Proceedings of the Ninth Prague Topological Symposium Contributed papers from the symposium held in Prague, Czech Republic, August 19–25, 2001

FUZZY FUNCTIONS AND AN EXTENSION OF THE CATEGORY L-TOP OF CHANG-GOGUEN L-TOPOLOGICAL SPACES

ALEXANDER P. ŠOSTAK

Abstract. We study FT OP (L), a fuzzy category with fuzzy functions in the role of morphisms. This category has the same objects as the category L-TOP of Chang-Goguen L-topological spaces, but an essentially wider class of morphisms—so called fuzzy functions introduced earlier in our joint work with U. Höhle and H. Porst.

Introduction

In research works where fuzzy sets are involved, in particular, in Fuzzy Topology, mostly certain usual functions are taken as morphisms: they can be certain mappings between corresponding sets, or between the fuzzy powersets of these sets, etc. On the other hand, in our joint works with U. Höhle and H.E. Porst [9], [10] a certain class of L-relations (i.e. mappings F : X × Y → L) was distinguished which we view as (L-)fuzzy functions from a set X to a set Y; these fuzzy functions play the role of morphisms in an L-fuzzy category of sets FSET (L), introduced in [10].

Later on we constructed a fuzzy category FT OP (L) related to topology with fuzzy functions in the role of morphisms, see [18]. Further, in [19] a certain uniform counterpart of FT OP (L) was introduced. Our aim here is to continue the study of FT OP (L). In particular, we show that the top frame FT OP (L)^{⊤} of the fuzzy category FT OP (L) is a topological category in H. Herrlich’s sense [1]) over the top frame FSET (L)^{⊤} of the fuzzy category FSET (L).

In order to make exposition self-contained, we start with Section 1 Prerequisities, where we briefly recall the three basic concepts which are essentially used in this work: they are the concepts of a GL-monoid (see e.g. [6], [8], etc.); of an L-valued set (see e.g. [7], etc.), and of an L-fuzzy category (see e.g. [14], [15], [16], etc.). In Section 2 we consider basic facts about fuzzy functions and introduce the L-fuzzy category FSET (L) [9], [10]. The properties of this fuzzy category and some related categories are the subject of Section 3. FSET (L) is used as the ground category for the L-fuzzy category FT OP (L) whose objects are Chang-Goguen L-topological spaces [4], [5], and whose morphisms are certain fuzzy functions, i.e. morphisms from FSET (L). Fuzzy category FT OP (L) is considered in Section 4. Its crisp top frame FT OP (L)^{⊤} is studied in Section 5. In particular, it is shown that FT OP (L)^{⊤} is a topological category over FSET (L)^{⊤}. Finally, in Section 6 we consider the behaviour of the topological property of compactness with respect to fuzzy functions — in other words in the context of the fuzzy category FT OP (L) and, specifically, in the context of the category FT OP (L)^{⊤}.

2000 Mathematics Subject Classification. 03E72, 18A05, 54A40.

Key words and phrases. Fuzzy category.

Partly supported by grant 01.0530 of Latvijas Zinātnes Padome.

This article will be revised and submitted for publication elsewhere.

Alexander P. Šostak, Fuzzy functions and an extension of the category L-Top of Chang-Goguen L-topological spaces, Proceedings of the Ninth Prague Topological Symposium, (Prague, 2001), pp. 271–294, Topology Atlas, Toronto, 2002; arXiv:math.GN/0204139.

Acknowledgements. The author is thankful to U. Höhle for many useful discussions which stimulated this work.

1. Prerequisities

1.1. GL-monoids. Let (L, ≤) be a complete infinitely distributive lattice, i.e. (L, ≤) is a partially ordered set such that for every subset A ⊂ L the join ⋁ A and the meet ⋀ A are defined and (⋁ A) ∧ α = ⋁{a ∧ α) | a ∈ A} and (⋀ A) ∨ α = ⋀{a ∨ α) | a ∈ A} for every α ∈ L. In particular, ⋁ L =: ⊤ and ⋀ L =: ⊥ are respectively the universal upper and the universal lower bounds in L. We assume that ⊥ ≠ ⊤, i.e. L has at least two elements.

A GL−monoid (see [6], [7], [8]) is a complete lattice enriched with a further binary operation ∗, i.e. a triple (L, ≤, ∗) such that:

(1) ∗ is monotone, i.e. α ≤ β implies α ∗ γ ≤ β ∗ γ, ∀α, β, γ ∈ L;

(2) ∗ is commutative, i.e. α ∗ β = β ∗ α, ∀α, β ∈ L;

(3) ∗ is associative, i.e. α ∗ (β ∗ γ) = (α ∗ β) ∗ γ, ∀α, β, γ ∈ L;

(4) (L, ≤, ∗) is integral, i.e. ⊤ acts as the unity: α ∗ ⊤ = α, ∀α ∈ L;

(5) ⊥ acts as the zero element in (L, ≤, ∗), i.e. α ∗ ⊥ = ⊥, ∀α ∈ L;

(6) ∗ is distributive over arbitrary joins, i.e. α ∗ (⋁_{j} β_{j}) = ⋁_{j}(α ∗ β_{j}), ∀α ∈ L, ∀{β_{j} : j ∈ J} ⊂ L;

(7) (L, ≤, ∗) is divisible, i.e. α ≤ β implies existence of γ ∈ L such that α = β ∗ γ.

It is known that every GL−monoid is residuated, i.e. there exists a further binary operation “⟼” (implication) on L satisfying the following condition:

α ∗ β ≤ γ ⟺ α ≤ (β ⟼ γ) ∀α, β, γ ∈ L.

Explicitly implication is given by

α ⟼ β = ⋁{λ ∈ L | α ∗ λ ≤ β}.

Below we list some useful properties of GL−monoids (see e.g. [6], [7], [8]):

FUZZY FUNCTIONS AND L-TOP 273

(i) α ⟼ β = ⊤ ⟺ α ≤ β; (ii) α ⟼ (⋀_{i} β_{i}) = ⋀_{i}(α ⟼ β_{i}); (iii) (⋁_{i} α_{i}) ⟼ β = ⋀_{i}(α_{i} ⟼ β);

(v) α ∗ (⋀_{i} β_{i}) = ⋀_{i}(α ∗ β_{i}); (vi) (α ⟼ γ) ∗ (γ ⟼ β) ≤ α ⟼ β; (vii) α ∗ β ≤ (α ∗ α) ∨ (β ∗ β). Important examples of GL-monoids are Heyting algebras and MV -alg-

ebras. Namely, a Heyting algebra is GL-monoid of the type (L, ≤, ∧, ∨, ∧) (i.e. in case of a Heyting algebra ∧ = ∗), cf. e.g. [12]. A GL-monoid is called an MV -algebra if (α ⟼ ⊥) ⟼ ⊥ = α ∀α ∈ L, [2], [3], see also [8, Lemma 2.14]. Thus in an MV -algebra an order reversing involution ^{c} : L → L can be naturally defined by setting α^{c} := α ⟼ ⊥ ∀α ∈ L.

If X is a set and L is a GL-monoid, then the fuzzy powerset L^{X} in an obvious way can be pointwise endowed with a structure of a GL-monoid. In particular the L-sets 1_{X} and 0_{X} defined by 1_{X}(x) := ⊤ and 0_{X}(x) := ⊥ ∀x ∈ X are respectively the universal upper and lower bounds in L^{X}.

In the sequel L denotes an arbitrary GL-monoid.

1.2. L-valued sets. Following U. Höhle (cf. e.g. [7]) by a (global) L-valued set we call a pair (X, E) where X is a set and E is an L-valued equality, i.e. a mapping E : X × X → L such that (1eq) E(x, x) = ⊤ (2eq) E(x, y) = E(y, x) ∀x, y ∈ X; (3eq) E(x, y) ∗ E(y, z) ≤ E(x, z) ∀x, y, z ∈ X. A mapping f : (X, E_{X}) → (Y, E_{Y}) is called extensional if

E_{X}(x, x′) ≤ E_{Y} (f(x), f(x′)) ∀x, x′ ∈ X.

Let SET (L) denote the category whose objects are L-valued sets and whose morphisms are extensional mappings between the corresponding L-valued sets. Further, recall that an L-set, or more precisely, an L-subset of a set X is just a mapping A : X → L. In case (X, E) is an L-valued set, its L-subset A is called extensional if

⋁_{x∈X} A(x) ∗ E(x, x′) ≤ A(x′) ∀x′ ∈ X.

1.3. L-fuzzy categories.

Definition 1.3.1 ([14], [15], [16], [17]). An L-fuzzy category is a quintuple C = (Ob(C), ω, M(C), μ, ∘) where C_{⊥} = (Ob(C), M(C), ∘) is a usual (classical) category called the bottom frame of the fuzzy category C; ω : Ob(C) ⟶ L is an L-subclass of the class of objects Ob(C) of C_{⊥} and μ : M(C) ⟶ L is an L-subclass of the class of morphisms M(C) of C_{⊥}. Besides ω and μ must satisfy the following conditions:

(1) if f : X → Y, then μ(f) ≤ ω(X) ∧ ω(Y);

(2) μ(g ∘ f) ≥ μ(g) ∗ μ(f) whenever composition g ∘ f is defined;

(3) if e_{X} : X → X is the identity morphism, then μ(e_{X}) = ω(X).

Given an L-fuzzy category C = (Ob(C), ω, M(C), μ, ∘) and X ∈ Ob(C), the intuitive meaning of the value ω(X) is the degree to which a potential object X of the L-fuzzy category C is indeed its object; similarly, for f ∈ M(C) the intuitive meaning of μ(f) is the degree to which a potential morphism f of C is indeed its morphism.

Definition 1.3.2. Let C = (Ob(C), ω, M(C), μ, ∘) be an L-fuzzy category. By an (L-fuzzy) subcategory of C we call an L-fuzzy category

C′ = (Ob(C), ω′, M(C), μ′, ∘)

where ω′ ≤ ω and μ′ ≤ μ. A subcategory C′ of the category C is called full if μ′(f) = μ(f)∧ω′(X)∧ω′(Y) for every f ∈ M_{C}(X, Y), and all X,Y ∈ Ob(C).

Thus an L-fuzzy category and its subcategory have the same classes of potential objects and morphisms. The only difference of a subcategory from the whole category is in L-fuzzy classes of objects and morphisms, i.e. in the belongness degrees of potential objects and morphisms.

Let C = (Ob(C), M(C), ∘) be a crisp category and D = (Ob(D), M(D), ∘) be its subcategory. Then for every GL-monoid L the category D can be identified with the L-fuzzy subcategory

D̃ = (Ob(C), ω′, M(C), μ′, ∘)

of C such that ω′(X) = ⊤ if X ∈ Ob(D) and ω′(X) = ⊥ otherwise; μ′(f) = ⊤ if f ∈ M(D) and μ′(f) = ⊥ otherwise. In particular, D̃_{⊤} = D.

On the other hand sometimes it is convenient to identify a fuzzy subcategory

C′ = (Ob(C), ω′, M(C), μ′, ∘)

of the fuzzy category

C = (Ob(C), ω, M(C), μ, ∘)

with the fuzzy category

D = (Ob(D), ω_{D}, M(D), μ_{D}, ∘)

where

Ob(D) := {X ∈ Ob(C) | ω′(X) ≠ ⊥}

and

M(D) := {f ∈ M(C) | μ′(f) ≠ ⊥}

and ω_{D} and μ_{D} are restrictions of ω′ and μ′ to Ob(D) and M(D) respectively.

2. Fuzzy functions and fuzzy category FSET (L)

As it was already mentioned above, the concept of a fuzzy function and the corresponding fuzzy category FSET (L) were introduced in [9], [10]. There were studied also basic properties of fuzzy functions. In this section we recall those definitions and results from [10] which will be needed in the sequel^{1}. Besides some new needed facts about fuzzy functions will be established here, too.

2.1. Fuzzy functions and category FSET (L).

Definition 2.1.1 (cf. [9, 2.1]). A fuzzy function^{2} F from an L-valued set (X, E_{X}) to (Y, E_{Y}) (in symbols F : (X, E_{X}) ↣ (Y, E_{Y}))) is a mapping F : X × Y → L such that

(1ff) F (x, y) ∗ E_{Y} (y, y′) ≤ F (x, y′) ∀x ∈ X, ∀y, y′ ∈ Y; (2ff) E_{X}(x, x′) ∗ F (x, y) ≤ F (x′, y) ∀x, x′ ∈ X, ∀y ∈ Y; (3ff) F (x, y) ∗ F (x, y′) ≤ E_{Y} (y, y′) ∀x ∈ X, ∀y, y′ ∈ Y.

Notice that conditions (1ff)–(2ff) say that F is a certain L−relation, while axiom (3ff) together with evaluation μ(F) (see Subsection 2.2) specify that the L-relation F is a fuzzy function.

Remark 2.1.2. Let F : (X, E_{X}) ↣ (Y, E_{Y}) be a fuzzy function, X′ ⊂ X, Y′ ⊂ Y, and let the L-valued equalities E_{X′} and E_{Y′} on X′ and Y′ be defined as the restrictions of the equalities E_{X} and E_{Y} respectively. Then defining a mapping F′ : X′×Y′ → L by the equality F′(x, y) = F (x, y) ∀x ∈ X′, ∀y ∈ Y′ a fuzzy function F′ : (X′, E_{X′}) ↣ (Y′, E_{Y′}) is obtained. We refer to it as the restriction of F to the subspaces (X′, E_{X′}) (Y′, E_{Y′})

Given two fuzzy functions F : (X, E_{X}) ↣ (Y, E_{Y}) and G : (Y, E_{Y}) ↣ (Z, E_{Z}) we define their composition G ∘ F : (X, E_{X}) ↣ (Z, E_{Z}) by the formula

(G ∘ F)(x, z) = ⋁_{y∈Y} (F (x, y) ∗ G(y, z))).

In [10] it was shown that the composition G∘F is indeed a fuzzy function and that the operation of composition is associative. Further, if we define the identity morphism by the corresponding L-valued equality: E_{X} : (X, E_{X}) ↣ (X, E_{X}), we come to a category FSET (L) whose objects are L-valued sets and whose morphisms are fuzzy functions F : (X, E_{X}) ↣ (Y, E_{Y}).

2.2. Fuzzy category FSET (L). Given a fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}) let

μ(F) = inf_{x} sup_{y} F (x, y).

Thus we define an L-subclass μ of the class of all morphisms of FSET (L). In case μ(F) ≥ α we refer to F as a fuzzy α-function.

1Actually, the subject of [9], [10] was a more general fuzzy category L-FSET (L) containing FSET (L) as a full subcategory. However, since for the merits of this work the category FSET (L) is of importance, when discussing results from [9], [10] we reformulate (simplify) them for the case of FSET (L) without mentioning this every time explicitly

2Probably, the name an L-fuzzy function would be more adequate here. However, since the GL-monoid L is considered to be fixed, and since the prefix “L” appears in the text very often, we prefer to say just a fuzzy function

If F : (X, E_{X}) ↣ (Y, E_{Y}) and G : (Y, E_{Y}) ↣ (Z, E_{Z}) are fuzzy functions, then μ(G ∘ F) ≥ μ(G) ∗ μ(F) [10]. Further, given an L-valued set (X, E) let ω(X, E) := μ(E) = inf_{x} E(x, x) = ⊤. Thus a fuzzy category FSET (L)= (F SET (L), ω, μ) is obtained.

Example 2.2.1. Let ∗ = ∧ and E_{Y} be a crisp equality on Y, i.e. E_{Y} (y, y′) = ⊤ iff y = y′, and E_{Y} (y, y′) = ⊥ otherwise. Then every fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}) such that μ(F) = ⊤ is uniquely determined by a usual function f : X → Y. Indeed, let f(x) = y iff F (x, y) = ⊤. Then condition (3ff) implies that there cannot be f(x) = y, f(x) = y′ for two different y, y′ ∈ Y and condition μ(F) = ⊤ guarantees that for every x ∈ X one can find y ∈ Y such that f(x) = y. If besides E_{X} is crisp, then, vice versa, every mapping f : X → Y can be viewed as a fuzzy mapping F : (X, E_{X}) ↣ (Y, E_{Y}) (since the conditions of extensionality (2ff) and (3ff) are automatically fulfilled in this case)

Remark 2.2.2. If F′: (X′, E_{X′}) ↣ (Y, E_{Y}) is the restriction of F: (X, E_{X}) ↣ (Y, E_{Y}) (see Remark above) and μ(F) ≥ α, then μ(F′) ≥ α. However, generally the restriction F′: (X′, E_{X′}) ↣ (Y′, E_{Y′}) of F: (X, E_{X}) ↣ (Y, E_{Y}) may fail to satisfy condition μ(F′) ≥ α.

2.3. Images and preimages of L-sets under fuzzy functions. Given a fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}) and L-subsets A : X → L and B : Y → L of X and Y respectively, we define the fuzzy set F^{→}(A) : Y → L (the image of A under F) by the equality F^{→}(A)(y) = ⋁_{x} F (x, y) ∗ A(x) and the fuzzy set F^{←}(B) : X → L (the preimage of B under F) by the equality F^{←}(B)(x) = ⋁_{y} F (x, y) ∗ B(y).

Note that if A ∈ L^{X} is extensional, then F^{→}(A) ∈ L^{Y} is extensional (by (2ff)) and if B ∈ L^{Y} is extensional, then F^{←}(B) ∈ L^{X} is extensional (by (3ff)).

Proposition 2.3.1 (Basic properties of images and preimages of L-sets under fuzzy functions).

(1) F^{→}(⋁_{i∈I}(A_{i}) = ⋁_{i∈I} F^{→}(A_{i}) ∀{A_{i} : i ∈ I} ⊂ L^{X};

(2) F^{→}(A_{1} ⋀ A_{2}) ≤ F^{→}(A_{1}) ⋀ F^{→}(A_{2}) ∀A_{1}, A_{2} ∈ L^{X};

(3) If L-sets B_{i} are extensional, then

⋀_{i∈I} F^{←}(B_{i}) ∗ μ(F)^{2} ≤ F^{←}(⋀_{i∈I} B_{i}) ≤ ⋀_{i∈I} F^{←}(B_{i}) ∀{B_{i} : i ∈ I} ⊂ L^{Y} .

In particular, if μ(F) = ⊤, then F^{←}(⋀_{i∈I} B_{i}) = ⋀_{i∈I} F^{←}(B_{i}) for every family of extensinal L-sets {B_{i} : i ∈ I} ⊂ L^{Y}.

(4) F^{←}(⋁_{i∈I} B_{i}) = ⋁_{i∈I} F^{←}(B_{i}) ∀{B_{i} : i ∈ I} ⊂ L^{Y}.

(5) A ∗ μ(F)^{2} ≤ F^{←}(F^{→}(A)) for every A ∈ L^{X}.

(6) F^{→}(F^{←}(B)) ≤ B for every extensional L-set B ∈ L^{Y}.

(7) F^{←}(c_{Y}) ≥ μ(F)∗c where c_{Y} : Y → L is the constant function taking value c ∈ L. In particular, F^{←}(c_{Y}) = c if μ(F) = ⊤.

Proof. (1).

(⋁_{i} F^{→}(A_{i}))(y) = ⋁_{i} ⋁_{x}(F (x, y) ∗ A_{i}(x))
= ⋁_{x} ⋁_{i}(F (x, y) ∗ A_{i}(x))
= ⋁_{x}(F (x, y) ∗ (⋁_{i} A_{i})(x))
= F^{→}(⋁_{i} A_{i})(y).

(2). The validity of (2) follows from the monotonicity of F^{→}.

(3). To prove property 3 we first establish the following inequality

(2.3.1.1)  ⋁_{y∈Y} (F (x, y))^{2} ≥ ( ⋁_{y∈Y} F (x, y))^{2}.

Indeed, by a property (vii) of a GL-monoid

(⋁_{y∈Y} F (x, y))^{2} = (⋁_{y∈Y} F (x, y)) ∗ (⋁_{y′∈Y} F (x, y′))
= ⋁_{y,y′∈Y} (F (x, y) ∗ F (x, y′))
≤ ⋁_{y,y′∈Y} (F (x, y)^{2} ∨ F (x, y′)^{2})
= ⋁_{y∈Y} (F (x, y))^{2}.

In particular, it follows from (2.3.1.1) that

(2.3.1.2)  ∀x ∈ X ⋁_{y∈Y} (F (x, y))^{2} ≥ μ(F)^{2}.

Now, applying (2.3.1.2) and taking into account extensionality of L-sets B_{i}, we proceed as follows:

(⋀_{i} F^{←}(B_{i}))(x) ∗ (μ(F))^{2}
≤ (⋀_{i}(⋁_{y_{i}∈Y} (F (x, y_{i}) ∗ B_{i}(y_{i}))) ∗ ⋁_{y∈Y} (F (x, y))^{2}
= ⋁_{y∈Y} ((F (x, y))^{2} ∗ ⋀_{i}(⋁_{y_{i}∈Y} F (x, y_{i}) ∗ B_{i}(y_{i}))
= ⋁_{y∈Y} (F (x, y) ∗ (⋀_{i}(F (x, y) ∗ (⋁_{y_{i}∈Y} F (x, y_{i}) ∗ B_{i}(y_{i}))))
= ⋁_{y∈Y} (F (x, y) ∗ (⋀_{i}(⋁_{y_{i}∈Y} (F (x, y) ∗ F (x, y_{i}))) ∗ B_{i}(y_{i})))
≤ ⋁_{y∈Y} (F (x, y) ∗ (⋀_{i}(⋁_{y_{i}∈Y} E(y, y_{i}) ∗ B_{i}(y_{i}))))
≤ ⋁_{y∈Y} (F (x, y) ∗ (⋀_{i} B_{i}(y)))
= F^{←}(⋀_{i} B_{i}))(x),

and hence

(⋀_{i} F^{←}(B_{i})) ∗ (μ(F))^{2} ≤ F^{←}(⋀_{i} B_{i})).

To complete the proof notice that the inequality

F^{←}(⋀_{i∈I} B_{i}) ≤ ⋀_{i∈I} F^{←}(B_{i})

is obvious.

(4). The proof of (4) is similar to the proof of (1) and is therefore omitted.

(5). Let A ∈ L^{X}, then

F^{←}(F^{→}(A))(x) = ⋁_{y}(F (x, y) ∗ F^{→}(A)(y))
= ⋁_{y}(F (x, y) ∗ (⋁_{x′} F (x′, y) ∗ A(x′)))
≥ ⋁_{y}(F (x, y)^{2} ∗ A(x))
≥ (μ(F))^{2} ∗ A(x)

for every x ∈ X, and hence 5 holds.

(6). To show property 6 assume that B ∈ L^{Y} is extensional. Then

F^{→}(F^{←}(B))(y) = ⋁_{x}(F (x, y) ∗ F^{←}(B)(x))
= ⋁_{x} F (x, y) ∗ (⋁_{y′} F (x, y′) ∗ B(y′))
= ⋁_{x∈X,y∈Y} (F (x, y) ∗ F (x, y′) ∗ B(y′))
≤ E_{Y} (y, y′) ∗ B(y′)
≤ B(y),

and hence F^{→}(F^{←}(B)) ≤ B.

(7). The proof of property 7 is straightforward and therefore omitted. □

Comments 2.3.2.

(1) Properties 1,2 and 4 were proved in [10, Proposition 3.2]. Here we reproduce these proofs in order to make the article self-contained.

(2) The inequality in item 2 of the previous proposition obviously cannot be improved even in the crisp case.

(3) One can show that the condition of extensionality cannot be omitted in items 3 and 6.

(4) The idea of the proof of Property 3 was communicated to the author by U. Höhle in Prague at TOPOSYM in August 2001.

(5) In [10] there was established the following version of Property 3 without the assumption of extensionality of L-sets B_{i} in case L is completely distributive:

(⋀_{i∈I} F^{←}(B_{i}))^{5} ≤ F^{←}(⋀_{i∈I} B_{i}) ≤ ⋀_{i∈I} F^{←}(B_{i}) ∀{B_{i} : i ∈ I} ⊂ L^{Y}

and

⋀_{i∈I} F^{←}(B_{i}) = F^{←}(⋀_{i∈I} B_{i}) ∀{B_{i} : i ∈ I} ⊂ L^{Y} , in case ∗ = ∧

2.4. Injectivity, surjectivity and bijectivity of fuzzy functions.

Definition 2.4.1. A fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}) is called injective, if

(inj) F (x, y) ∗ F (x′, y) ≤ E_{X}(x, x′) ∀x, x′ ∈ X, ∀y ∈ Y.

Definition 2.4.2. Given a fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}), we define its degree of surjectivity by the equality:

σ(F) := inf_{y} sup_{x} F (x, y)

In particular, a fuzzy function F is called α-surjective, if σ(F) ≥ α.

In case F is injective and α-surjective, it is called α-bijective.

Remark 2.4.3. Let (X, E_{X}), (Y, E_{Y}) be L-valued sets and (X′, E_{X′}), (Y′, E_{Y′}) be their subspaces. Obviously, the restriction F′: (X′, E_{X′}) ↣ (Y′, E_{Y′}) of an injection F: (X, E_{X}) ↣ (Y, E_{Y}) is an injection. The restriction F′: (X, E_{X}) ↣ (Y′, E_{Y′}) of an α-surjection F: (X, E_{X}) ↣ (Y, E_{Y}) is an α-surjection. On the other hand, generally the restriction F′: (X′, E_{X′}) ↣ (Y′, E_{Y′}) of an α-surjection F: (X, E_{X}) ↣ (Y, E_{Y}) may fail to be an αsurjection.

A fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}) determines a fuzzy relation F^{−1} : X × Y → L by setting F^{−1}(y, x) = F (x, y) ∀x ∈ X, ∀y ∈ Y.

Proposition 2.4.4 (Basic properties of injections, α-surjections and α-bi-jections).

(1) F^{−1} : (Y, E_{Y}) ↣ (X, E_{X}) is a fuzzy function iff F is injective (actually F^{−1} satisfies (3ff) iff F satisfies (inj))

(2) F is α-bijective iff F^{−1} is α-bijective.

(3) If F is injective, and L-sets A_{i} are extensional, then

(⋀_{i} F^{→}(A_{i})) ∗ (σ(F))^{2} ≤ F^{→}(⋀_{i} A_{i}) ≤ ⋀_{i} F^{→}(A_{i}) ∀{A_{i} : i ∈ I} ⊂ L^{X}.

In particular, if F is ⊤-bijective, then

(⋀_{i} F^{→}(A_{i}) = F^{→}(⋀_{i} A_{i}) ∀{A_{i} : i ∈ I} ⊂ L^{X}.

(4) F^{→}(F^{←}(B)) ≥ σ(F)^{2} ∗ B. In particular, if F is ⊤-surjective and B is extensional, then F^{→}(F^{←}(B)) = B.

(5) F^{→}(c_{X}) ≥ σ(F) ∗ c where c_{X} : X → L is the constant function with value c. In particular, F^{→}(c_{X}) = c if σ(F) = ⊤.

Proof. Properties 1 and 2 follow directly from the definitions.

(2). The proof of Property 3 is analogous to the proof of item 3 of Proposition 2.3.1:

First, reasoning as in the proof of (2.3.1.1) we establish the following inequality

(2.4.4.1)  ⋁_{x∈X}(F (x, y))^{2} ≥ ( ⋁_{x∈X} F (x, y))^{2}.

In particular, from here it follows that

(2.4.4.2)  ∀y ∈ Y ⋁_{x∈X}(F (x, y)^{2}) ≥ σ(F)^{2}.

Now, applying (2.4.4.2) and taking into account extensionality of L-sets A_{i}, we proceed as follows:

(⋀_{i} F^{→}(A_{i}))(x) ∗ (σ(F))^{2}
≤ (⋀_{i}(⋁_{x_{i}∈X}(F (x_{i}, y) ∗ A_{i}(x_{i}))) ∗ ⋁_{x∈X}(F (x, y))^{2}
= ⋁_{x∈X}((F (x, y))^{2} ∗ ⋀_{i}(⋁_{x_{i}∈X} F (x_{i}, y) ∗ A_{i}(x_{i}))
= ⋁_{x∈X}(F (x, y) ∗ (⋀_{i}(F (x, y) ∗ (⋁_{x_{i}∈X} F (x_{i}, y) ∗ A_{i}(x_{i}))))
= ⋁_{x∈X}(F (x, y) ∗ (⋀_{i}(⋁_{x_{i}∈X}(F (x, y) ∗ F (x_{i}, y))) ∗ A_{i}(x_{i})))
≤ ⋁_{x∈X}(F (x, y) ∗ (⋀_{i}(⋁_{x_{i}∈X} E_{X}(x, x_{i}) ∗ A_{i}(x_{i}))))
≤ ⋁_{x∈X}(F (x, y) ∗ (⋀_{i} A_{i}(x)))
= F^{→}(⋀_{i} A_{i}))(y),

and hence

(⋀_{i} F^{→}(A_{i})) ∗ (σ(F))^{2} ≤ F^{→}(⋀_{i} A_{i})).

To complete the proof notice that the inequality

F^{→}(⋀_{i∈I} A_{i}) ≤ ⋀_{i∈I} F^{→}(A_{i})

is obvious.

(4). Let B ∈ L^{Y}, then

F^{→}(F^{←}(B))(y) = ⋁_{x}(F (x, y) ∗ F^{←}(B)(x))
= ⋁_{x} F (x, y) ∗ (⋁_{y′} F (x, y′) ∗ B(y))
≥ ⋁_{x} F (x, y) ∗ F (x, y) ∗ B(y)
≥ σ(F)^{2} ∗ B(y),

and hence the first inequality in item 4 is proved. From here and Proposition 2.3.1 (6) the second statement of item 4 follows.

(5) The proof of the last property is straightforward and therefore omitted. □

Question 2.4.5. We do not know whether inequalities in items 3, 4 and 5 can be improved.

Comments 2.4.6.

(1) Properties 1 and 2 were first established in [10].

(2) In [10] the following version of Property 3 was proved: If L is completely distributive and F is injective, then

(⋀_{i} F^{→}(A_{i}))^{5} ≤ F^{→}(⋀_{i} A_{i}) ≤ ⋀_{i} F^{→}(A_{i}) ∀{A_{i} : i ∈ I} ⊂ L^{X}

and

F^{→}(⋀_{i} A_{i}) = ⋀_{i} F^{→}(A_{i}) in case ∧ = ∗

No extensionality is assumed in these cases.

(3) In case of an ordinary function f : X → Y the equality

f^{→}(⋀_{i∈I}(A_{i})) = ⋀_{i∈I} f^{→}(A_{i})

holds just under assumption that f is injective. On the other hand, in case of a fuzzy function F to get a reasonable counterpart of this property we need to assume that F is bijective. The reason for this, as we see it, is that in case of an ordinary function f, when proving the equality, we actually deal only with points belonging to the image f(X), while the rest of Y \f(X) does not play any role. On the other hand, in case of a fuzzy function F : X ↣ Y the whole Y is “an image of X to a certain extent”, and therefore, when operating with images of L-sets, we need to take into account, to what extent a point y is in the “image” of X.

3. Further properties of fuzzy category FSET (L)

In this section we continue to study properties of the fuzzy category FSET (L). As different from the previous section, were our principal interest was in the “set-theoretic” aspect of fuzzy functions, here shall be mainly interested in their properties of “categorical nature”.

First we shall specify the two (crisp) categories related to FSET (L): namely, its bottom frame FSET (L)^{⊥} (=FSET (L) (this category was introduced already in Section 2) and its top frame FSET (L)^{⊤}. The last one will be of special importance for us. By definition its morphisms F satisfy condition μ(F) = ⊤, and as we have seen in the previous section, fuzzy functions satisfying this condition “behave themselves much more like ordinary functions” than general fuzzy functions. Respectively, the results which we are able to establish about FSET (L)^{⊤} and about topological category FT OP (L)^{⊤}based on it, are more complete and nice, then their more general counterparts.

Second, note that the “classical” category SET (L) of L-valued sets can be naturally viewed as a subcategory of FSET (L)^{⊤}. In case L = {0, 1} obviously the two categories collapse into the category SET of sets. On the other hand, starting with the category SET (=SET({0, 1})) (i.e. L = {0, 1}) of sets and enriching it with respective fuzzy functions, we obtain again the category SET as FSET (L)^{⊤} and obtain the category of sets and partial functions as FSET (L)^{⊥}.

3.1. Preimages of L-valued equalities under fuzzy functions. Let an L-valued set (Y, E_{Y}), a set X and a mapping F : X × Y → L be given. We are interested to find the largest L-valued equality E_{X} on X for which F : (X, E_{X}) → (Y, E_{Y}) is a fuzzy function. This L-valued equality will be called the preimage of E_{Y} under F and will be denoted F^{←}(E_{Y}).

Note first that the axioms

(1ff) F (x, y) ∗ E_{Y} (y, y′) ≤ F (x, y′), and

(3ff) F (x, y) ∗ F (x, y′) ≤ E(y, y′) do not depend on the L-valued equality on X and hence we have to demand that the mapping F originally satisfies them. To satisfy the last axiom

(2ff) E_{X}(x, x′) ∗ F (x, y) ≤ F (x′, y) in an “optimal way” we define

E_{X}(x, x′) := ⋀_{y}((F (x, y) ⟼ F (x′, y)) ∧ (F (x′, y) ⟼ F (x, y))).

Then E_{X} : X ×X → L is an L-valued equality on X. Indeed, the validity of properties E_{X}(x, x) = ⊤ and E_{X}(x, x′) = E_{X}(x′, x) is obvious. To establish the last property, i.e. E_{X}(x, x′) ∗ E_{X}(x′, x′′) ≤ E_{X}(x, x′′), we proceed as follows:

E_{X}(x, x′) ∗ E_{X}(x′, x′′)
= ⋀_{y}((F (x, y) ⟼ F (x′, y)) ∧ (F (x′, y) ⟼ F (x, y)))
∗ ⋀_{y}((F (x′, y) ⟼ F (x′′, y)) ∧ (F (x′′, y) ⟼ F (x′, y)))
≤ ⋀_{y}((F (x, y) ⟼ F (x′, y)) ∧ (F (x′, y) ⟼ F (x, y))
∗(F (x′, y) ⟼ F (x′′, y)) ∧ (F (x′′, y) ⟼ F (x′, y)))
≤ ⋀_{y}((F (x, y) ⟼ F (x′, y)) ∗ (F (x′, y) ⟼ F (x, y))
∧((F (x′, y) ⟼ F (x′′, y)) ∗ (F (x′′, y) ⟼ F (x′, y)))
≤ ⋀_{y}((F (x, y) ⟼ F (x′′, y)) ∧ (F (x′′, y) ⟼ F (x, y)))
= E_{X}(x, x′′).

Further, just from the definition of E_{X} it is clear that F satisfies the axiom (2ff) and hence it is indeed a fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}). Moreover, from the definition of E_{X} it is easy to note that it is really the largest L-valued equality on X for which F satisfies axiom (2ff).

Finally, note that the value μ(F) is an inner property of the mapping F : X × Y → L and does not depend on L-valued equalities on these sets.

Question 3.1.1. We do not know whether the preimage F^{←}(E_{Y}) is the initial structure for the source F : X ↣ (Y, E_{Y}) in FSET (L). Namely, given an L-valued set (Z, E_{Z}) and a “fuzzy quasi-function” G : (Z, E_{Z}) ↣ X is it true that composition F ∘ G : (Z, E_{Z}) ↣ (Y, E_{Y}) is a fuzzy function if and only if G : (Z, E_{Z}) ↣ (X, E_{X}) is a fuzzy function? By a fuzzy quasifunction we mean that G satisfies properties (1ff) and (3ff) which do not depend on the equality on X.

3.2. Images of L-valued equalities under fuzzy functions. Let an Lvalued set (X, E_{X}), a set Y and a mapping F : X × Y → L be given. We are interested to find the smallest L-valued equality E_{Y} on Y for which F : (X, E_{X}) → (Y, E_{Y}) is a fuzzy function. This L-valued equality will be called the image of E_{X} under F and will be denoted F^{→}(E_{X}).

Note first that the axiom

(2ff) E_{X}(x, x′) ∗ F (x, y) ≤ F (x′, y) does not depend on the L-valued equality on Y and hence we have to demand that the mapping F originally satisfies it. Therefore we have to bother that F satisfies the remaining two axioms:

(1ff) F (x, y) ∗ E_{Y} (y, y′) ≤ F (x, y′), and (3ff) F (x, y) ∗ F (x, y′) ≤ E(y, y′)

These conditions can be rewritten in the form of the double inequality:

\begin{aligned}F (x, y) ∗ F (x, y′) & ≤ & E_{Y} (y, y′) \\ & ≤ & (F (x, y′) ⟼ F (x, y)) ∧ (F (x, y) ⟼ F (x, y′)). \end{aligned}

Defining E_{Y} by the equality

E_{Y} (y, y′) = ⋁_{x}(F (x, y) ∗ F (x, y′)),

we shall obviously satisfy both of them. Moreover, it is clear that E_{Y} satisfies property (3ff) and besides E_{Y} cannot be diminished without loosing this property. Hence we have to show only that E_{Y} is indeed an L-valued equality. However, to prove this we need the assumption that σ(F) = ⊤, that is F is ⊤-surjective. Note that

E_{Y} (y, y) = ⋁_{x}(F (x, y) ∗ F (x, y)) ≥ (σ(F))^{2},

and hence the first axiom is justified in case σ(F) = ⊤.

The equality E_{Y} (y, y′) = E_{Y} (y′, y) is obvious.

Finally, to establish the last property, we proceed as follows. Let y, y′, y′′ ∈ Y. Then

E_{Y} (y, y′) ∗ E_{Y} (y′, y′′)
= ⋁_{x}(F (x, y) ∗ F (x, y′)) ∗ ⋁_{x}(F (x, y′) ∗ F (x, y′′))
= ⋁_{x,x′}(F (x, y) ∗ F (x, y′) ∗ F (x′, y′) ∗ F (x′, y′′))
≤ ⋁_{x,x′}(F (x, y) ∗ E_{X}(x, x′) ∗ F (x′, y′′))
≤ ⋁_{x}(F (x, y) ∗ F (x, y′′))
= E(y, y′′).

Question 3.2.1. We do not know whether the image F^{→}(E_{X}) is the final structure for the sink F : (X, E_{X}) ↣ Y in FSET (L) in case σ(F) = ⊤. Namely, given an L-valued set (Z, E_{Z}) and a “fuzzy almost-function” G : Y ↣ (Z, E_{Z}) is it true that composition F ∘ G : (Z, E_{Z}) ↣ (Y, E_{Y}) is a fuzzy function if and only if G : (Y, E_{Y}) ↣ (Z, E_{Z}) is a fuzzy function? By a fuzzy almost-function we mean that G satisfies property (2ff) which does not depend on the equality on Y.

3.3. Products in FSET (L)^{⊤}. Let Y = {(Y_{i}, E_{i}) : i ∈ I} be a family of L-valued sets and let Y = ∏_{i} Y_{i} be the product of the corresponding sets. We introduce the L-valued equality E : Y × Y → L on Y by setting E_{Y} (y, y′) = ⋀_{i∈I} E_{i}(y_{i}, y′_{i}) where y = (y_{i})_{i∈I}, y′ = (y′_{i})_{i∈I}. Further, let p_{i} : Y → Y_{i} be the projection. Then the pair (Y, E) thus defined with the family of projections p_{i} : Y → Y_{i}, i ∈ I, is the product of the family Y in the category FSET (L)^{⊤}.

To show this notice first that, since the morphisms in this category are fuzzy functions, a projection p_{i_{0}} : Y → Y_{i_{0}} must be realized as the fuzzy function p_{i_{0}} : Y × Y_{i_{0}} → L such that p_{i_{0}}(y, y_{i_{0}}^{0}) = ⊤ if and only if the i_{0}-coordinate of y is y_{i_{0}}^{0} and p_{i_{0}}(y, y_{i_{0}}^{0}) = ⊥ otherwise.

Next, let F_{i} : (X, E_{X}) ↣ (Y_{i}, E_{i}), i ∈ I be a family of fuzzy functions. We define the fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}) by the equality:

F (x, y) = ⋀_{i∈I} F_{i}(x, y_{i}).

It is obvious that μ(F) = ⊤ and hence F is in FSET (L)^{⊤}. Finally, notice that the composition

(X, E_{X}) ⟶^{F} (Y, E_{Y}) ⟶^{p_{i_{0}}} (Y_{i_{0}}, E_{i_{0}})

is the fuzzy function

F_{i_{0}} : (X, E_{X}) ↣ (Y_{i_{0}}, E_{i_{0}}).

Indeed, let x^{0} ∈ X and y_{i_{0}}^{0} ∈ Y_{i_{0}}. Then, taking into account that μ(F_{i}) = ⊤ for all i ∈ I, we get

(p_{i_{0}} ∘ F)(x^{o}, y_{i_{0}}^{0}) = ⋁_{y∈Y} (p_{i_{0}}(y, y_{i_{0}}^{0}) ∧ F (x^{0}, y))
= ⋁_{y∈Y} (p_{i_{0}}(y, y_{i_{0}}^{0}) ∧ ⋀_{i∈I} F_{i}(x^{0}, y_{i}))
= F_{i_{0}}(x^{0}, y_{i_{0}}).

Question 3.3.1. We do not know whether products in FSET (L) can be defined in a reasonable way.

3.4. Coproducts in FSET (L). Let X = {(X_{i}, E_{i}) : i ∈ I} be a family of L-valued sets, let X = ⋃ X_{i} be the disjoint sum of sets X_{i}. Further, let q_{i} : X_{i} → X be the inclusion map. We introduce the L-equality on X_{0} by setting E(x, x′) = E_{i}(x, x′) if (x, x′) ∈ X_{i} × X_{i} for some i ∈ I and E(x, x′) = ⊥ otherwise (cf. [7]). An easy verification shows that (X, E) is the coproduct of X in FSET (L) and hence, in particular, in FSET (L)^{⊤}.

Indeed, given a family of fuzzy functions F_{i} : (X_{i}, E_{i}) → (Y, E_{Y}), let the fuzzy function

⊕_{i∈I}F_{i} : (X, E) → (Y, E_{Y})

be defined by

⊕_{i∈I}F_{i}(x, y) = F_{i_{0}}(x, y) whenever x ∈ X_{i_{0}}.

Then for x = x_{i_{0}} ∈ X_{i_{0}} we have

\begin{aligned}(⊕_{i∈I}F_{i} ∘ q_{i_{0}})(x, y) & = & ⋁_{x′∈X}(q_{i_{0}}(x, x′) ∧ (⊕_{i∈I}F_{i}(x′, y))) \\ & = & F_{i_{0}}(x, y). \end{aligned}

3.5. Subobjects in FSET (L). Let (X, E) be an L-valued set, let Y ⊂ X and let e : Y → X be the natural embedding. Further, let E_{Y} := e^{←}(E) be the preimage of the L-valued equality E. Explicitly, in this case this means that E_{Y} (y, y′) = E(y, y′) for all y, y′ ∈ Y. One can easily see that (Y, E_{Y}) is a subobject of (X, E) in the fuzzy category FSET (L).

4. Fuzzy category FT OP (L)

4.1. Basic concepts.

Definition 4.1.1 (see [18], cf. also [4], [5], [11]). A family τ_{X} ⊂ L^{X} of extensional L-sets^{3} is called an L-topology on an L-valued set (X, E_{X}) if it is closed under finite meets, arbitrary joins and contains 0_{X} and 1_{X}. Corresponding triple (X, E_{X}, τ_{X}) will be called an L-valued L-topological space or just an L-topological space for short. A fuzzy function F : (X, E_{X}, τ_{X}) ↣ (Y, E_{Y} , τ_{Y}) is called continuous if F^{←}(V) ∈ τ_{X} for all V ∈ τ_{Y}.

L-topological spaces and continuous fuzzy mappings between them form the fuzzy category which will be denoted FT OP (L). Indeed, let

F: (X, E_{X}, τ_{X}) ↣ (Y, E_{Y} , τ_{Y}) and G: (Y, E_{Y} , τ_{Y}) ↣ (Z, E_{Z}, τ_{Z})

be continuous fuzzy functions and let W ∈ τ_{Z}. Then

\begin{aligned}(G ∘ F)^{←}(W)(x) & = & ⋁_{z}((G ∘ F)(x, z) ∗ W (z)) \\ & = & ⋁_{z,y}(F (x, y) ∗ G(y, z) ∗ W (z)). \end{aligned}

On the other hand, G^{←}(W)(y) = ⋁_{z} G(y, z) ∗ W (z) and

\begin{aligned}F^{←}(G^{←}(W))(x) & = & ⋁_{y} F (x, y) ∗ (⋁_{z}(G(y, z) ∗ W (z))) \\ & = & ⋁_{z,y}(F (x, y) ∗ G(y, z) ∗ W (z)). \end{aligned}

Thus (G ∘ F)^{←}(W) = G^{←}(F^{←}(W)) for every W, and hence composition of continuous fuzzy functions is continuous. Besides, we have seen already before that μ(G∘F) ≥ μ(G)∗μ(F). Finally, E\overleftarrow{X}(B) = B for every extensional B ∈ L^{X} and hence the identity mapping E_{X} : (X, E_{X}, τ_{X}) ↣ (X, E_{X}, τ_{X}) is continuous.

Remark 4.1.2. In case when L-valued equality E_{X} is crisp, i.e. when X is an ordinary set, the above definition of an L-topology on X reduces to the “classical” definition of an L-topology in the sense of Chang and Goguen, [4], [5].

Remark 4.1.3. Some (ordinary) subcategories of the fuzzy category FT OP (L) will be of special interest for us. Namely, let FT OP (L)^{⊥} =: FTOP(L) denote the bottom frame of FT OP (L), let FT OP (L)^{⊤} be the top frame of FT OP (L), and finally let L-TOP(L) denote the subcategory of FT OP (L) whose morphisms are ordinary functions. Obviously the “classical” category L-TOP of Chang-Goguen L-topological spaces can be obtained as a full subcategory L-TOP(L) whose objects carry crisp equalities. Another way to obtain L-TOP is to consider fuzzy subcategory of FT OP (L)whose objects carry crisp equalities and whose morphisms satisfy condition μ(F) > ⊥.

3Since L-topology is defined on an L-valued set X the condition of extensionality of elements of L-topology seems natural. Besides the assumption of extensionality is already implicitly included in the definition of a fuzzy function.

In case when L is an MV -algebra and involution ^{c} : L → L on L is defined in the standard way, i.e. α^{c} := α ⟼ ⊥ we can reasonably introduce the notion of a closed L-set in an L-topological space:

Definition 4.1.4. An L-set A in an L-topological space (X, E_{X}, τ_{X}) is called closed if A^{c} ∈ τ_{X} where A^{c} ∈ L^{X} is defined by the equality

A^{c}(x) := A(x) ⟼ ⊥ ∀x ∈ X.

Let C_{X} denote the family of all closed L-sets in (X, E_{X}, τ_{X}). In case when L is an MV -algebra the families of sets τ_{X} and C_{X} mutually determine each other:

A ∈ τ_{X} ⟺ A^{c} ∈ C_{X}.

4.2. Analysis of continuity. Since the operation of taking preimages F^{←} commutes with joins, and in case when μ(F) = ⊤ also with meets (see Proposition 2.3.1), one can easily verify the following

Theorem 4.2.1. Let (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}) be L-topological spaces, β_{Y} be a base of τ_{Y}, ξ_{Y} its subbase and let F : X ↣ Y be a fuzzy function. Then the following are equivalent: (1con) F is continuous; (2con) for every V ∈ β_{Y} it holds F^{←}(V) ∈ τ_{X}; (3con) F^{←}(Int_{Y} (B)) ≤ Int_{X}(F^{←}(B)), for every B ∈ L^{Y} where Int_{X} and

Int_{Y} are the corresponding L-interior operators on X and Y respectively.

In case when μ(F) = ⊤ these conditions are equivalent also to the following (4con) for every V ∈ ξ_{Y} it holds F^{←}(V) ∈ τ_{X}.

In case when L is an MV-algebra one can characterize continuity of a fuzzy function by means of closed L-sets and L-closure operators:

Theorem 4.2.2. Let (L, ≤, ∨, ∧, ∗) be an MV-algebra, (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}) be L-topological spaces and F : X ↣ Y be a fuzzy function. Further, let C_{X}, C_{Y} denote the families of closed L-sets and cl_{X}, cl_{Y} denote the closure operators in (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}) respectively. Then the following two conditions are equivalent: (1con) F is continuous; (5con) For every B ∈ C_{Y} it follows F^{←}(B) ∈ C_{X}. In case when μ(F) = ⊤, the previous conditions are equivalent to the following: (6con) For every A ∈ L^{X} it holds F^{→}(cl_{X}(A)) ≤ cl_{Y} (F^{→}(A)).

Proof. In case when L is equipped with an order reversing involution, as it is in our situation, families of closed and open L-sets mutually determine each other. Therefore, to verify the equivalence of (1con) and (5con) it is sufficient to notice that for every B ∈ L^{Y} and every x ∈ X it holds

F^{←}(B^{c})(x) = ⋁_{y}(F (x, y) ∗ (B(y) ⟼ ⊥))
= ⋁_{y}(F (x, y) ∗ B(y) ⟼ ⊥)
= (⋁_{y}(F (x, y) ∗ B(y)))^{c}
= (F^{←}(B))^{c}(x),

and hence

F^{←}(B^{c}) = (F^{←}(B))^{c} ∀B ∈ L^{Y} ,

i.e. operation of taking preimages preserves involution.

To show implication (5con) ⟹ (6con) under assumption μ(F) = ⊤ let A ∈ L^{X}. Then, according to Proposition 2.3.1 (5),

A ≤ F^{←}(F^{→}(A)) ≤ F^{←}(cl_{Y} (F^{→}(A))),

and hence, by (5con), also

cl_{X}(A) ≤ F^{←}(cl_{Y} (F^{→}(A))).

Now, by monotonicity of the image operator and by Proposition 2.3.1 (6) (taking into account that cl_{X}A is extensional as a closed L-set), we get:

F^{→}(cl_{X}(A)) ≤ F^{→}(F^{←}(cl_{Y} (F^{→}(A)))) ≤ cl_{Y} (F^{→}(A)).

Conversely, to show implication (6con) ⟹ (5con) let B ∈ C_{Y} and let F^{←}(B) := A. Then, by (6con),

F^{→}(cl_{X}(A)) ≤ cl_{Y} (F^{→}(A)) ≤ cl_{Y} (B) = B.

In virtue of Proposition 2.3.1 (5) and taking into account that μ(F) = ⊤, it follows from here that cl_{X}(A) ≤ F^{←}(B) = A, and hence cl_{X}(A) = A. □

4.3. Fuzzy α-homeomorphisms and fuzzy α-homeomorphic spaces. The following definition naturally stems from Definitions 2.4.1, 2.4.2 and 4.1.1 and item 2 of Proposition 2.3.1:

Definition 4.3.1. Given L-topological spaces (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}), a fuzzy function F : X ↣ Y is called a fuzzy α-homeomorphism if μ(F) ≥ α, σ(F) ≥ α, it is injective, continuous, and the inverse fuzzy function F^{−1} : Y ↣ X is also continuous. Spaces (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}) are called fuzzy α-homeomorphic if there exists a fuzzy α-homeomorphism F : (X, E_{X}, τ_{X}) ↣ (Y, E_{Y} , τ_{Y}).

One can easily verify that composition of two fuzzy α-homeomorphisms is a fuzzy α^{2}-homeomorphism; in particular, composition of fuzzy ⊤-homeo-morphisms is a fuzzy ⊤-homeomorphism, and hence fuzzy ⊤-homeomorph-

isms determine the equivalence relation ≈^{⊤} on the class of all L-topological spaces. Besides, since every (usual) homeomorphism is obviously a fuzzy ⊤-homeomorphism, homeomorphic spaces are also fuzzy ⊤-homeomorphic:

(X, E_{X}, τ_{X}) ≈ (Y, E_{Y} , τ_{Y}) ⟹ (X, E_{X}, τ_{X}) ≈^{⊤} (Y, E_{Y} , τ_{Y}).

The converse generally does not hold:

Example 4.3.2. Let L be the unit interval [0, 1] viewed as an MV -algebra (i.e. α∗β = max{α+β −1, 0}, let (X, ϱ) be an uncountable separable metric space such that ϱ(x, x′) ≤ 1 ∀x, x′ ∈ X, and let Y be its countable dense subset. Further, let the L-valued equality on X be defined by E_{X}(x, x′) := 1 − ϱ(x, x′) and let E_{Y} be its restriction to Y. Let τ_{X} be any L-topology on an L-valued set (X, E_{X}) (in particular, one can take τ_{X} := {c_{X} | c ∈ [0, 1]}). Finally, let a fuzzy function F : (X, E_{X}) ↣ (Y, E_{Y}) be defined by F (x, y) := 1 − ϱ(x, y). It is easy to see that F is a ⊤-homeomorphism and hence

(X, E_{X}, τ_{X}) ≈^{⊤} (Y, E_{Y} , τ_{Y}). On the other hand (X, E_{X}, τ_{X}) ≉(Y, E_{Y} , τ_{Y})

just for set-theoretical reasons.

5. Category FT OP (L)^{⊤}

Let FT OP (L)^{⊤} be the top-frame of FT OP (L), i.e. FT OP (L)^{⊤} is a category whose objects are the same as in FT OP (L), that is L-topological spaces, and morphisms are continuous fuzzy functions F : (X, E_{X}, τ_{X}) ↣ (Y, E_{Y} , τ_{Y}) such that μ(F) = ⊤.

Note, that as different from the fuzzy category FT OP (L), FT OP (L)^{⊤} is a usual category. Applying Theorem 4.2.1, we come to the following result:

Theorem 5.0.3. Let (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}) be L-topological spaces, β_{Y} be a base of τ_{Y}, ξ_{Y} its subbase and let F : X ↣ Y be a fuzzy function. Then the following conditions are equivalent: (1con) F is continuous; (2con) for every V ∈ β_{Y} it holds F^{←}(V) ∈ τ_{X}; (3con) F^{←}(Int_{Y} (B)) ≤ Int_{X}(F^{←}(B)), where Int_{X} and Int_{Y} are the cor-

responding L-interior operators on X and Y respectively; (4con) for every V ∈ ξ_{Y} it holds F^{←}(V) ∈ τ_{X}.

In case when L is an MV-algebra, we get from 4.2.2

Theorem 5.0.4. Let (L, ≤, ∨, ∧, ∗) be an MV-algebra, (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}) be L-topological spaces and F : X ↣ Y be a a morphism in FT OP (L)^{⊤}. Further, let C_{X}, C_{Y} denote the families of closed L-sets and cl_{X}, cl_{Y} denote the closure operators on (X, E_{X}, τ_{X}) and (Y, E_{Y} , τ_{Y}) respectively. Then the following two conditions are equivalent: (1con) F is continuous; (5con) For every B ∈ C_{Y} it follows F^{←}(B) ∈ C_{X}; (6con) For every A ∈ L^{X} it holds F^{→}(cl_{X}(A)) ≤ cl_{Y} (F^{→}(A)).

Theorem 5.0.5.

FT OP (L)^{⊤}is a topological category over the category FSET (L)^{⊤}.

Proof. Since intersection of any family of L-topologies is an L-topology, FT OP (L)^{⊤} is fiber complete. Therefore we have to show only that any structured source in FSET (L)^{⊤} F : (X, E_{X}) ↣ (Y, E_{Y} , τ_{Y}) has a unique initial lift. Let

τ_{X} := F^{←}(τ_{Y}) := {F^{←}(V) | V ∈ τ_{Y} }.

Then from theorem 2.3.1 it follows that τ_{X} is closed under taking finite meets and arbitrary joins. Furthermore, obviously F^{←}(0_{Y}) = 0_{X} and taking into account condition μ(F) = ⊤ one easily establishes that F^{←}(1_{Y}) = 1_{X}. Therefore, taking into account that preimages of extensional L-sets are extensional, (see Subsection 2.3) we conclude that the family τ_{X} is an L-topology on X.

Further, just from the construction of τ_{X} it is clear that F : (X, E_{X}, τ_{X}) ↣ (Y, E_{Y} , τ_{Y}) is continuous and, moreover, τ_{X} is the weakest L-topology on X with this property.

Let now (Z, E_{Z}, τ_{Z}) be an L-topological space and H : (Z, E_{Z}) ↣ (X, E_{X}) a fuzzy function such that the composition G := H ∘ F : (Z, E_{Z}, τ_{Z}) ↣ (Y, E_{Y} , τ_{Y}) is continuous. To complete the proof we have to show that H is continuous.

Indeed, let U ∈ τ_{X}. Then there exists V ∈ τ_{Y} such that U = F^{←}(V). Therefore

H^{←}(U) = H^{←}(F^{←}(V)) = G^{←}(V) ∈ τ_{Z}

and hence H is continuous. □

5.1. Products in FT OP (L)^{⊤}. Our next aim is to give an explicite description of the product in FT OP (L)^{⊤}.

Given a family Y = {(Y_{i}, E_{i}, τ_{i}) : i ∈ I} of L-topological spaces, let (Y, E) be the product of the corresponding L-valued sets {(Y_{i}, E_{i}) : i ∈ I} in FSET (L)^{⊤}and let p_{i} : Y → Y_{i} be the projections.

Further, for each U_{i} ∈ τ_{i} let Û_{i} := p_{i}^{−1}(U_{i}). Then the family ξ := {Û_{i} : U_{i} ∈ τ_{i}, i ∈ I} is a subbase of an L-topology τ on the product L-valued set (X, E_{X}) which is known to be the product L-topology for L-topological spaces {(Y_{i}, τ_{i}) | i ∈ I} in the category L-TOP. In its turn the triple (Y, E, τ) is the product of L-topological spaces {(Y_{i}, E_{i}, τ_{i}) : i ∈ I} in the category FT OP (L)^{⊤}. Indeed, let (Z, E_{Z}, τ_{Z}) be an L-topological space and {F_{i} : Z ↣ Y_{i} | i ∈ I} be a family of continuous fuzzy mappings. Then, defining a mapping F : Z × Y → L by F (z, y) = ∧_{i∈I}F_{i}(z, y_{i}) we obtain a fuzzy function F : Z ↣ Y such that μ(F) = ⋀_{i∈I} μ(F_{i}) = ⊤ and besides it can be easily seen that for every i_{0} ∈ I, every z ∈ Z, and for every U_{i_{0}} ∈ τ_{i_{0}} it holds:

F^{←}(Û_{i_{0}})(z) = ⋁_{y∈Y} ((⋀_{i∈I} F_{i}(z, y_{i})) ∗ U_{i}(y_{i}))
= ⋁_{y_{i}∈Y_{i}}(⋀_{i∈I}(F_{i}(z, y_{i})) ∗ U_{i_{0}}(y_{i_{0}}))
= ⋀_{i∈I}(⋁_{y_{i}∈Y_{i}} F_{i}(z, y_{i})) ∧ ⋁_{y_{i_{0}}∈X_{i_{0}}} (F_{i_{0}}(z, y_{i_{0}}) ∗ U_{i_{0}}(y_{i_{0}}))
= ⊤ ∧ F\overleftarrow{i_{0}}(U_{i_{0}})(z)
= F\overleftarrow{i_{0}}(U_{i_{0}})(z).

Hence continuity of all F_{i} guarantees the continuity of F : (Z, E_{Z}, τ_{Z}) ↣ (Y, E, τ). Thus (Y, E, τ) is indeed the product of Y = {(Y_{i}, E_{i}, τ_{i}) : i ∈ I} in FSET (L)^{⊤}.

Question 5.1.1. We do not know whether products in the fuzzy category FT OP (L) exist.

5.2. Subspaces in FT OP (L)^{⊤}. Let (X, E, τ) be an L-valued L-topological space, Y ⊂ X and let (Y, E_{Y}) be the subobject of the L-valued set (X, E) in the category FSET (L). Further, let τ_{Y} be the subspace L-topology, that is (Y, τ_{Y}) is a subspace of the L-topological space (X, τ) in the category L-TOP. Then it is clear that the triple (Y, E_{Y} , τ_{Y}) is the subobject of (X, E, τ) in the category FT OP (L)^{⊤}(and in the fuzzy category FT OP (L) as well).

5.3. Coproducts in FT OP (L)^{⊤}. Given a family X := {(X_{i}, E_{i}, τ_{i})} of L-topological spaces let (X, E) be the direct sum of the corresponding Lvalued sets (X_{i}, E_{i}) in FSET (L). Further, let τ be the L-topology on X determined by the subbase ⋃_{i∈I} τ_{i} ⊂ 2^{X}. In other words (X, τ) is the coproduct of L-topological spaces (X_{i}, τ_{i}) in the category L-TOP. Then the triple (X, E, τ) is the coproduct of the family X := {(X_{i}, E_{i}, τ_{i})} in the category FT OP (L)^{⊤}(and in the fuzzy category FT OP (L)as well). Indeed, let q_{i} : (X_{i}, E_{i}, τ_{i}) → (X, E, τ), i ∈ I denote the canonical embeddings. Further, consider an L-topological space (Y, E_{Y} , τ_{Y}) and a family of continuous fuzzy functions F_{i} : (X_{i}, E_{i}, τ_{i}) ↣ (Y, E_{Y} , τ_{Y}). Then, by setting F (x, y) := F_{i}(x_{i}, y) whenever x = x_{i} ∈ X_{i}, we obtain a continuous fuzzy function F : (X, E, τ) ↣ (Y, E_{Y} , τ_{Y}) (i.e. a mapping F : X × Y → L) such that F_{i} = q_{i} ∘ F for every i ∈ I.

5.4. Quotients in FT OP (L)^{⊤}. Let (X, E_{X}, τ_{X}) be an L-topological space, let q : X → Y be a surjective mapping. Further, let q^{→}(E_{X}) =: E_{Y} be the image of the L-valued equality E_{X} and let τ_{Y} = {V ∈ L^{Y} | q^{−1}(V) ∈ τ_{X}}, that is τ_{Y} is the quotient L-topology determined by the mapping q : (X, τ) → Y in the category L-TOP. Then (Y, E_{Y} , τ_{Y}) is the quotient object in the category FT OP (L)^{⊤}. Indeed, consider a fuzzy function F : (X, E_{X}, τ_{X}) ↣ (Z, E_{Z}, τ_{Z}) and let G : (Y, E_{Y}) ↣ (Z, E_{Z}) be a morphism in FSET (L) such that q ∘ G = F. Then an easy verification shows that the fuzzy function G : (Y, E_{Y} , τ_{Y}) ↣ (Z, E_{Z}, τ_{Z}) is continuous (i.e. a morphism in FT OP (L)^{⊤}) if and only if F : (X, E_{X}, τ_{X}) → (Z, E_{Z}, τ_{Z}) is continuous (i.e. a morphism in FT OP (L)^{⊤}).

Our next aim is to consider the behaviour of some topological properties of L-valued L-topological spaces in respect of fuzzy function. In this work we restrict our interest to the property of compactness. Some other topological properties, in particular, connectedness and separation properties will be studied in a subsequent work.

6. Compactness

6.1. Preservation of compactness by fuzzy functions. One of the basic facts of general topology — both classic and “fuzzy”, is preservation of compactness type properties by continuous mappings. Here we present a counterpart of this fact in FT OP (L). However, since in literature on fuzzy topology different definitions of compactness can be found, first we must specify which one of compactness notions will be used.

Definition 6.1.1. An L-topological space (X, E, τ) will be called (α, β)- compact where α, β ∈ L, if for every family U ⊂ τ such that ⋁ U ≥ α there exists a finite subfamily U_{0} ⊂ U such that ⋁ U_{0} ≥ β. An (α, α)-compact space will be called just α-compact.^{4}

Theorem 6.1.2. Let (X, E_{X}, τ_{X}), (Y, E_{Y} , τ_{Y}) be L-topological spaces, F : X ↣ Y be a continuous fuzzy function such that μ(F) ≥ β, and σ(F) ≥ γ. If X is α ∗ β-compact, then Y is (α, α ∗ β ∗ γ)-compact.

Proof. Let V ⊂ τ_{Y} be such that ⋁ V ≥ α. Then, applying Proposition 2.3.1 (4), (7) and taking in view monotonicity of F^{←}, we get

⋁_{V ∈V} F^{←}(V) = F^{←}( ⋁_{V ∈V} V) ≥ F^{←}(α) ≥ α ∗ β.

Now, since (X, E_{X}, τ_{X}) is α ∗ β-compact, it follows that there exists a finite subfamily V_{0} ⊂ V such that

⋁_{V ∈V_{0}} F^{←}(V) ≥ α ∗ β.

Applying Propositions 2.3.1 (6),(4) and 2.4.4 (5) we obtain:

⋁_{V ∈V_{0}} V ≥ F^{→}(F^{←}( ⋁_{V ∈V_{0}} V )) = F^{→}( ⋁_{V ∈V_{0}}(F^{←}(V))) ≥ F^{→}(α∗β) ≥ α∗β∗γ.

□

Corollary 6.1.3. Let (X, E_{X}, τ_{X}), (Y, E_{Y} , τ_{Y}) be L-topological spaces, F : X ↣ Y be a fuzzy function such that μ(F) = ⊤ and σ(F) = ⊤. If X is α-compact, then Y is also α-compact.

4Note that Chang’s definition of compactness [4] for a [0, 1]-topological space is equivalent to our 1-compactness. An [0, 1]-topological space is compact in Lowen’s sense [13] if it is (α, β)-compact for all α ∈ [0, 1] and all β < α

6.2. Compactness in case of an MV -algebra. In case L is an MValgebra one can characterize compactness by systems of closed L-sets:

Proposition 6.2.1. Let (X, E_{X}, τ_{X}) be an L-topological space and let C_{X} be the family of its closed L-sets. Then the space (X, E_{X}, τ_{X}) is (α, β)-compact if and only if for every A ⊂ C_{X} the following implication follows:

if ⋀_{A∈A_{0}} A ≰ β^{c} for every finite family A_{0} ⊂ A, then ⋀_{A∈A} A ≰ α^{c}.

Proof. One has just to take involutions “⟼ ⊥” in the definition of (α, β)- compactenss and apply De Morgan law. □

6.3. Perfect mappings: case of an MV-algebra L. In order to study preservation of compactness by preimages of fuzzy functions we introduce the property of (α, β)-perfectness of a fuzzy function. Since we shall operate with closed L-sets, from the beginning it will be assumed that L is an MV - algebra.

First we shall extend the notion of compactness for L-subsets of L-topolog-ical spaces. We shall say that an L-set S : X → L is (α, β)-compact if for every family A of closed L-sets of X the following implication holds:

if S ∧ ( ⋀_{A∈A_{0}} A) ≰ β^{c} for every finite A_{0} ∈ A then S ∧ ( ⋀_{A∈A} A) ≰ α^{c}.

Further, since the preimage F^{←}(y_{0}) : X → L of a point y_{0} ∈ Y under a fuzzy function F : X ↣ Y is obviously determined by the equality

F^{←}(y_{0})(x) = ⋁_{y∈Y} F (x, y) ∗ y_{0}(y) = F (x, y_{0}),

the general definition of (α, β)-compactness of an L-set in this case means the following:

The preimage F^{←}(y_{0}) of a point y_{0} under a fuzzy function F is (α, β)- compact if for every family A of closed sets of X the following implication holds:

⋁_{x}(F (x, y_{0}) ∧ ( ⋀_{A∈A_{0}} A(x))) ≰ β^{c} ∀A_{0} ⊂ A, |A_{0}| < ℵ_{0}

implies

⋁_{x}(F (x, y_{0}) ∧ ( ⋀_{A∈A} A(x))) ≰ α^{c}.

Now we can introduce the following

Definition 6.3.1. A continuous fuzzy mapping F: (X, E_{X}, τ_{X}) ↣ (Y, E_{Y} , τ_{Y}) is called (α, β)-perfect if

• F is closed, i.e. F^{→}(A) ∈ C_{Y} for every A ∈ C_{X};

• the preimage F^{←}(y) of every point y ∈ Y is (α, β)-compact.

Theorem 6.3.2. Let F : (X, E_{X}, τ_{X}) ↣ (Y, E_{Y} , τ_{Y}) be an (α, γ)-perfect fuzzy function such that μ(F) = ⊤ and σ(F) = ⊤. If the space (Y, E_{Y} , τ_{Y}) is (γ, β)-compact, then the space (X, E_{X}, τ_{X}) is (α, β)-compact.

Proof. Let A be a family of closed L-sets in X such that ⋀_{A∈A_{0}} A ≰ β^{c}. Without loss of generality we may assume that A is closed under taking finite meets. Let B := B_{A} := F^{→}(A) and let B := {B_{A} : A ∈ A}. Then, since μ(F) = ⊤, by 2.3.1 (7) it follows that B ≰ β^{c} ∀B ∈ B, and moreover, since A is assumed to be closed under finite meets,

B_{A_{1}} ∧ … ∧ B_{A_{n}} = F^{→}(A_{1}) ∧ … ∧ F^{→}(A_{n})
≥ F^{→}(A_{1} ∧ … ∧ A_{n})
= F^{→}(A),

for some A ∈ A, and hence ⋀_{B∈B_{0}}(B) ≰ β^{c} for every finite subfamily B_{0} ⊂ B. Hence, by (γ, β)-compactness of the space (Y, E_{Y} , τ_{Y}) we conclude that ⋀_{B∈B}(B) ≰ γ^{c}, and therefore there exists a point y_{0} ∈ Y such that F^{→}(A)(y_{0}) = B_{A}(y_{0}) ≰ γ^{c} for all A ∈ A. Now, applying (α, γ)-compactness of the preimage F^{←}(y_{0}) and recalling that A was assumed to be closed under taking finite meets, we conclude that

⋁_{x}(F (x, y_{0}) ∧ ( ⋀_{A∈A} A(x))) ≰ α^{c}.

and hence, furthermore,

⋀_{A∈A} A ≰ α^{c}.

□

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Department of Mathematics, University of Latvia, Riga, Latvia

E-mail address: sostaks@com.latnet.lv