Download I-fuzzy Alexandrov topologies and specialization orders

Fuzzy Sets and Systems 158 (2007) 2359 – 2374
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I -fuzzy Alexandrov topologies and specialization orders夡
Fang Jinming∗
Department of Mathematics, Ocean University of China, Qing Dao 266071, PR China
Received 25 April 2006; received in revised form 26 March 2007; accepted 7 May 2007
Available online 16 May 2007
Abstract
The aim of this paper is to investigate the relationships between certain generalized topological structures and fuzzy order structures
on a universe set X. In details, a fuzzy preordered relation induces an I-fuzzy Alexandrov topology; and an I-fuzzy topology induces
a fuzzy preordered relation, which turns out to be a specialization order in I-fuzzy setting. In addition, the Representation Theorem
of fuzzy preorders by I-fuzzy topologies is obtained. These results coincide with the conclusions obtained in literature in the version
of two-valued logic.
© 2007 Elsevier B.V. All rights reserved.
Keywords: Fuzzy preorder; Triangular norm; I-fuzzy topology; Alexandrov topology; Specialization order
1. Introduction
The motivation for this work starts from three concepts in mathematics. They are triangular t-norm, fuzzy relation
and fuzzy topology.
A triangular norm, a t-norm in short, on the unit interval I = [0, 1] is a binary operator ∗: [0, 1] × [0, 1] → [0, 1]
which is symmetric, associative, order-preserving on each place and has 1 as the unit element.
A fuzzy relation on a universe set X is a map R : X × X → [0, 1]. For each (x, y) ∈ X × X, the real number
R(x, y) is interpreted as the degree to which x is less than y; therefore, a fuzzy relation R has a kind of many-valued
sense to express imprecise information between elements of X. A fuzzy relation R is called reflexive if R(x, x)=1 for
each x in X; symmetric if R(x, y) = R(y, x) for all (x, y) ∈ X × X; ∗-transitive if R(x, y) ∗ R(y, z) R(x, z) for all
x, y, z ∈ X. A reflexive and ∗-transitive fuzzy relation is called a ∗-fuzzy preorder, and a symmetric fuzzy preorder is
called a fuzzy equivalence. Clearly, a classical relation on a set X could be considered as a fuzzy relation if needed.
An I-topology [16,17] on a set X is a crisp subset of I X , where I is the unit interval, which is closed with respect to
finite meets and arbitrary joins and which contains all the constant functions from X to [0, 1]. However, in a completely
different direction, Höhle [5] created the notion of a topology as an L-subset of a powerset (in his case, 2X ). Then
Kubiak [14] and Šostak [21] jointly and independently extended Höhle’s notion to an L-subset of LX (also see [5,6]),
夡
Project supported by Foundation of Nature Science of Shan Dong Province and China.
∗ Fax: +86 05322718159.
E-mail address: [email protected].
0165-0114/$ - see front matter © 2007 Elsevier B.V. All rights reserved.
doi:10.1016/j.fss.2007.05.001
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F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
where L is any appropriate lattice. Thus, we have a concept of L-fuzzy topology on a set X. Precisely, when L = I , an
I-fuzzy topology of the paper on a set X is a map : I X → [0, 1] satisfying the following conditions:
(1) () = 1 for all constant functions from X to [0, 1];
X
(2) (U
∧ V )(U) ∧ (V ), ∀U, V ∈ I ;
(3) ( j ∈J Uj ) j ∈J (Uj ), ∀j ∈ J, Uj ∈ I X .
Thus, the real number (U ) for a fuzzy subset U can be viewed as the degree to which U is open. Hence in a personal
viewpoint, I-fuzzy topology on a universe set X has many-valued sense which reflects internal requirement of fuzziness
of topology. If a map : 2X → [0, 1] satisfies the similar conditions of I-fuzzy topology on a set X, then is called a
fuzzifying topology [5,23,24] on X. Obviously, an I-topology on a set X can be viewed as a special kind of I-fuzzy
topology on the set X, which maps an element U of I X to 1 for U ∈ and 0 otherwise. The reader should note that
I-fuzzy topology of the paper is always in the sense above, which should not be confused with the concept of I-fuzzy
topology in the sense that an I-fuzzy topology (see [7]) is a map : I X → I satisfying
(O1) (1X ) = 1;
(O2) (U
) ∗ (V ), ∀U, V ∈ I X ;
∗ V )(U
(O3) ( j ∈J Uj ) j ∈J (Uj ), ∀j ∈ J, Uj ∈ I X ,
where ∗ is a certain binary operation on I.
Recall that in classical mathematics, for any fixed set X we can obtain a topology on X determined by a given
classical preorder on X, precisely, all upper sets with respect to forms an Alexandrov topology. Conversely, we
can also obtain a preorder on X determined by a given topology on X, called a specialization order usually. This process
determined one by another had been investigated in the fuzzy setting in the literature. For example:
(1) The authors [15] proved the set ∗ (R) of all upper fuzzy subsets (equivalently, extensional fuzzy sets = generators,
or eigenvectors [3,8,9,11,22]) of a fuzzy preorder R forms an I-topology on X. In [18], the authors investigated the
relationships between I-topology and fuzzy preorder from the viewpoint of rough sets, but they only focused on
finite sets.
(2) The author [22] offered a powerful method to generate fuzzy preorders and fuzzy equivalences by I-topology, or
even by a family of fuzzy subsets in X in general. For more details obtained fuzzy preorder by I-topologies, please
refer to [15]. In [15], the authors established the relationships between fuzzy preorders, crisp topological spaces
and I-topological spaces from a categorical point of view.
The aim of this paper is to investigate the relationships between I-fuzzy structures and fuzzy order structures on a
universe set X. In details, a fuzzy preordered relation induces an I-fuzzy Alexandrov topology, which is an I-fuzzified
set of all upper sets indeed; and an I-fuzzy topology induces a fuzzy preordered relation, which turns out to be a
specialization order in I-fuzzy setting. In addition, the Representation Theorem of fuzzy preorders by I-fuzzy topologies
is obtained. These results coincide with the conclusions obtained in literature in the version of two-valued logic.
It should be noted that the relationship between fuzzy preorders and fuzzy topological spaces has also been investigated from the viewpoint of category and that of modifiers, see e.g. [13,15]. However, as the reader shall see, the
topics in these papers are different from this article. In article [15], the authors focused on I-topologies in the sense of
Lowen [16,17], not I-fuzzy topologies in this paper. And in [13], the authors constructed an L-fuzzy topology on a set
from a crisp preorder on X, where L is a complete lattice. But the focus of this paper is the relationship between fuzzy
preordered sets and I-fuzzy topological spaces.
The contents of this paper are organized as follows: Section 2 offers some preparations including some definitions
and results needed in the paper. In Section 3, we focus on the problem how to obtain an I-fuzzy topology through a
fuzzy relation on a set. Therefore, the notion of an I-fuzzified set of all upper sets (dually, an I-fuzzified set of all lower
sets) of a fuzzy preordered set (X, R) is introduced, and it is showed that an I-fuzzified set of all upper sets induced by
a fuzzy preorder R is precisely an I-fuzzy topology of the fuzzy preordered set (X, R), called an I-fuzzy Alexandrov
topology in the paper. Categorically speaking, we get a concrete functor from the category of fuzzy preordered sets to
that of I-fuzzy topological spaces. Moreover, a general method to induce fuzzy relations is also obtained in this section,
which coincides with the powerful method introduced in [15,22] in the version of two-valued logic. In addition, if the
fuzzy preorder R on a set X is fuzzy equivalence, we obtain the notion of ∗-pseudo-discrete I-fuzzy topology on X.
In Section 4, we concern the problem how to get fuzzy preorder through an I-fuzzy topology on a set X. In details,
F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
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the notion of specialization order of an I-fuzzy topological space is introduced, and surprisingly, the fuzzy preordered
relation induced by the general method of Section 3 must be the specialization order of an I-fuzzy topology. Thus,
we also obtain a concrete functor from the category of I-fuzzy topological spaces to that of fuzzy preordered sets.
Furthermore, the Representation Theorem of fuzzy preorders by I-fuzzy topologies is obtained. Finally, we show that
the resulting two concrete functors in Sections 3 and 4 form a pair of adjoint functor in case of t-norm ∗ = ∧.
2. Preliminaries
The aim of this section is to offer a collection of known results of fuzzy relations on a set X and triangular norms on
the unit interval I = [0, 1]. Let us start with the definition of triangular norm.
Definition 2.1 (Klement et al. [12] and Schweizer and Sklar [20]). A triangular norm (a t-norm, in short) is a binary
operation ∗ on the unit interval [0, 1], i.e., a function ∗: [0, 1] × [0, 1] → [0, 1], such that for all x, y, z ∈ [0, 1] the
following axioms are satisfied:
(T1)
(T2)
(T3)
(T4)
(commutativity) x ∗ y = y ∗ x;
(monotonicity) x ∗ y x ∗ z if y z;
(associativity) (x ∗ y) ∗ z = x ∗ (y ∗ z);
(boundary condition) x ∗ 1 = x.
A t-norm ∗ is said to be left continuous if for each x ∈ [0, 1], the function x ∗ (−): [0, 1] → [0, 1] has a right adjoint.
That is, there is a function x −→ (−): [0, 1] → [0, 1] such that x ∗ y z if and only if x y −→ z.
In complete analogy with classical (or crisp) relation on a universe set X, a fuzzy relation on a set X is a map R:
X × X → I. A fuzzy relation R is called
(1) reflexive if R(x, x) = 1 for all x ∈ X;
(2) symmetric if R(x, y) = R(y, x) for all x, y ∈ X;
(3) ∗-transitive if R(x, y) ∗ R(y, z) R(x, z) for all x, y, z ∈ X and a t-norm ∗ on [0, 1].
Moreover, we have the following terminology with respect to fuzzy relation.
Definition 2.2. Let X be a universe set and ∗ a t-norm on [0, 1]. A fuzzy relation R on X is called
(1) ∗-fuzzy preorder if it is a reflexive and ∗-transitive fuzzy relation;
(2) ∗-fuzzy equivalence if it is a symmetric ∗-fuzzy preorder;
(3) ∗-fuzzy partial order if it is a ∗-fuzzy preorder and for any pair(x, y) ∈ X × X, R(x, y) = R(y, x) = 1 implies
that x = y.
If the t-norm ∗ is clear from the context, we will omit ∗, just a fuzzy preorder, a fuzzy equivalence and a fuzzy partial
order. A pair (X, R) is said to be a fuzzy preordered set (a fuzzy partial order set) if R is a fuzzy preorder (partial order).
Clearly, a classical preorder on a set X can be viewed as a fuzzy preorder R on X whose range is {0, 1}. Note
that when my paper is accepted, under the Editors and Reviewers’s suggestion, we find the paper [2] which gives
representation and construction results for fuzzy weak orders, and the readers may read it if they like.
Definition 2.3 (Lai and Zhang [15]). A function f: (X, RX ) → (Y, RY ) between fuzzy preordered sets is called monotone if RX (x, y)RY (f (x), f (y)) for all x, y ∈ X. The category of all fuzzy preordered sets and monotone functions
is denoted by FPrOrd. FEQ for the full subcategory of FPrOrd composed of objects of the form (X, R) with R a
fuzzy equivalence.
For a given left continuous t-norm ∗ on [0, 1], the resulting binary function −→: [0, 1] × [0, 1] → [0, 1], given by
−→ (x, y) = (x −→ y), is called the residuation operation or the implication operation, with respect to ∗. The binary
function ←→: [0, 1] × [0, 1] → [0, 1], given by ←→ (x, y) = (x ←→ y) = min{x −→ y, y −→ x}, is called the
biresiduation operation on [0, 1]. The following results on t-norms and residuation operations can be found in [12,15]
and others.
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F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
Proposition 2.4. Suppose that ∗ is a left continuous t-norm and −→ is the residuation operation with respect to ∗.
Then
(I1)
(I2)
(I3)
(I4)
(I5)
(I6)
(I7)
(I8)
(I9)
(I10)
(I11)
(I12)
x −→ y = 1 if and only if x y;
1 −→ x = x;
(x −→
y) ∗ (y −→z) x −→ z;
x
−
→
j ∈J yj = j ∈J (x −→ yj ), hence, x −→ y x −→ z if y z;
y
= j ∈J yj −→ y, hence, x −→ z y −→ z if x y;
j ∈J
j −→ y x ∗ j ∈J yj = j ∈J x ∗ yj ;
x −→ (y −→ z) = y −→ (x −→ z);
(x −→ y) ∗ (u −→ v)x ∗ u −→ y ∗ v;
(x ←→ y) ∗ (u ←→ v)(x ∗ u) ←→ (y ∗ v);
x (x −→ y) −→ y;
x −→ y = ∈[0,1] (( −→ x) −→ ( −→ y));
x −→ y = ∈[0,1] ((y −→ ) −→ (x −→ )).
By (I1) and (I3), the residuation operation −→: [0, 1] × [0, 1] → [0, 1] with respect to a t-norm ∗ is a fuzzy preorder
on [0, 1] and called the canonical fuzzy preorder on [0, 1]. Clearly, the residuation operation −→: [0, 1]×[0, 1] → [0, 1]
is order-reversing in the first place and order-preserving in the second place by (I5) and (I4), respectively.
Let X be a universe set, and the family of all fuzzy subsets in X will be denoted by I X . By 0X and 1X , we denote
the constant fuzzy set on X taking the value 0 and 1, respectively. For terminology in I-topology, we refer to [16] and
an I-topology on a set is always supposed to be in Lowen’s sense [16], and in classical topology, we refer to [10]. The
following definitions and results will be used frequently in the sequel.
Definition 2.5 (Höhle [5–7], Kubiak [14] and Šostak [21]). An I-fuzzy topology on a set X is a map : I X → I such
that
(I-FO1) () = 1 for every constant fuzzy subset : X → [0, 1];
X
(I-FO2) (U
∧ V ) (U) ∧ (V ), ∀U, V ∈ I ;
(I-FO3) ( j ∈J Uj ) j ∈J (Uj ), ∀j ∈ J, Uj ∈ I X .
And is called an I-fuzzy saturated topology if, in addition, satisfies
(I-FOS) ( j ∈J Uj ) j ∈J (Uj ), ∀j ∈ J, Uj ∈ I X .
If is an I-fuzzy topology on X, then we say that (X, ) is an I-fuzzy topological space (I-fts, in short). A continuous
map between two I-fts’s (X, ) and (Y, ) is a mapping f : X → Y such that (U ◦ f )(U ) for all U ∈ I Y . The
category of I-fuzzy topological spaces and their continuous mappings is denoted by I-FTop [5–7,19].
In this paper, a concrete category is a pair (C, U), where C is a category and U: C → Set is a forgetful functor. So,
every object in a concrete category can be regarded as a structured set. We simply write C for (C, U) if the forgetful
functor is evident. All the categories involved in this paper are concrete categories. A concrete functor between concrete
categories (C, U) and (D, V) is a functor F: C → D such that U = V ◦ F. It means that F only changes the structures
on the underlying sets, leaving the underlying sets and morphisms unchanged.
Proposition 2.6 (Adamek et al. [1]). Suppose that F: C → D and G: D → C are concrete functors. Then the following
conclusions are equivalent:
(1) {idY : F ◦ G(Y ) → Y |Y ∈ D} is a natural transformation from the functor F ◦ G to the identity functor on D; and
{idX : X → G ◦ F(X)|X ∈ C} is a natural transformation from the identity functor on C to G ◦ F.
(2) For each Y ∈ D, idY : F ◦ G(Y ) → Y is a D-morphism; and for each X ∈ C, idX : X → G ◦ F(X) is a C-morphism.
In this case, (F, G) is called a Galois correspondence [1].
If (F, G) is a Galois correspondence, then it is easy to check that F is a left adjoint of G or equivalently, G is a right
adjoint of F.
F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
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We end this section with the observation that all algebraic operations on the unit interval [0, 1] can be extended to
the set I X pointwisely. That is, for U, V ∈ I X , ∈ [0, 1] and x ∈ X,
(1)
(2)
(3)
(4)
U V if and only if U (x) V (x);
(U ∗ V )(x) = U (x) ∗ V (x);
( ∗ U )(x) = ∗ U (x);
(U −→ )(x) = U (x) −→ .
X
Obviously,
(I
lattice and for a family of fuzzy subsets F ⊆ I X , F is given
, ) is still a completely
distributive
by F(x) = {U (x)|U ∈ F} and F(x) = {V (x)|V ∈ F} for x ∈ X. Finally, we do not distinguish a number
and a constant function : X → [0, 1] such that (x) = for all x ∈ X.
3. Fuzzy preorders and I -fuzzified set of all upper sets
In this section, we focus on the problem how to obtain an I-fuzzy topology through a fuzzy relation on a set. To
realize the targets, the notion of an I-fuzzified set of all upper sets (dually, an I-fuzzified set of all lower sets) of a fuzzy
preordered set (X, R) is proposed firstly.
Definition 3.1. An I-fuzzified set of all upper sets of a fuzzy preordered set (X, R) is a map ∇(R): I X → I defined by
∀U ∈ I X , ∇(R)(U ) =
R(x, y) −→ (U (x) −→ U (y)).
(x, y)∈X×X
Dually, an I-fuzzified set of all lower sets of a fuzzy preordered set (X, R) is a map (R): I X → I defined by
R(x, y) −→ (D(y) −→ D(x)).
∀D ∈ I X , (R)(D) =
(x, y)∈X×X
A fuzzy subset U is called an upper set (resp., a lower set) if ∇(R)(U ) = 1 (resp., (R)(U ) = 1).
Remark 3.2. (1) For a fuzzy subset U ∈ I X , the real number ∇(R)(U ) (resp., (R)(U )) is viewed as the degree to
which U is an upper set (resp., a lower set).
(2) A fuzzy set : X → [0, 1] of a fuzzy preordered set (X, R) is called an upper set (see Definition 3.6 [15]) if
(x) ∗ R(x, y)(y) for any x, y ∈ X. Thus, a fuzzy subset U is an upper set if and only if ∇(R)(U ) = 1. Hence, the
notion of upper set [15] coincides with that of upper set introduced in this paper.
Example 3.3. (1) Suppose that (X, R) is a fuzzy preordered set and z ∈ X. Then the fuzzy set [z]R : X → [0, 1]
given by [z]R (x) = R(z, x) is called the principal upper set [15] generated by z ∈ X. Similarly, the fuzzy subset [z]R :
X → [0, 1] given by [z]R (x) = R(x, z) is a lower set and called the principal lower set [15] generated by z ∈ X. In
this case, we have both ∇(R)([z]R ) = 1 and (R)([z]R ) = 1. That is, the degree to which [z]R (resp., [z]R ) is an upper
set (resp., a lower set) is 1.
(2) Suppose (X, R) is a classical preordered set. It is easy to check that for each U ∈ I X , ∇(R)(U ) = 1 ((R)(U ) =
1) in (X, R) viewed as a fuzzy preordered set if and only if U is order-preserving (order-reversing).
For a crisp preordered set X, the family of all upper subsets of X is clearly a topology on X, called the Alexandrov
topology usually and denoted (). We write simply (X) for the topological space (X, ( )). According to this
idea, we can find the result below.
Theorem 3.4. For a given fuzzy preorder R on X, ∇(R), the I-fuzzified set of all upper sets of (X, R) has the following
properties: for all F ⊆ I X , U, V ∈ I X and ∈ [0, 1],
(i)
(ii)
(iii)
(iv)
(v)
∇(R)() = 1 for every
constant mapping from X to [0, 1];
∇(R)(F)∇(R)(
F), where ∇(R)(F) = {∇(R)(U )| U ∈ F};
∇(R)(F) ∇(R)( F);
∇(R)(U )∇(R)( ∗ U );
∇(R)(U )∇(R)( −→ U ).
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F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
Proof. (i): Trivial and omitted. We show (ii) and (iii). Suppose that F is a family of fuzzy subset in X. Then
∇(R)
F =
R(x, y) −→
F (x) −→
F (y)
(x, y)∈X×X
=
R(x, y) −→
(x, y)∈X×X
=
U (x) −→
F (y) (by (I5))
U ∈F
R(x, y) −→ U (x) −→
F (y) (by (I4))
(x, y)∈X×X U ∈F
R(x, y) −→ (U (x) −→ U (y)) (by (I4))
U ∈F (x, y)∈X×X
=
∇(R)(U ).
U ∈F
Thus we obtain (ii). For (iii), we have
∇(R)
F =
R(x, y) −→
F (x) −→
F (y)
(x, y)∈X×X
=
R(x, y) −→
U ∈F
(x, y)∈X×X
F (x) −→ (U (y))
U ∈F
R(x, y) −→ (U (x) −→ U (y))
U ∈F (x, y)∈X×X
=
∇(R)(U ),
U ∈F
by (I4) and (I5) in Proposition 2.4.
(iv) For ∈ [0, 1], U ∈ I X ,
∇(R)(U ) =
R(x, y) −→ (U (x) −→ U (y))
(x, y)∈X×X
=
R(x, y) −→ (( −→ ) ∗ (U (x) −→ U (y)))
(x, y)∈X×X
R(x, y) −→ ( ∗ U (x) −→ ∗ U (y))
(by (I8))
(x, y)∈X×X
= ∇(R)( ∗ U ).
(v) For any U ∈ I X and ∈ [0, 1],
∇(R)( −→ U ) =
R(x, y) −→ (( −→ U (x)) −→ ( −→ U (y)))
(x, y)∈X×X
R(x, y) −→ (U (x) −→ U (y))
(x, y)∈X×X
= ∇(R)(U ),
by (I11) of Proposition 2.4.
Following Theorem 3.4, if we assume R is a fuzzy equivalence, we obtain a further result as follows.
Theorem 3.5. Let R be a fuzzy equivalence on X. Then ∇(R) satisfies (i)–(v) in Theorem 3.4 and in addition,
(vi) ∀ ∈ [0, 1], U ∈ I X , ∇(R)(U )∇(R)(U −→ ).
F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
2365
Proof. In fact, using (I12) of Proposition 2.4, we have
∇(R)(U −→ ) =
R(x, y) −→ ((U (x) −→ ) −→ (U (y) −→ ))
(x, y)∈X×X
R(y, x) −→ (U (y) −→ U (x))
(y, x)∈X×X
= ∇(R)(U ).
Remark 3.6. For ∈ [0, 1], U ∈ I X , the form of U −→ is worth a comment. U −→ can be considered as the
negation of U, which is pointed out by authors in [15]. Thus this consideration agrees with the definition of negation
in classical sense. In fact, given U ⊆ X, 1U −→ 0 = 1(X−U ) , where −→ is the implication for the Boolean algebra
2 = {0, 1}. Theorem 3.5 tells us that the degree to which a fuzzy set U is open is less than the degree to which its
negation U −→ is open. Note that the conclusions of Theorem 3.5 generalize those of Theorem 3.17 in [15], which
assert that U −→ ∈ ∗ (R) for all U ∈ ∗ (R), ∈ [0, 1], and it was proved in [11] firstly.
Proposition 3.7. Suppose that (X, R) is a fuzzy preordered set. Then
(a) ∀ ∈ [0, 1], U ∈ I X , ∇(R)(U )(R)(U −→ );
(b) ∀ ∈ [0, 1], U ∈ I X , (R)(U )∇(R)(U −→ ).
Proof. We prove the conclusion (a) for example. Indeed, it follows from (I12) that
(R)(U −→ ) =
R(x, y) −→ ((U (x) −→ ) −→ (U (y) −→ ))
(x, y)∈X×X
R(x, y) −→ (U (y) −→ U (x))
(x, y)∈X×X
= ∇(R)(U ).
Remark 3.8. The conclusion (a) (resp.,(b)) in Proposition 3.7 implies that the degree to which a fuzzy set U is an
upper set (resp., a lower set) is less than that to which its negation is a lower (resp., an upper) set. Specially, in the case
of ∇(R)(U ) = 1 (resp., (R)(U ) = 1), we get that if a fuzzy set U is an upper set (resp., a lower set), then its negation
U −→ is a lower set (resp., an upper set) [15].
Definition 3.9. Let (X, R) be any fuzzy preordered set and ∇(R): I X → [0, 1] be the I-fuzzified set of all upper
sets of (X, R). Then ∇(R) is an I-fuzzy topology on X (see Theorem 3.4) and it will be called an I-fuzzy Alexandrov
topology on X induced by the fuzzy preorder R. Write ∇(X) for the I-fuzzy topological space (X, ∇(R)) briefly.
Recall that for a crisp topological space (X) determined by a crisp preordered set X, if X is a poset, then (X) is
a T0 -topological space. Given a preordered set X, if the preorder on X is an equivalence, it is known that (X) is a
pseudo-discrete topological space, i.e., the complement of every open set in (X) is also a open set. Now to get their
generalized form (see Theorem 3.11 below) in I-fuzzy setting, we propose two concepts firstly.
(1) An I-fuzzy topological space (X, ) is said to be ∗-pseudo-discrete, or simply pseudo-discrete if the t-norm ∗ is
clear from the context, if (U ) (U −→ ) for all ∈ [0, 1] and U ∈ I X . Note that the underlying idea is that the
degree to which a set U is open is less than that to which its complement X − U is open. Thus, this definition agrees
well with the definition of pseudo-discrete spaces in classical setting.
(2) The degree to which (X, ) is T0 is defined by
⎡
⎢
T0 (X, ) =
⎣
x=y
⎤
U (x)U (y)
(U ) ∨
V (y)V (x)
⎥
(V )⎦ .
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F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
Lemma 3.10 (Lai and Zhang [15]). Suppose that R is a fuzzy preorder on X. The following statements are
equivalent:
(1) R is a fuzzy partial order;
(2) For any x, y ∈ X, x = y if and only if R(z, x) = R(z, y) for all z ∈ X.
Theorem 3.11. Suppose that R is a fuzzy preorder on X. Then
(1) If R is a fuzzy partial order, then T0 (∇(X)) = 1;
(2) If R is a fuzzy equivalence, then ∇(X) is pseudo-discrete.
Proof. The proof of (1): Let R be a fuzzy partial order and x, y ∈ X with x = y. It follows from Lemma 3.10 (2) that
there exists some z ∈ X such that R(z, x) = R(z, y). We consider the case of R(z, x)R(z, y) for example. Thus,
U(x,y) := [z]R : X → I such that U(x, y) (u) = R(z, u) for all u ∈ X satisfies ∇(R)(U(x, y) ) = 1 by Example 3.3 (1).
It follows from U(x, y) (x) = R(z, x)R(z, y) = U(x, y) (y) that
T0 (∇(X)) =
∇(R)(U ) ∨
∇(R)(V )
x=y
U (x)U (y)
V (y)V (x)
∇(R)(U(x, y) ) = 1.
x=y
The proof of (2) is from Theorem 3.5.
Proposition 3.12. Suppose that f: (X, RX ) → (Y, RY ) is a monotone function between two fuzzy preordered sets.
Then f: ∇(X) → ∇(Y ) is continuous.
Proof. For any U ∈ I Y , we have
∇(RX )(U ◦ f ) =
RX (x, y) −→ (U ◦ f (x) −→ U ◦ f (y))
(x, y)∈X×X
RY (f (x), f (y)) −→ (U ◦ f (x) −→ U ◦ f (y))
(x, y)∈X×X
RY (u, v) −→ (U (u) −→ V (v))
(u, v)∈Y ×Y
= ∇(RY )(U ),
since RX (x, y) R(f (x), f (y)) for all pair (x, y) ∈ X × X. By the definition of continuity of f, the conclusion is
true. By the above proposition, we obtain a concrete functor ∇: FPrOrd −→ I -FTop from the category of fuzzy
preordered sets to that of I-fuzzy topological spaces.
In [22], the author provides a powerful method to generate fuzzy preorders and fuzzy equivalences by a family of
fuzzy subsets on X, which is called the Valverde orders on X by Lai and Zhang [15]. That is,
Theorem 3.13 (Valverde [22]). Let F ⊆ I X be a family of fuzzy subsets in X. Then
∗ (F)(x, y) =
U (x) −→ U (y)
U ∈F
is a fuzzy preorder on X. And,
R(x, y) =
U (x) ←→ U (y)
U ∈F
is a fuzzy equivalence on X. In this case, ∗ (F) is called the Valverde order on X determined by F.
F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
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On account of this idea, we could obtain a more general method to generate fuzzy preorder on a set X.
Definition 3.14. Let : I X → I be a map and define a fuzzy relation R as follows:
∀(x, y) ∈ X × X, R (x, y) =
(U ) −→ (U (x) −→ U (y)).
U ∈I X
Then R is reflexive fuzzy relation, and we denote the transitive closure of R by (), call it -order determined
by .
Theorem 3.15. Let : I X → I be a map and ∗ be idempotent, i.e., ∗ = ∧. Then R : X × X → I defined by
∀(x, y) ∈ X × X, R (x, y) =
(U ) −→ (U (x) −→ U (y)),
U ∈I X
is a ∧-fuzzy preorder on X, in the other words, R = (). Furthermore,
R(x, y) =
(U ) −→ (U (x) ←→ U (y))
U ∈I X
is a ∧-fuzzy equivalence on X.
Proof. For the conclusion that R is a fuzzy preorder on X, it suffices to prove that R is ∧-transitive, i.e., ∀x, y, z ∈ X,
R (x, y) ∧ R (y, z) R (x, z).
This inequality can be proved by the following formulas:
R (x, z) =
(W ) −→ (W (x) −→ W (z))
W ∈I X
(W ) ∧ (W ) −→ (W (x) −→ W (y)) ∧ (W (y) −→ W (z))
W ∈I X
(by (I3) of Proposition 2.4)
[(W ) −→ (W (x) −→ W (y))] ∧ [(W ) −→ (W (y) −→ W (z))]
W ∈I X
(by (I8) of Proposition 2.4)
[(U ) −→ (U (x) −→ U (y))] ∧ [(V ) −→ (V (y) −→ V (z))]
U,V ∈I X
⎡
⎣
⎤
⎡
(U ) −→ (U (x) −→ U (y))⎦ ∧ ⎣
U ∈I X
= R (x, y) ∧ R (y, z).
⎤
(V ) −→ (V (y) −→ V (z))⎦
V ∈I X
Obviously,
R(x, y) =
(U ) −→ (U (x) ←→ U (y))
U ∈I X
is both reflexive and symmetric. Its ∧-transitivity can be obtained as follows: ∀x, y, z ∈ X,
(W ) −→ (W (x) ←→ W (z))
R(x, z) =
W ∈I X
(W ) ∧ (W ) −→ (W (x) ←→ W (y)) ∧ (W (y) ←→ W (z))
W ∈I X
(by (I3) of Proposition 2.4)
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F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
[(W ) −→ (W (x) ←→ W (y))] ∧ [(W ) −→ (W (y) ←→ W (z))]
W ∈I X
(by (I8) of Proposition 2.4)
[(U ) −→ (U (x) ←→ U (y))] ∧ [(V ) −→ (V (y) ←→ V (z))]
U,V ∈I X
⎡
=⎣
U ∈I X
⎤
⎡
(U ) −→ (U (x) ←→ U (y))⎦ ∧ ⎣
= R(x, y) ∧ R(y, z).
⎤
(V ) −→ (V (y) ←→ V (z))⎦
V ∈I X
Remark 3.16. (1) When : I X → 2={0,1}⊆ I , can be viewed as a family of fuzzy subsets F = −1 (1)={U |
(U ) = 1} ⊆ I X . Thus, () coincides with the Valverde order on X determined by F.
(2) If : I I → I such that (idI ) = 1 and otherwise = 0, then () is the canonical fuzzy preorder on [0, 1]
named by Lai and Zhang [15].
Example 3.17. By Theorem 3.15, R : X × X → I defined by
∀(x, y) ∈ X × X, R (x, y) =
(U ) −→ (U (x) −→ U (y))
U ∈I X
is a ∧-fuzzy preorder on X, in the other words, R = () when ∗ is idempotent (i.e., ∗ = ∧). We point out the
condition, which the t-norm ∗ is idempotent, is necessary. Otherwise, the conclusion of Theorem 3.15 need not be
true. In fact, let X = [0, 1], A : X → [0, 1] such that A(x) = 2x − 1 if 21 < x 1 and = 0 otherwise. Assume that
: [0, 1]X → [0, 1] such that (A) = 78 and =0 otherwise. Thus, appealing to definition of R , we obtain a concrete
form of R as follows:
∀(x, y) ∈ X × X, R (x, y) = (A) −→ (A(x) −→ A(y)).
Now we could show that R is not transitive if ∗, where x ∗ y = max{x + y − 1, 0} for x, y ∈ [0, 1], is the Lukasiewicz
t-norm on X, which is not idempotent. Indeed, let x = 78 , y = 68 and z = 58 , we have that R (x, y) = 78 , R (y, z) = 78
and R (x, z) = 58 . Hence,
3
4
= R (x, y) ∗ R (y, z)R (x, z) = 58 .
It means that R is not transitive with respect to the Lukasiewicz t-norm ∗.
4. Fuzzy orders and I-fuzzy topologies
In this section, for a given I-fuzzy topological space (X, ), we consider the problem how to induce a fuzzy order
on X by the I-fuzzy topology .
Recall that the specialization order on X induced by a classical topological space (X, T ) is as follows: x y if
x ∈ U implies y ∈ U for each open set U of X, or equivalently, x ∈ {y}, where {y} is the closure of {y} in (X, T ).
Following this idea and Definition 3.14, we propose a definition of the specialization order of an I-fuzzy topological
space.
Definition 4.1. Let (X, ) be any I-fuzzy topological space. The -order () determined by is called the specialization order of (X, ). We will write (X) for the fuzzy preordered set (X, ()) briefly.
Remark 4.2. (1) For (x, y) ∈ X × X and a real number , the condition of ()(x, y) can be interpreted as the
statement that in -level, x is less than or equal to y if “x is in U” implies “y is in U” for any U in .
(2) If an I-fuzzy topology on a set X reduces to an I-topology in the sense of Lowen [16,17], i.e., : I X → 2 =
{0, 1} ⊆ [0, 1], this definition is in accord with that of the specialization order of I-topological space in the sense of
Lai and Zhang [15]. Clearly, this definition is also in accordance with that of the classical specialization order.
F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
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An I-fuzzy topology on X is said to be generated by : I X → I if is the coarsest I-fuzzy topology on X containing
, i.e., , and an I-fuzzy topology on X with the property of implies . In this case, is said to be a
subbase [7] of .
Theorem 4.3. Let (X, ) be any I-fts and be generated by : I X → I . Then ()(x, y) = ()(x, y) for each
(x, y) ∈ X × X, i.e., () = ().
Proof. Let
R (x, y) =
(U ) −→ (U (x) −→ U (y))
U ∈I X
for (x, y) ∈ X × X. Then R is a fuzzy relation on X. Let ∇(R ) be the I-fuzzy Alexandrov topology on X induced
by R . We have, for any U ∈ I X ,
R (x, y) −→ (U (x) −→ U (y))
∇(R )(U ) =
(x, y)∈X×X
=
⎡
⎣
⎤
(V ) −→ (V (x) −→ V (y))⎦ −→ (U (x) −→ U (y))
V ∈I X
(x, y)∈X×X
[(U ) −→ (U (x) −→ U (y))] −→ (U (x) −→ U (y))
(x, y)∈X×X
(by (I5) of Proposition 2.4)
(U ) (by (I10) of Proposition 2.4)
i.e., ∇(R ). It follows that ∇(R ) since is the coarsest I-fuzzy topology on X containing . On account of
, we obtain that
(U ) −→ (U (x) −→ U (y))
R (x, y) =
U ∈I X
(U ) −→ (U (x) −→ U (y)) = R (x, y)
(by (I5) of Proposition 2.4)
U ∈I X
U ∈I X
=
U ∈I X
∇(R )(U ) −→ (U (x) −→ U (y)) (by (I5) of Proposition 2.4)
⎡
⎣
⎤
R (s, t) −→ (U (s) −→ U (t))⎦ −→ (U (x) −→ U (y))
(s, t)∈X×X
[R (x, y) −→ (U (x) −→ U (y))] −→ (U (x) −→ U (y))
U ∈I X
(by (I5) of Proposition 2.4)
R (x, y) (by (I10) of Proposition 2.4),
for each pair (x, y) ∈ X × X, i.e., R = R . Therefore, ∀(x, y) ∈ X × X,
()(x, y) = ()(x, y)
or
() = ().
Remark 4.4. By Definition 3.14, we have obtained the general method induced fuzzy preorder on a set X by a map :
I X → [0, 1]. Theorem 4.3 tells us the -order determined by is just the specialization order of the I-fuzzy topological
space (X, ), where is a subbase of . Hence, the -order is nothing but the specialization order of I-fuzzy topological
spaces.
Referring to Proposition 3.11(2), we have:
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F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
Theorem 4.5. If (X, ) is a pseudo-discrete I-fuzzy topological space, then R is a both reflexive and symmetric fuzzy
relation on X.
Proof. It suffices to show that R is symmetric. We will show
R (x, y) =
(U ) −→ (U (x) ←→ U (y))
U ∈I X
for each (x, y) ∈ X × X and hence R is symmetric, as desired. In general, clearly
R (x, y) (U ) −→ (U (x) ←→ U (y))
U ∈I X
by the definition of R . The proof of converse inequality is given as follows: for any U ∈ I X , if U (x) U (y), then
(U ) −→ (U (x) −→ U (y)) = (U ) −→ (U (x) ←→ U (y));
otherwise if U (x) < U (y), letting V ∈ [0, 1]X such that V (z) = U (z) −→ U (x) for any z ∈ X, then applying to the
pseudo-discrete property of and V (x) = 1,
(V ) −→ (V (x) −→ V (y)) (U ) −→ (U (y) −→ U (x))
= (U ) −→ (U (y) ←→ U (x)).
Therefore,
R (x, y) (U ) −→ (U (x) ←→ U (y)).
U ∈I X
Proposition 4.6. If f: (X, X ) → (Y, Y ) is a continuous function between two I-fuzzy topological spaces, then f :
(X) → (Y ) is monotone.
Proof. It follows from the continuity of f that Y (V )X (V ◦ f ) for each V ∈ I Y . Thus ∀(x, y) ∈ X × X,
(Y )(f (x), f (y)) RY (f (x), f (y))
=
Y (V ) −→ (V (f (x)) −→ V (f (y)))
V ∈I Y
X (V ◦ f ) −→ (V ◦ f (x) −→ V ◦ f (y))
V ∈I Y
X (U ) −→ (U (x) −→ U (y))
U ∈I X
= RX (x, y).
Because (Y )(f (−), f (−)) : X × X → [0, 1] is ∗-transitive fuzzy relation containing RX , it follows that
(Y )(f (x), f (y)) (X )(x, y) from the definition of (X ) for any pair (x, y) ∈ X × X. This implies that f:
(X) → (Y ) is monotone. From the above proposition, we obtain a concrete functor : I-FTop → FPrOrd from the category of I-fuzzy
topological spaces to that of fuzzy preordered sets.
At the end of the section, we want to investigate the relationships between the concrete functor ∇ : FPrOrd → I FTop and : I-FTop → FPrOrd. Before doing it, let us study the question how to represent fuzzy relation by I-fuzzy
topology. The following theorem is called the Representation Theorem of fuzzy preorders by Lai and Zhang [15], and
this conclusion was also obtained in [22].
Theorem 4.7 (Valverde [22] and Lai and Zhang [15] (Representation Theorem of fuzzy preorders by I-topologies)).
F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
2371
Suppose R: X × X → [0, 1] is a fuzzy preorder on a set X. Then
U (x) −→ U (y)
[z]R (x) −→ [z]R (y) =
R(x, y) =
∇(R)(U )=1
z∈X
=
[z]R (y) −→ [z]R (x) =
U (y) −→ U (x).
(R)(U )=1
z∈X
By means of I-fuzzified set of all upper sets and I-fuzzified set of all lower sets, the above Representation Theorem
can be generalized as follows:
Theorem 4.8 (Representation Theorem of fuzzy preorders by I-fuzzy topologies). Let R be a fuzzy preorder on a set X.
Then
∇(R)(U ) −→ (U (x) −→ U (y))
R(x, y) =
U∈ IX
=
(R)(U ) −→ (U (y) −→ U (x)).
U∈ IX
Proof. We prove the first equality for example. At first, by the definition of I-fuzzified set of all upper sets, we have
R(x, y) [R(x, y) −→ (U (x) −→ U (y))] −→ (U (x) −→ U (y))
U∈ IX
(by (I10) of Proposition 2.4)
⎡
⎤
⎣
R(s, t) −→ (U (s) −→ U (t))⎦ −→ (U (x) −→ U (y))
U ∈I X
=
(s, t)∈X×X
∇(R)(U ) −→ (U (x) −→ U (y))
U ∈I X
for each (x, y) ∈ X × X. Hence, by Theorem 4.7,
∇(R)(U ) −→ (U (x) −→ U (y))
R(x, y) U∈ IX
U (x) −→ U (y)
∇(R)(U )=1
= R(x, y)
for all (x, y) ∈ X × X. So, we obtain the first equality.
Corollary 4.9. Let R be a fuzzy preorder on a set X. Then (∇(R)) = R, is a left inverse of ∇.
Proof. By the proof of Theorem 4.8, we obtain that
∇(R)(U ) −→ (U (x) −→ U (y)) = R∇(R)
R(x, y) =
U∈ IX
whenever R is a fuzzy preorder. In the present case, R = (∇(R)) since R∇(R) is ∗-transitive by the transitivity
of R. Remark 4.10. Following the conclusion of Theorem 4.7, Lai and Zhang [15] pointed out that every fuzzy preorder
is a Valverde order. By Theorem 4.8 and especially Corollary 4.9, we assert that every fuzzy preorder is an -order
determined by some I-fuzzy topology deeply.
Lemma 4.11. Let be an I-fuzzy topology on a set X and a t-norm ∗ be idempotent, i.e., ∗ = ∧. Then ∇(()).
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F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
Proof. By Theorem 3.15, () = R is true. Then, it follows that ∇(R ) from the proof of Theorem 4.3. Hence
∇(()), as desired. By Corollary 4.9, if R is a fuzzy preorder on a set X, then the identity map idX : (X, R) → (X, (∇(R))) is
monotone. Moreover, by Lemma 4.11, if is an I-fuzzy topology on a set Y and a t-norm ∗ = ∧, then the identity map
idY : (Y, ∇(())) → (X, ) is continuous. Therefore, by Proposition 2.6, we obtain a conclusion which reflects the
closed connection between fuzzy orders and I-fuzzy topologies.
Theorem 4.12. If the t-norm ∗ = ∧, then (∇, ) is a Galois correspondence between the category of fuzzy preordered
sets and that of I-fuzzy topological spaces, and is a left inverse of ∇.
Remark 4.13. Let be an I-fuzzy topology on a set X and fix the T-norm ∗=∧ on the unit interval [0, 1]. By
Theorem 3.15, we know that () is ∧-fuzzy preorder on X and by Lemma 4.11, the inequality ∇(()) is
true. But it need not be true that ∇(()) = 1 even if satisfies the conditions of (i)–(v) in Theorem 3.4. This
assertion is supported by Example 4.15.
Remark 4.14. We point out that if is an I-topology on a set in the sense of Lowen [16], with the conditions of (a)–(e)
in Theorem 3.15 [15], the authors had proved ∗ (∗ ()) = , where ∗ () denote the Valverde order determined
by and ∗ (∗ ()) the family of all uppersets of (X, ∗ ()) in the sense of [15]. Also, if : 2X → [0, 1] satisfies
the conditions of (i)–(iii) in Theorem 3.4, namely saturated fuzzifying topology, then the author of the paper with his
cooperator [4] had
proved that R = , where R : X × X → [0, 1] is the ∧-fuzzy preorder determined by so
that R (x, y) = (x,y)∈A×(X−A) (1 − (A)) for any x, y ∈ X and R : 2X → [0, 1] denote the fuzzifying topology
determined by the ∧-fuzzy preorder R so that R (U ) = (u,v)∈U ×(X−U ) (1 − R (u, v)) for any U ∈ 2X .
Example 4.15. Suppose that ∗ is the Lukasiewicz t-norm on I =[0,1] and R(x, y)=x −→ y for any x, y ∈ I is the
canonical fuzzy preorder on [0,1] with respect to ∗. Precisely, x ∗ y = max{x + y − 1, 0}, R(x, y) = min{1 − x + y, 1}.
Lai and Zhang [15] had pointed out a function U : I → I is an upper set of ([0, 1], R) if and only if it satisfies the
following conditions:
(1) U : I → I is an increasing function;
(2) U is 1-Lipschitz. That is, U (x) − U (y) x − y for all x y.
Suppose that a fuzzy subset V : I → I is defined by
0
if 0 x 21 ,
V (x) =
2x − 1 if 21 < x 1.
Then V is not an upper set since it is not 1-Lipschitz. Let be the family of all upper sets of the preordered set [0,1]
with the canonical fuzzy preorder R(x, y) = x −→ y on [0,1] with respect to ∗. Thus, is a [0,1]-topology on the
unit interval [0,1] satisfying (a)–(e) of Theorem 3.15 in [15]. Then, when regarded as an I -fuzzy topology, satisfies
(i)–(v) in Theorem 3.4 of the paper. We assert that ∇(()) = is not true in general. In fact, when regarded as an
I -fuzzy topology, (U ) = 1 whenever U is an upper set and = 0 otherwise. Thus, (V ) = 0 because V is not an upper
set. But ∇(())(V ) = 21 by the following computations:
()(s, t) −→ (V (s) −→ V (t))
∇(())(V ) =
(s,t)∈I ×I
=
(s −→ t) −→ (V (s) −→ V (t))
(s,t)∈I ×I
(by Theorem 4.8 [15], () = R)
1 The author is indebted to the referees for their comments which help to assert that the equality ∇(()) = is negative under the conditions
therein.
F. Jinming / Fuzzy Sets and Systems 158 (2007) 2359 – 2374
=
2373
(s −→ t) −→ ((2s − 1) −→ V (t))
(s,t)∈( 21 ,1]×[0,1]
(by the definition of V )
min{1, 1 − s + t} −→ min{1, 1 − (2s − 1) + V (t)}
=
(s,t)∈( 21 ,1]×[0,1]
(by the definition of −→)
min{1, 1 − (1 − s + t) + (1 − 2s + 1)}
=
(s,t)∈( 21 ,1]×[0, 21 ]
∧
min{1, 1 − s + t} −→ min{1, 1 − (2s − 1) + (2t − 1)}
(s,t)∈( 21 ,1]×( 21 ,1]
=
min{1, 2 − s − t}
(s,t)∈( 21 ,1]×[0, 21 ]
∧
min{1, 1 − s + t} −→ min{1, 1 − 2s + 2t}
(s,t)∈( 21 ,1]×( 21 ,1]
=
1
2
∧
min{1, 1 − (1 − s + t) + (1 − 2s + 2t)}
1
2 <t<s 1
=
1
2
∧
min{1, 1 − s + t} =
1
2
∧
1
2
= 21 .
1
2 <t<s 1
Acknowledgement
The author is grateful to Area Editor B. De Baets and the referees for their valuable suggestions. Finally, a special
words of thanks to my wife Wang Yuan and son Fang Zhou for their encouragements.
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