzy closure L—systems on L—ordered sets
  7. References

Recall that a fuzzy set A in X describes the truth degree of the fact that xX belongs to A. In this sense, fuzzy sets can be interpreted as a generalization of the classical subsets in the fuzzy setting. Apparently, the notion of fuzzy closure system in Definition 3.1 is developed in the classical way. A natural question arises that whether fuzzy closure systems can be defined in term of fuzzy sets. To this end, we propose the following definition.

Definition 4.1 Let ((X, ≈), e) be an L—ordered set. PLX is called a fuzzy closure (respectively, interior) L—system if for any xX, there exists equation image such that

  • (1)
    equation image (respectively, equation image;
  • (2)
    equation image (respectively, equation image);
  • (3)
    equation image (respectively, equation image) for any y, zX whenever P(y)=P(z)=1.

We call x̃ the upper (respectively, lower) limit of x under P.

Remark 4.2

  • (1)
    Suppose PLX is a fuzzy closure (respectively, interior) L—system. For any xX, the upper (respectively, lower) limit of x under P is unique.
  • (2)
    Obviously, every fuzzy closure L—system P on L—ordered set ((X, ≈), e) is a fuzzy interior L—system on the dual L—ordered set ((X, ≈), e−1). Moreover, for any xX, the upper limit of x under P w.r.t. ((X, ≈), e) is the lower limit of x under P w.r.t. ((X, ≈), e−1). Analogous properties hold for the fuzzy interior L—systems.

Proposition 4.3 Let ((X, ≈), e) be an L—ordered set, PLX a fuzzy closure L—system and equation image. Then the following are equivalent:

  • (1)
    x̃ is the upper limit of x under P.
  • (2)
    equation image, equation image, and equation image holds for any zX whenever P(z)=1.

Proof. Straightforward.

Dually, we have

Proposition 4.4 Let ((X, ≈), e) be an L—ordered set, PLX a fuzzy interior L—system and equation image. Then the following are equivalent:

  • (1)
    x̃ is the lower limit of x under P.
  • (2)
    equation image, equation image, and equation image holds for any zX whenever P(z)=1.

Now we are going to study the relationship between fuzzy closure (respectively, interior) systems and fuzzy closure (respectively, interior) L—systems on a common L—ordered set. For this purpose, we define the following mappings.

Given a fuzzy closure L—system PLX and a fuzzy closure system MX on L—ordered set ((X, ≈), e), define MPX by MP = {xX|P(x) = 1} and define PMLX by PM(x)=e(mx, x) for any xX, where mxM is the least upper bound of x in M. For a fuzzy interior L—system QLX and a fuzzy interior system NX, define NQX by NQ = {xX|Q(x) = 1} and QNLX by QN(x)=e(x, nx) for any xX, where nxN is the greatest lower bound of x in N. The theorem below shows that there is a bijective correspondence between fuzzy closure systems and fuzzy closure L—systems.

Theorem 4.5 Let ((X, ≈), e) be an L—ordered set, PLX a fuzzy closure L—system and MX a fuzzy closure system. Then MP is a fuzzy closure system and PM is a fuzzy closure L—system. Moreover, it holds that equation image and equation image, i.e., the mappings equation image and equation image are mutually inverse.

Proof. Suppose PLX is a fuzzy closure L—system. For any xX, by Definition 4.1, it is obvious that equation image and equation image. In addition, since P(x)=1 iff xMP, for any m, nMP, i.e., P(m)=P(n)=1, we have equation image. Therefore, MP is a fuzzy closure system and x̃ is exactly the least upper bound of x in MP.

On the other hand, suppose MX is a fuzzy closure system and xX. PM is well-defined from Remark 4.2 (1). To show PM is a fuzzy closure L—system on X, it is straightforward to check that the least upper bound mx of x in M is exactly the upper limit of x under the system PM.

It is trivial to check that equation image and equation image.

Dually,

Theorem 4.6 Let ((X, ≈), e) be an L—ordered set, QLX a fuzzy interior L—system and NX a fuzzy interior system. Then NQ is a fuzzy interior system and QN is a fuzzy interior L—system. Moreover, it holds that equation image and equation image, i.e., the mappings equation image and equation image are mutually inverse.

Finally, we investigate the relationship between fuzzy closure (respectively, interior) L—systems and fuzzy closure (respectively, interior) operators. Let ((X, ≈), e) be an L—ordered set, for a fuzzy closure L—system PLX and a fuzzy closure operator f: XX, define equation image by associating every xX with the upper limit x̃ of x under P and define PfLX by Pf(x)=e(f(x), x) for any xX. Dually, given a fuzzy interior L—system QLX and a fuzzy interior operator g: XX, define equation image by associating every xX with the lower limit x̃ of x under Q and define QgLX by Qg(x)=e(x, g(x)) for any xX. It is clear that equation image, equation image and equation image, equation image. By Theorem 3.8 and Theorem 4.5, we have

Corollary 4.7 Let ((X, ≈), e) be an L—ordered set, PLX a fuzzy closure L—system and f: XX a fuzzy closure operator. Then equation image is a fuzzy closure operator and Pf is a fuzzy closure L—system. Moreover, it holds that equation image and equation image, i.e., the mappings equation image and equation image are mutually inverse.

Dually, we have

Corollary 4.8 Let ((X, ≈), e) be an L—ordered set, QLX a fuzzy interior L—system and g: XX a fuzzy interior operator. Then equation image is a fuzzy interior operator and Qg is a fuzzy interior L—system. Moreover, it holds that equation image and equation image, i.e., the mappings equation image and equation image are mutually inverse.

From Theorem 3.8, Theorem 4.5, and Corollary 4.7, we have

Theorem 4.9 The diagram in Figure 1 commutes.

thumbnail image

Figure 1. Connection between fuzzy closure systems (operators) and Galois connections.

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By Theorem 3.8, Theorem 4.6 and Corollary 4.8, one can get a similar commutative diagram corresponding to the fuzzy interior operators, interior systems and interior L—systems.

References

  1. Top of page
  2. Abstract
  3. Introduction
  4. L—ordered sets and fuzzy closure operators
  5. Fuzzy closure systems on L—ordered sets
  6. Fuzzy closure L—systems on L—ordered sets
  7. References
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