Fuzzy Function and the Generalized Extension Principle

Irina Perﬁlieva

1,2

and Alexandr

Sostak

University of Ostrava, Institute for Research and Applications of Fuzzy Modeling,

NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic

Institute of Mathematics and CS, University of Latia, LV-1459, Riga, Latvia

Keywords:

Fuzzy Function, Extensional Function, Fuzzy Equivalence, Extension Principle.

Abstract:

The aim of this contribution is to develop a theory of such concepts as fuzzy point, fuzzy set and fuzzy function

in a similar style as is common in classical mathematical analysis. We recall some known notions and propose

new ones with the purpose to show that, similarly to the classical case, a (fuzzy) set is a collection of (fuzzy)

points or singletons. We show a relationship between a fuzzy function and its ordinary “skeleton” that can

be naturally associated with the original function. We show that any fuzzy function can be extended to the

domain of fuzzy subsets and this extension is analogous to the Extension Principle of L. A. Zadeh.

1 INTRODUCTION

The notion of fuzzy function has at least two differ-

ent meanings in fuzzy literature. On the one side (see

e.g., (H´ajek, 1998; Klawonn, 2000; Demirci, 1999;

Demirci, 2002; H¨ohle et al., 2000;

Sostak, 2001)), a

fuzzy function is a special fuzzy relation with a gen-

eralized property of uniqueness. According to this

approach, each element from the ordinary domain of

thus deﬁned fuzzy function is associated with a cer-

tain fuzzy set. Thus, a fuzzy function establishes a

“point”-to-“fuzzy set” correspondence.

On the other hand (see (Nov´ak, 1989; Perﬁlieva,

2004; Perﬁlieva, 2011; Perﬁlieva et al., 2012)), a

fuzzy function is a mapping between two universes

of fuzzy sets, i.e. establishes a “fuzzy set”-to-“fuzzy

set” correspondence. This approach is implicitly used

in many papers devoted to fuzzy IF-THEN rule mod-

els where the latter are actually partially given fuzzy

functions.

In this contribution, we show that both viewpoints

can be connected by a natural generalization of the

Extension Principle of L. Zadeh (Zadeh, 1975). In

details, a fuzzy function as a mapping is an extension

of a fuzzy function as a relation to the domain of fuzzy

sets. The similar approach has been used in (

Sostak,

2001).

In order to establish the above mentioned exten-

sion, we introduce various spaces of fuzzy objects

with fuzzy equivalence relations on them. We show

that similar to the classical case, a (fuzzy) set is a col-

lection of (fuzzy) points or (fuzzy) singletons.

Last, but not least, we analyze a relationship be-

tween a surjective fuzzy function and its ordinary

core function. The similar study has been attempted

in (Demirci, 1999) for a perfect fuzzy function and

in (Klawonn, 2000) for one particular example of a

fuzzy function. We propose a solution in the general

case.

The present paper is organized as follows. In Sec-

tion 2, we give preliminary information about ex-

tension principle, residuated lattices, fuzzy sets and

fuzzy spaces. Fuzzy functions and two approaches to

this notion are discussed in Section 3. Section 3 con-

tains also main results of the paper.

2 PRELIMINARIES

2.1 Extension Principle and Its

Relational Form

An extension principle has been proposed by L.

Zadeh (Zadeh, 1975) in 1975 and since then it is

widely used in the fuzzy set theory and its applica-

tions. Let us recall the principle and propose its rela-

tion form which will be later on used in a relationship

to fuzzy function.

Assume that X,Y are universal sets and f : X →

Y is a function with the domain X. Let moreover,

F (X), F (Y) be respective universes of fuzzy sets on

169

Perﬁlieva I. and Šostak A..

Fuzzy Function and the Generalized Extension Principle.

DOI: 10.5220/0005132701690174

In Proceedings of the International Conference on Fuzzy Computation Theory and Applications (FCTA-2014), pages 169-174

ISBN: 978-989-758-053-6

 2014 SCITEPRESS (Science and Technology Publications, Lda.)

X and Y identiﬁed with their membership functions,

i.e. F (X) = {A : X → [0, 1]} and similarly, F (Y).

By the extension principle, f induces a function f

→

F (X) → F (Y) such that for all A ∈ F (X),

→

(A)(y) = sup

y= f(x)

A(x). (1)

Let R

be a binary relation on X × Y which corre-

sponds to the function f, i.e.

(x, y) = 1 ⇔ y = f (x).

Then it is easy to see that (1) can be equivalently rep-

resented by

→

(A)(y) =

y∈Y

(A(x) · R

(x, y)). (2)

Expression (2) is the relational form of the extension

principle. The meaning of expression (2) becomes

more general when A is an L-fuzzy set (see Deﬁni-

tion 3 below), binary relation R

is a fuzzy relation,

and multiplication · changes to a monoidal operation

(see Section 2.2). In Section 3, we will discuss the

proposed generalization and its relationship to fuzzy

functions.

2.2 Residuated Lattice

Our basic algebra of operations is a residuated lattice.

Deﬁnition 1. A residuated lattice is an algebra

L = hL, ∨, ∧, ∗, →, 0, 1i.

with a support L and four binary operations and two

constants such that

• hL, ∨, ∧, 0, 1i is a lattice where the ordering ≤ de-

ﬁned using operations ∨, ∧ as usual, and 0, 1 are

the least and the greatest elements, respectively;

• hL, ∗, 1i is a commutative monoid, that is, ∗ is a

commutative and associative operation with the

identity a∗1 = a;

• the operation → is a residuation operation with

respect to ∗, i.e.

a∗b ≤ c ⇐⇒ a ≤ b → c.

A residuated lattice is complete if it is complete as

a lattice.

The following is a binary operation of biresidua-

tion on L :

x ↔ y = (x → y) ∧(y → x).

The well known examples of residuated lattices

are: boolean algebra, G¨odel, Łukasiewicz and prod-

uct algebras. In the particular case L = [0, 1], multi-

plication ∗ is a left continuous t-norm.

From now on we ﬁx a complete residuated lattice

L .

2.3 L-fuzzy Sets, Fuzzy Relations and

Fuzzy Spaces

Below, we recall deﬁnitions of principal notions in the

fuzzy set theory.

Fuzzy Sets with Crisp Equality. Let X be a non-

empty universal set, L a complete residuated lattice.

An (L-)fuzzy set A of X (fuzzy set, shortly) is a map

A : X → L that establishes a relationship between ele-

ments of X and degrees of membership to A.

Fuzzy set A is normal if there exists x

∈ X such

that A(x

) = 1. The (ordinary) set Core(A) = {x ∈

X | A(x) = 1} is a core of the normal fuzzy set A.

The (ordinary) set Supp(A) = {x ∈ X | A(x) > 0} is a

support set of fuzzy set A.

A class of L-fuzzy sets of X will be denoted

. The couple (L

, =) is called an ordinary fuzzy

space on X. The elements of (L

, =) are fuzzy sets

equipped with a crisp equality relation, i.e. for all

A, B ∈ L

A = B if and only if (∀x ∈ X)A(x) = B(x).

In (L

, =), we strictly distinguish between fuzzy sets

even if their membership functions differ in one point.

On (L

, =), we can deﬁne the structure of resid-

uated lattice using pointwise operations over fuzzy

sets. Moreover, the underlying lattice hL

, ∨, ∧, 0, 1i

is complete, where the bottom 0 and the top 1 are con-

stant fuzzy sets, respectively.

A class of normal L-fuzzy sets of X will be de-

noted N (X). The space (N (X), =) is a subspace of

, =).

By identifying a point u ∈ X with a fuzzy subset

: X → L such that I

(u) = 1 and I

(x) = 0 whenever

x 6= u we may view X as a subspace of (L

, =) and as

a subspace of (N (X), =)

Space with Fuzzy Equivalence. Fuzzy Points. Let

X, Y be universal sets. Similarly to L-valued fuzzy

sets, we deﬁne (binary) (L-)fuzzy relations as fuzzy

sets of X ×Y. If X = Y, then a fuzzy set of X × X is

called a (binary) (L-)fuzzy relation on X.

A binary fuzzy relation E on X is called fuzzy

equivalence on X (see (Klawonn and Castro, 1995;

H¨ohle, 1998; De Baets and Mesiar, 1998))

if for all

x, y,z ∈ X, the following holds:

1. E(x, x) = 1, reﬂexivity,

2. E(x, y) = E(y, x), symmetry,

3. E(x, y) ∗ E(y, z) ≤ E(x, z), transitivity.

Fuzzy equivalence appears in the literature under the

names similarity or indistinguishability as well.

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If fuzzy equivalence E fulﬁlls

1. E(x, y) = 1 if and only if x = y,

then it is called separated or a fuzzy equality on X.

Let us remark that fuzzy equivalence E creates

fuzzy sets on X, we will call them E-fuzzy points

X or simply fuzzy points if E is clear from the context.

Every E-fuzzy point is a class of fuzzy equivalence E

of just one point of X. In more details, if t ∈ X, then

E-fuzzy point E

is a fuzzy set E

: X → L such that

for all x ∈ X, E

(x) = E(t, x). It is easy to see that E

is a normal fuzzy set and t ∈ Core(E

The set of all E-fuzzy points of X will be denoted

= {E

| t ∈ X}.

Obviously, P

⊆ L

and (P

, =) is a subspace of

, =). If E is a fuzzy equivalence on X, then it may

happen that the same element, say E

from (P

, =)

has different representations, i.e. there exists u ∈ X

such that E

= E

. It can be shown that this holds true

if and only if E(t, u) = 1, or u ∈ Core(E

On the other side, if E is a fuzzy equality on X,

then the core of every E-fuzzy point consists of one

element and thus, a representation of any E-fuzzy

point in the form E

is unique.

Space with Fuzzy Equivalence and Crisp Equal-

ity. Fuzzy Singletons and Sub-singletons. Let us

equip the space X with both crisp = and fuzzy E

equalities and denote it by (X, =, E). In this space, we

are able to distinguish degrees of coincidence E(t, u)

between any two elements t,u from X. As we dis-

cussed above, crisp and fuzzy equalities put into the

correspondence with each element t of X its charac-

teristic function I

and its E-fuzzy point E

. Both are

normal fuzzy sets in L

with the same one-element

core. Let us consider fuzzy sets S

∈ L

, that are in

between I

and E

, i.e. for all x ∈ X,

(x) ≤ S

(x) ≤ E

(x). (3)

We will call them fuzzy singletons. In (Klawonn,

2000), fuzzy singletons were introduced as normal

fuzzy sets S

∈ L

with {t} as a one-element core,

i.e. S

(t) = 1, and such that for all x, y ∈ X,

(x) ∗ S

(y) ≤ E(x, y), (4)

where ∗ is the monoidal operation from a chosen

residuated lattice L. It is easy to show that this is

equivalent to our deﬁnition. Indeed, if S

fulﬁlls (3),

then it is normal, it has {t} as a one-element core, and

for all x, y ∈ X,

(x) ∗ S

(y) ≤ E(t, x) ∗ E(t, y) ≤ E(x, y).

This notion was introduced in (Klawonn, 2000)

On the other side, if S

has {t} as a one-element core

and fulﬁlls (4), then for all x ∈ X, I

(x) ≤ S

(x) and

(x) = S

(x) ∗ S

(t) ≤ E(t, x) = E

(x).

From (4) and the discussion above it follows that E-

fuzzy point E

is the greatest fuzzy singleton with the

one-element core {t}. The space of all fuzzy single-

tons, considered in (X, =, E), will be denoted by S

Obviously, S

⊆ L

and (S

, =) is a subspace of

, =).

Let us discard normality in the deﬁnition of fuzzy

singleton and deﬁne fuzzy sub-singleton as a fuzzy set

U ∈ L

, such that there exists t ∈ X, so that

0 < U(x) ≤ E

(x), x ∈ X. (5)

In order to stress that a fuzzy sub-singleton is con-

nected with a certain fuzzy point E

, we will denote

it as U

. Similarly to the above, we can prove that

any fuzzy sub-singleton fulﬁlls (4). The space of all

fuzzy sub-singletons, considered in (X, =, E), will be

denoted by U

. Obviously, S

⊆ U

⊆ L

and

, =) is a subspace of (L

, =).

Extensional Hulls. Let again our space be (X,=

, E) – a space with fuzzy equivalence and crisp equal-

ity. We remind (Klawonn, 2000) that fuzzy set A is

extensional (with respect to E) if for all x, y ∈ X,

A(x) ∗ E(x,y) ≤ A(y).

The smallest extensional fuzzy set A

containing

fuzzy set A is called the extensional hull of A. It is

not difﬁcult to prove the following representation of

Lemma 1. The extensional hull A

of every fuzzy set

A ∈ L

can be represented as follows:

(y) = sup

x∈X

A(x) ∗ E(x,y). (6)

Representation (6) has been obtained in many pa-

pers (see e.g.,(H¨ohle, 1998)), therefore will not prove

this again.

Lemma 1 has two important corollaries.

Corollary 1. Extensional hull of element t ∈ X iden-

tiﬁed with I

is equal to fuzzy point E

Corollary 2. Extensional hull of fuzzy singleton S

∈

, t ∈ X, is equal to the corresponding fuzzy point

Decomposition of a Fuzzy Set into Fuzzy Sub-

singletons

Theorem 1. Let A ∈ L

be a non-zero fuzzy set. Then

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171

• A can be represented as a supremum of fuzzy sub-

singletons U

, t ∈ Supp(A), such that

(x) = A(x) ∧E

(x), x ∈ X, (7)

• A can be represented as a supremum of fuzzy sub-

singletons W

, t ∈ Supp(A), such that

(x) = A(x) ∗ E

(x), x ∈ X, (8)

In both cases, for all x ∈ X,

A(x) = sup

t∈Supp(A)

(x) = sup

t∈Supp(A)

(A(x)∧E

(x)), (9)

and

A(x) = sup

t∈Supp(A)

(x) = sup

t∈Supp(A)

(A(x) ∗ E

(x)).

(10)

Proof. At ﬁrst, we will prove that for all t ∈ Supp(A),

and W

are fuzzy sub-singletons, i.e. U

and W

are non-zero and less than E

. The ﬁrst assertion fol-

lows from the assumption t ∈ Supp(A), so that

(t) =A(t) ∧E

(t) = A(t) > 0,

(t) =A(t) ∗ E

(t) = A(t) > 0.

The second assertion easily follows from (7) and (8).

To prove (9) and (10), we ﬁrst notice that both of

them are trivially valid for x 6∈ Supp(A). Therefore,

we assume that x ∈ Supp(A). Then (9) follows from

the two inequalities below:

A(x) = sup

t∈Supp(A)

(x) =

= sup

t∈Supp(A)

(A(x) ∧E

(x)) ≥ A(x) ∧E

(x) = A(x),

and

A(x) = sup

t∈Supp(A)

(x) = sup

t∈Supp(A)

(A(x) ∧E

(x)) ≤

≤ sup

t∈Supp(A)

A(x) = A(x).

To prove (10), we recall that in every complete

residuated lattice the following holds true:

sup

t∈Supp(A)

(A(x) ∗ E

(x)) = A(x) ∗ sup

t∈Supp(A)

(x).

Because for x ∈ Supp(A), sup

t∈Supp(A)

(x) = 1, we eas-

ily get

A(x) = sup

t∈Supp(A)

(x) = sup

t∈Supp(A)

(A(x) ∗ E

(x)) =

= A(x) ∗ sup

t∈Supp(A)

(x) = A(x).

3 FUZZY FUNCTIONS

The notion of fuzzy function has many deﬁnitions in

the literature, see e.g. (H´ajek, 1998; Klawonn, 2000;

Demirci, 2002; Perﬁlieva, 2004). In (H´ajek, 1998;

Klawonn, 2000; Demirci, 2002), a fuzzy function is

considered as a special fuzzy relation. Below, we re-

mind the notion of fuzzy function as it appeared (in-

dependently) in (Klawonn, 2000), (H¨ohle et al., 2000)

and (Demirci, 2002):

Deﬁnition 2. Let E, F be fuzzy equivalences on X

and Y, respectively. A fuzzy function is a binary fuzzy

relation ρ on X ×Y such that for all x, x

′

∈ X, y, y

′

∈ Y

the following axioms hold true:

1. ρ(x, y) ∗ E(x,x

′

) ≤ ρ(x

′

, y),

2. ρ(x, y) ∗ F(y, y

′

) ≤ ρ(x, y

′

3. ρ(x, y) ∗ ρ(x, y

′

) ≤ F(y, y

′

A fuzzy function is called perfect (Demirci, 1999), (cf

also (H

ohle et al., 2000, Section 3.2)) if it additionally

fulﬁlls

1. for all x ∈ X, there exists y∈ Y, such that ρ(x, y) =

A fuzzy function is called (strong) surjective

(Demirci, 1999), cf also (H¨ohle et al., 2000, Section

4.2) if

1. for all y ∈ Y, there exists x ∈ X, such that ρ(x, y) =

Actually, a fuzzy function ρ establishes a double ex-

tensional correspondencebetween the space (X, =, E)

and the space of (Y, =, F) (axioms FF.1, FF.2) which

is weakly functional (axioms FF.3). Moreover, it is a

point-to-(fuzzy set) mapping between X and L

such

that for all x ∈ X, ρ(x, ·) is a fuzzy set on Y. If for

all x ∈ X, ρ(x, ·) is a normal fuzzy set then ρ is per-

fect, and there is an ordinary function g : X → Y such

that for all y ∈ Y, ρ(x, y) = F(g(x), y) (see (Demirci,

2002)). This means that every F-fuzzy point F

g(x)

Y determined by g(x) is a fuzzy value of ρ at x ∈ X.

In our study, we will consider the case where ρ is

surjective and deﬁned everywhere on X, i.e.

(∀x ∈ X)(∃y ∈ Y) ρ(x, y) > 0. (11)

In this case, we will propose an analytic representa-

tion of ρ and use ρ in the generalized extension prin-

ciple. Moreover, we will discover a relationship be-

tween a fuzzy function, its ordinary core function and

its extension to a mapping over the domain of fuzzy

sets.

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3.1 Fuzzy Function and Its Core

In this Section, we will show that each surjective

fuzzy function ρ on X × Y determines a correspond-

ing ordinary core function g : X

′

→ Y, where X

′

⊆ X,

such that at any x

′

∈ X

′

, the value ρ(x

′

, ·) is equal to

the F-fuzzy point F

g(x

′

)(·)

. The proofs of the below

given Theorems 2 and 3 are in (Perﬁlieva, 2011).

Theorem 2. Let fuzzy relations E on X and F on

Y be fuzzy equivalences and moreover, F be a fuzzy

equality. Let fuzzy relation ρ on X ×Y be a surjective

fuzzy function. For every y ∈ Y, let us choose and ﬁx

∈ Core(ρ(x, y)). Denote X

′

= {x

| x

∈ X, y ∈ Y}.

Then the following fuzzy relation on X

′

(x, x

′

) =

y∈Y

(ρ(x, y) ↔ ρ(x

′

, y)), (12)

is a fuzzy equivalence E

′

such that

1. E ≤ E

′

and ρ is a fuzzy function with respect to

fuzzy equivalences E

′

and F,

2. for all x ∈ X, y ∈ Y,

ρ(x, y) = E

′

(x, x

), (13)

3. for all y, y

′

∈ Y,

′

, x

′

) = F(y, y

′

), (14)

4. the mapping g : X

′

→ Y such that g(x

) = y is sur-

jective and extensional with respect to E

′

and F,

i.e. for all x,t ∈ X

′

(x,t) ≤ F(g(x),g(t)). (15)

Corollary 3. Fuzzy equivalence E

′

, given by (12), is

the greatest one (in the sense of ≤) that fulﬁls Theo-

rem 2.

Corollary 4. Fuzzy equivalence E

′

, given by (12),

covers X, i.e. for all x ∈ X there exists x

∈ X

′

such

that E

′

(x, x

) > 0.

Proof. By (11), for arbitrary x ∈ X there exists y ∈ Y,

such that ρ(x, y) > 0. By (13), ρ(x, y) = E

′

(x, x

), and

therefore, E

′

(x, x

) > 0.

The meaning of the assertions below is that a sur-

jective fuzzy function ρ is indeed a fuzziﬁed version

of its core function g: X

′

→ Y, where X

′

⊆ X. If x ∈ X,

then the fuzzy value of ρ(x, ·) is a “linear”-like com-

bination of F-fuzzy points F

g(x

′

)

(·). In particular, if

′

∈ X

′

- domain of g, then the fuzzy value of ρ(x

′

, ·)

is equal to the corresponding F-fuzzy point F

g(x

′

)

(·).

Theorem 3. Let fuzzy relations E, E

′

, F, ρ and func-

tion g : X

′

→ Y where X

′

= {x

| y ∈ Y} fulﬁl assump-

tions and conclusions of Theorem 2. Then

1. for all x ∈ X, y ∈ Y,

ρ(x, y) =

′

∈X

′

(x) ∗ F

g(x

′

)

(y)), (16)

2. for all t ∈ X

′

, y ∈ Y,

ρ(t, y) = F

g(t)

(y). (17)

3.2 Generalized Extension Principle

In this Section, we will show that every fuzzy func-

tion ρ that establishes a point-to-(fuzzy set) mapping

between X and L

can be extended (via the Gener-

alized extension principle) to a (fuzzy set)-to-(fuzzy

set) mapping between L

and L

. We will use ex-

pression (2), where we replace · by ∗ and use fuzzy

relation ρ instead of ordinary R

. Moreover, we will

use our representation (10) of a non-zero fuzzy set

and show that the extended mapping between L

′

and

is fully determined by its reduction to a certain set

of fuzzy sub-singletons, and in particular, to sets of

′

-fuzzy points of X.

Deﬁnition 3 (Generalized extension principle). Let

L be a complete residuated lattice and (L

, =),

, =) fuzzy spaces. Let E, F be fuzzy equivalences

on X and Y, respectively, and fuzzy relation ρ on

X × Y be a fuzzy function. Then ρ induces the map

→

: L

→ L

such that for every A ∈ L

→

(A)(y) =

x∈X

(A(x) ∗ ρ(x, y)). (18)

Theorem 4. Let fuzzy relations E on X and F on Y be

fuzzy equivalences and moreover, F be a fuzzy equal-

ity. Let fuzzy relation ρ on X ×Y be a surjective fuzzy

function and E

′

be fuzzy equivalence given by (12).

Then for any A ∈ L

→

(A) =

t∈Supp(A)

→

), (19)

where W

is a fuzzy sub-singleton (8) in the space

(X, =, E

′

In particular, if A is represented as a supremum of

fuzzy points E

′

, i.e. A =

t∈Supp(A)

′

, then

→

(A)(y) =

t∈Supp(A)

→

′

)(y) =

t∈Supp(A)

ρ(t, y).

(20)

Proof. By Theorem 1, A can be represented as a

supremum of fuzzy sub-singletons W

, t ∈ Supp(A),

where W

(x) = A(x) ∗ E

(x), x ∈ X. Thence,

FuzzyFunctionandtheGeneralizedExtensionPrinciple

173

→

(A)(y) =

x∈X

(A(x) ∗ ρ(x, y)) =

x∈X





t∈Supp(A)

(x)





∗ ρ(x, y) =

t∈Supp(A)

x∈X

(x)∗ρ(x, y) =

t∈Supp(A)

→

)(y).

To prove (20), we ﬁrst decompose

→

(A)(y) =

t∈Supp(A)

→

′

)(y),

and then continue with the following chain of equali-

ties

→

′

)(y) =

x∈X

′

(x) ∗ ρ(x, y)) =

x∈X

′

(x) ∗

′

∈X

′

(x) ∗ F

g(x

′

)

(y)) =

′

∈X

′

x∈X

′

(x) ∗ E

′

(x)

∗ F

g(x

′

)

(y) =

′

∈X

′

(x) ∗ F

g(x

′

)

(y) = ρ(t, y),

where we made use of representation ρ by (16).

4 CONCLUSION

In this contribution we started a mathematical analy-

sis of basic concepts in the fuzzy set theory such as

fuzzy point, fuzzy set and fuzzy function. We intro-

duced various spaces of elements equipped with crisp

and fuzzy equivalences with the purpose to show that

similar to the classical case, a (fuzzy) set is a collec-

tion of (fuzzy) points or singletons. We recalled the

notion of a fuzzy function as a special fuzzy relation

and showed that similarly to the classical case, any

fuzzy function can be extended to the domain of fuzzy

subsets and this extension is similar to the Extension

Principle of L. Zadeh. We clariﬁed a relationship be-

tween a fuzzy function and its ordinary core function.

ACKNOWLEDGEMENTS

The support of the ESF project 2013/0024/1DP/

1.1.1.2.0/13/APIA/VIAA/045 is kindly announced.

Further support comes from the European Regional

Development Fund in the IT4Innovations Centre of

Excellence project (CZ.1.05/1.1.00/02.0070).

REFERENCES

De Baets, B. and Mesiar, R. (1998). T-partitions. Fuzzy Sets

Systems, 97:211–223.

Demirci, M. (1999). Fuzzy functions and their fundamental

properties. Fuzzy Sets and Systems, 106:239 – 246.

Demirci, M. (2002). Fundamentals of m-vague algebra

and m-vague arithmetic operations. Int. Journ. Un-

certainty, Fuzziness and Knowledge-Based Systems,

10:25–75.

H´ajek, P. (1998). Metamathematics of Fuzzy Logic. Kluwer,

Dordrecht.

H¨ohle, U. (1998). Many-valued equalities, singletons and

fuzzy partitions. Soft Computing, 2:134–140.

H¨ohle, U., Porst, H., and

Sostak, A. (2000). Fuzzy func-

tions: a fuzzy extension of the category set and some

related categories. Applied General Topology, 1:115–

127.

Klawonn, F. (2000). Fuzzy points, fuzzy relations and fuzzy

functions. In Nov´ak, V. and Perﬁlieva, I., editors, Dis-

covering the World with Fuzzy Logic, pages 431–453.

Springer, Berlin.

Klawonn, F. and Castro, J. L. (1995). Similarity in fuzzy

reasoning. Mathware Soft Comput., pages 197–228.

Nov´ak, V. (1989). Fuzzy Sets and Their Applications. Adam

Hilger, Bristol.

Perﬁlieva, I. (2004). Fuzzy function as an approximate so-

lution to a system of fuzzy relation equations. Fuzzy

Sets and Systems, 147:363–383.

Perﬁlieva, I. (2011). Fuzzy function: Theoretical and practi-

cal point of view. In Proc. 7th conf. European Society

for Fuzzy Logic and Technology, EUSFLAT2011, Aix-

Les-Bains, France, July 18-22, 2011, pages 480–486.

Atlantis Press.

Perﬁlieva, I., Dubois, D., Prade, H., Godo, L., Esteva, F.,

and Ho

d´akov´a, P. (2012). Interpolation of fuzzy data:

analytical approach and overview. Fuzzy Sets and Sys-

tems, 192:134158.

Sostak, A. (2001). Fuzzy functions and an extension of the

category l-top of chang-goguen l-topological spaces.

In Proc. 9th Prague Topological Symp., pages 271–

294, Prague, Czech Republlic.

Zadeh, L. A. (1975). The concept of a linguistic variable

and its application to approximate reasoning I, II, III.

Information Sciences, 8-9:199–257, 301–357, 43–80.

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