Measure of Roughness for Rough Approximation of Fuzzy Sets and Its

Topological Interpretation

Alexander

Sostak

Institute of Mathematics and Computer Science of the University of Latvia

Raina bulv. 29, Riga, LV-1459, Latvia

Keywords:

L-fuzzy Set, L-relation, L-fuzzy Rough Set, cl-monoid, Measure of Inclusion, L-fuzzy Topology, L-fuzzy

Co-topology.

Abstract:

We deﬁne the measure of upper and the measure of lower rough approximation for L-fuzzy subsets of a

set equipped with a reﬂexive transitive fuzzy relation R. In case when the relation R is also symmetric,

these measures coincide and we call their value by the measure of roughness of rough approximation. Basic

properties of such measures are studied. A realization of measures of rough approximation in terms of L-fuzzy

topologies is presented.

1 INTRODUCTION

The concept of a rough subset of a set equipped with

an equivalence relation was introduced by Z. Pawlak

(Pawlak 1982). Rough sets found important applica-

tions in real-world problems, and also arouse interest

among ”pure” mathematicians as an interesting math-

ematical notion having deep relations with other fun-

damental mathematical concepts, in particular, with

topology. Soon after Pawlak’s work, the concept of

roughness was extended to the context of fuzzy sets;

D. Dubois’ and H. Prade’s paper (Dubois and Prade

1990) was the ﬁrst work in this direction. At present

there is a vast literature where fuzzy rough sets are

investigated and applied. In particular, fuzzy rough

sets are studied and used in (Kortelainen 1994, Ciuci

2009, Yao 1998, Qin and Pai 2005, Qin and Pai 2008,

Hao and Li 2011, Radzikowska and Kerre 2002, Ti-

wari and Srivastava 2013, Mi and HU 2013, Yu and

Zhou 2014) just to mention a few of numerous works

dealing with (fuzzy) rough sets. However, as far as

we know, there were no attempts undertaken to mea-

sure the degree of roughness of a fuzzy set. To state

it in another way, to measure, ”how much rough” is a

given (fuzzy) subset of a set equipped with a (fuzzy)

relation. We undertake such attempt in this paper.

Namely, given an L-fuzzy subset A of a set equipped

with a reﬂexive transitive L-relation we assign to A

an element α ∈ L showing how much this set differs

from its upper and lower rough approximations.

The structure of the paper is as follows. In the

next section we recall two notions fundamental for

our work, namely a cl-monoid and an L-relation. In

the third section we introduce the measure of inclu-

sion of one fuzzy set into another, and describe the

behavior of this measure.

In Section 4 we deﬁne operators of upper and

lower rough approximation for an L-fuzzy subset of

a set endowed with an L-relation. Note that similarly

deﬁned operators under differemt assumptions appear

also in the previous researches, see e.g. (J¨arvinen and

Kortelainen, 2007, Qin and Pai 2005, Sostak 2010,

Sostak 2012)

In Section 5 we deﬁne the measures of upper

K (A) and lower T (A) rough approximation for an L-

fuzzy subset A of a set endowed with an L-relation.

Essentially, K (A) is the measure of inclusion of the

upper approximation of an L-fuzzy set A into A, while

T (A) is the measure of inclusion of A into its lower

approximation. By showing K (A) = T (A) whenever

R is symmetric, we come to the measure of roughness

R A(A) of an L-fuzzy set A.

In Section 6 we interpret the operator of measur-

ing roughness of rough approximation as an L-fuzzy

ditopology (that is a pair of a an L-fuzzy topology T

and an L-fuzzy K co-topology) on a set X and discuss

some issues of this interpretation.

In the last, Conclusion, section we discuss some

directions for the prospective work.

Šostak A..

Measure of Roughness for Rough Approximation of Fuzzy Sets and Its Topological Interpretation.

DOI: 10.5220/0005080400610067

In Proceedings of the International Conference on Fuzzy Computation Theory and Applications (FCTA-2014), pages 61-67

ISBN: 978-989-758-053-6

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

2 PRELIMINARIES

2.1 cl-monoids

Let (L,≤, ∧, ∨) denote a complete lattice, that is a

lattice in which arbitrary suprema (joins) and inﬁma

(meets) exist. In particular, the top 1

and the bottom

elements in L exist and 0

6= 1

Deﬁnition 2.1. (Birkhoff 1995) A tuple (L, ≤, ∧, ∨,∗)

is called a cl-monoid if (L, ≤, ∧, ∨) is a complete lat-

tice and the binary operation ∗ : L × L → L satisﬁes

conditions:

(0∗) ∗ is monotone: α ≤ β =⇒ α ∗ γ ≤ β ∗ γ

for all α, β, γ ∈ L;

(1∗) ∗ is commutative: α∗ β = β∗ α

for all α, β ∈ L;

(2∗) ∗ is associative: (α ∗ β) ∗ γ = α∗ (β∗ γ)

for all α, β, γ ∈ L;

(3∗) ∗ distributes over arbitrary joins:

α∗ (

i∈I

) =

i∈I

(α∗ β

)

for all α ∈ L, for all { β

| i ∈ I} ⊆ L,

(4∗) α ∗ 1

= α, α∗ 0

= 0

for all α ∈ L.

(Note, that a cl-monoid can be deﬁned also as an

integral commutative quantale in the sense of (Rosen-

thal 1990).)

Remark 2.2. In case L = [0, 1] the operation ∗ :

[0, 1] × [0, 1] → [0, 1] satisfying properties (0∗), (1∗),

(2∗) and (4∗) (deﬁned in a slightly different form) for

the ﬁrst time appeared in K. Menger’s papers, see e.g.

(Menger 1979)under the name a triangular norm, or a

t-norm. Latert-normswere thoroughly studied by dif-

ferent authors, see e.g. ( Schweitzer and Sclar 1983,

Klement, Messiar and Pap 2000) A t-norm satisfying

property (3∗) is called lower semicontinuous.

In a cl-monoid a further binary operation 7→, re-

sidium, is deﬁned:

α 7→ β =

{λ ∈ L | λ ∗ α ≤ β}.

Residuation is connected with operation ∗ by the Ga-

lois connection:

α∗ β ≤ γ ⇐⇒ α ≤ (β 7→ γ),

see e.g. (H¨ohle 1992). In the following proposition

we collect well-known properties of the residium:

Proposition 2.3. (see e.g. H¨ohle 1992.)

(17→) (

) 7→ β =

(α

7→ β)

for all {α

| i ∈ I} ⊆ L, for all β ∈ L;

(27→) α 7→ (

) =

(α 7→ β

)

for all α ∈ L, for all {β

| i ∈ I} ⊆ L,;

(37→) 1

7→ α = α for all α ∈ L;

(47→) α 7→ β = 1

whenever α ≤ β;

(57→) α ∗ (α 7→ β) ≤ β for all α, β ∈ L;

(67→) (α 7→ β) ∗ (β 7→ γ) ≤ α 7→ γ for all α, β, γ ∈ L;

(77→) α 7→ β ≤ (α∗ γ 7→ β ∗ γ) for all α, β, γ ∈ L.

In the sequel we will need the following two lem-

mas:

Lemma 2.4. Let (L, ≤, ∧, ∨,∗) be a cl-monoid. Then

for every {α

| i ∈ I} ⊆ L and every {β

| i ∈ I} ⊆ L it

holds:





7→





≥

(α

7→ β

Indeed, applying Proposition 2.3 we have:

7→

(

7→ β

) ≥

(α

7→ β

Lemma 2.5. Let (L, ≤, ∧, ∨,∗) be a cl-monoid. Then

for every {α

| i ∈ I} ⊆ L, and every {β

| i ∈ I} ⊆ L it

holds:





7→





≥

(α

7→ β

Proof. Applying Proposition 2.3 we have

(α

7→ β

) ∗ α

≤ β

for each i ∈ I. Let c =

(α

7→ β

). Then c∗ α

≤ β

for each i ∈ I. Taking suprema on the both sides of

the above inequality over i ∈ I we get c∗

≤

and hence, by the Galois connection,

(α

7→ β

). ≤

7→

2.2 L-relations

The concept of a fuzzy relation (or an [0, 1]-relation

in our terminology) was ﬁrst introduced by Zadeh and

then redeﬁned and studied by different authors.

Deﬁnition 2.6. (Zadeh 1971, Valverde 1985) An L-

relation on a set X is a mapping R : X × X → L.

(r) L-relation R is called reﬂexive if

R(x, x) = 1 for each x ∈ X;

(s) L-relation R is called symmetric if

R(x, y) = R(y, x) for all x, y ∈ X;

(t) L-relation R is called transitive if

R(x, y) ∗ R(y, z) ≤ R(x, z) for all x, y, z ∈ X.

A reﬂexive symmetric transitive L-relation is called an

L-equivalence, or a similarity L-relation.

Let a lattice L be ﬁxed and let REL(L) be the

category whose objects are pairs (X, R), where X is

a set and R : X × X → L is a transitive reﬂexive L-

relation on it. Morphisms in REL(L) are mappings

f : (X, R

) → (Y, R

) such that

(x, x

′

) ≤ R

( f(x), f(x

′

)) for all x, x

′

∈ X.

FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications

3 THE MEASURE OF INCLUSION

OF L-fuzzy SETS

If (L, ≤, ∧, ∨, ∗) is a cl-monoid, and X is a set, then the

lattice and the monoidal structures of L can be point-

wise lifted to the L-powerset L

of X. Namely, given

A, B ∈ L

we set A ≤ B iff A(x) ≤ B(x) for all x ∈ X,

and deﬁne operations on L

by setting

(A∧ B)(x) = A(x) ∧ B(x), (A ∨ B)(x) = A(x) ∨ B(x),

(A∗ B)(x) = A(x) ∗ B(x)

for all x ∈ X. One can easily notice that in this way

, ≤, ∧, ∨, ∗) becomes a cl-monoid.

Deﬁnition 3.1. By setting

A ֒→ B = inf

x∈X

(A(x) 7→ B(x))

for all A, B ∈ L

we obtain a mapping

֒→: L

× L

→ L.

Equivalently, ֒→ can be deﬁned by

A ֒→ B = inf(A 7→ B),

where the inﬁmum of the L-fuzzy set A 7→ B is taken

in the lattice L

. We call A ֒→ B by the measure of

inclusion of the L-fuzzy set A in the L-fuzzy set B.

As the next proposition shows, the measure of in-

clusion

֒→: L

× L

→ L

has properties in a certain sense resembling the prop-

erties of the residuation:

Proposition 3.2. Mapping ֒→: L

× L

→ L deﬁned

above satisﬁes the following properties:

(1֒→) (

) ֒→ B =

֒→ B)

for all {A

| i ∈ I} ⊆ L

, for all B ∈ L

;

(2֒→) A ֒→ (

) =

(A ֒→ B

)

for all A ∈ L

, for all {B

| i ∈ I} ⊆ L

;

(3֒→) A ֒→ B = 1

whenever A ≤ B;

(4֒→) 1

֒→ A = inf

A(x)

for all A ∈ L

;

(5֒→) A∗ (A ֒→ B) ≤ B

for all A, B ∈ L

;

(6֒→) (A ֒→ B) ∗ (B ֒→ C) ≤ A ֒→ C

for all A, B,C ∈ L

;

(7֒→) A ֒→ B ≤ (A∗C ֒→ B∗C)

for all A, B,C ∈ L

;

(8֒→) (

) ֒→ (

) ≥

֒→ B

)

for all {A

, B

∈ I} ⊆ L

;

(9֒→) (

) ֒→ (

) ≥

֒→ B

)

for all {A

, B

∈ I} ⊆ L

Proof. The proof can be done straightforward from

the deﬁnition of operation ֒→ and applying properties

of the residium 7→: L×L → L collected in Proposition

2.3, Lemma 1 and Lemma 2.

4 ROUGH APPROXIMATION OF

A FUZZY SET

Let R : X × X → L be a reﬂexive transitive L-

relation on a set X and A ∈ L

. By the rough ap-

proximation of the L-fuzzy set A we call the pair

(A), l

(A)) where u

: L

→ L

and l

: L

→ L

are respectively operators of upper and lower rough

approximations of A deﬁned below.

4.1 Upper Rough Approximation

Given a reﬂexive transitive L-relation R : X × X → L,

we deﬁne the upper rough approximation operator

: L

→ L

(A)(x) = sup

′



R(x, x

′

) ∗ A(x

′

)



∀A ∈ L

, ∀x ∈ X.

Theorem 4.1. The upper rough approximation oper-

ator satisﬁes the following properties:

(1u) u

) = 0

;

(2u) A ≤ u

(A) ∀A ∈ L

;

(3u) u

(

) =

) ∀{A

| i ∈ I} ⊆ L

;

(4u) u

(A)) = u

(A) ∀A ∈ L

;

Proof. Statement (1u) is obvious. Statement (2u)

follows easily taking into account reﬂexivity of the

L-relation R. We prove property (3u) as follows:

(

)(x) = sup

′

(R(x, x

′

) ∗ (

′

))) =

sup

′

(

R(x, x

′

)∗A

′

)) =

(sup

′

(R(x, x

′

)∗A

′

)))

)(x)) =



)



(x).

Finally, taking into account transitivity of the L-

relation we have:

(A))(x) = sup

′

(A)(x

′

) ∗ R(x, x

′

)) =

sup

′′

sup

′

(A(x

′′

) ∗ R(x, x

′

) ∗ R(x

′

, x

′′

)) ≤

sup

′′

A(x

′′

) ∗ R(x, x

′′

) = u

(A)(x).

Since the converse inequality follows from (2u), we

get property (4u).

4.2 Lower Rough Approximation

Induced by a Reﬂexive Transitive

L-relation

Given a reﬂexive transitive L-relation R : X × X → L,

we deﬁne a lower rough approximation operator l

→ L

(A)(x) = inf

′



R(x, x

′

) 7→ A(x

′

)



∀A ∈ L

∀x ∈ X.

Similar results can be found e.g. in (J¨arvinen and Ko-

rtelainen , Qin and Pei 2005, Sostak 2010.)

MeasureofRoughnessforRoughApproximationofFuzzySetsandItsTopologicalInterpretation

Theorem 4.2. The lower rough approximation oper-

ator satisﬁes the following properties:

(1l) l

) = 1

;

(2l) A ≥ l

(A) ∀A ∈ L

;

(3l) l

(

) =

) ∀{A

| i ∈ I} ⊆ L

;

(4l) l

(A)) = l

(A) ∀A ∈ L

;

Proof. Statement (1l) is obvious. Statement

(2l) follows easily taking into account reﬂexivity

of the L-relation R. We proveproperty (3l) as follows:

(

)(x) = inf

′



R(x, x

′

) 7→

′

)



inf

′



R(x, x

′

) 7→ A

′

)



inf

′



R(x, x

′

) 7→ A

′

)



Finally, taking into account transitivity of the L-

relation we have:

(A))(x) = inf

′

(R(x, x

′

) 7→ l

(A)(x

′

)) =

inf

′

(R(x, x

′

) 7→ inf

′

(R(x

′

, x

′′

) 7→ A(x

′′

))) =

inf

′

(inf

′′

(R(x, x

′

) ∗ R(x

′

, x

′′

) 7→ A(x

′′

))) ≥

inf

′′

(R(x, x

′′

) 7→ A(x

′′

)) = l

(A)(x).

Since the converse inequality follows from (2l), we

get property (4l).

5 THE MEASURE OF

ROUGHNESS OF AN L-FUZZY

SET

Let R : X × X → L be a reﬂexive transitive L-relation

on a set X. Given an L-fuzzy set A ∈ L

we deﬁne the

measure K (A) of its upper rough approximation by

K (A) = u

(A) ֒→ A

and the measure T (A) of its lower rough approxima-

tion by

T (A) = A ֒→ l

(A).

Theorem 5.1. If R is also symmetric, that is an equiv-

alence L-relation, then K (A) = T (A) for every L-

fuzzy set A.

Proof. For the measure of the upper rough approxi-

mation we have

K (A) = u

(A) ֒→ A = inf

(A)(x) 7→ A(x)) =

inf

sup

′

(A(x) ∗ R(x, x

′

) 7→ A(x

′

)) =

inf

′



A(x) ∗ R(x, x

′

) 7→ A(x

′

)



Similar results can be found e.g. in (J¨arvinen and Ko-

rtelainen , Qin and Pei 2005, Sostak 2010.)

On the other hand, for the lower rough approxima-

tions we have

A ֒→ l

(A) = inf

(A(x) 7→ l

(A)(x)) =

inf

(A(x) 7→ inf

′



R(x, x

′

) 7→ A(x

′

))



inf

′

(A(x) 7→ (R(x, x

′

) 7→ A(x

′

)) =

inf

′

(A(x) ∗ R(x, x

′

) 7→ A(x

′

Since R(x, x

′

) = R(x

′

, x) in case R is symmetric, to

complete the proof it is sufﬁcient to notice that

(α∗ β) 7→ γ = α 7→ (β 7→ γ)

for any α,β, γ ∈ L. Indeed

(α∗ β) 7→ γ =

{λ | λ ∗ (α∗ β) ≤ γ},

and

α 7→ (β 7→ γ) =

{λ | λ ∗ α ≤ β 7→ γ} =

{λ | (λ ∗ α) ∗ β 7→ γ};

the last equality is justiﬁed by Galois connection be-

tween 7→ and ∗.

The previous theorem allows us to introduce the

following deﬁnition:

Deﬁnition 5.2. Let R be an equivalence L-relation on

a set X and A ∈ L

. The measure of rough approxi-

mation of A is deﬁned by

R A(A) = u

(A) ֒→ A = A ֒→ l

(A).

In the next theorem we collect the main properties

of the operators K : L

→ L and T : L

→ L, and

hence also of the operator R A : L

→ L in case the

relation R is symmetric.

Theorem 5.3. Measures of roughness of upper and

lower rough approximation K , T : L

→ L has the

following properties:

1. K (0

) = 1

: X → L is the constant function

(x) = 0

∀x ∈ X);

2. T (1

) = 1

: X → L is the constant function

(x) = 1

∀x ∈ X);

3. K (u(A)) = 1

for every A ∈ L

;

4. T (l(A)) = 1

for every A ∈ L

;

5. K (

) ≥

K (A

)

for every family of L-fuzzy sets {A

| i ∈ I} ⊆ L

;

6. T (

) ≥

T (A

)

for every family of L-fuzzy sets {A

| i ∈ I} ⊆ L

FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications

Proof. (1) Referring to Theorem 4.1 and applying

Proposition 3.2 (3 ֒→) we have

K (A) = u

) ֒→ 0

= 0

֒→ 0

= 1

(2) Referring to Theorem 4.2 and applying Proposi-

tion 3.2 (3 ֒→), we have

T (A) = 1

֒→ l

) = 1

֒→ 1

= 1

(3) Referring to Theorem 4.1 and applying Proposi-

tion 3.2 (3 ֒→), we have

K (u(A))) = u

(A) ֒→ u

(A) = u

(A) ֒→ u

(A) = 1.

(4) Referring to Theorem 4.2 and applying Proposi-

tion 3.2 (3 ֒→), we have

T (l(A))) = l

(A) ֒→ l

(A)) = l

(A) ֒→ l

(A) = 1

′

(5) Referring to Theorem 4.1 and applying Proposi-

tion 3.2 (8 ֒→), we have

K (

) = u

(

) ֒→

≥

(u(A

) ֒→ A

(6) Referring to Theorem 4.2 and applying Proposi-

tion 3.2 (9 ֒→), we have

T (

) = (

) ֒→ l(

) =

֒→

l(A

) ≥

֒→ l(A

Theorem 5.4. Let R

: X × X → L and R

:Y ×Y →

L be reﬂexive transitive L-relations on sets X and Y

respectively. Further, let f : X → Y be a mapping such

that

(x, x

′

) ≤ R

( f(x), f(x

′

))

for every x, x

′

∈ X. Then

( f

−1

(B)) ≥ K

(B) and T

( f

−1

(B)) ≥ T

(B)

for every B ∈ L

Proof. follows from the next sequences of

(in)equalities:

( f

−1

(B)) = u

( f

−1

(B)) ֒→ f

−1

(B) =

inf

( f

−1

(B)(x) 7→ f

−1

(B)(x)) =

inf

(sup

′

B( f(x))∗ R(x, x

′

) 7→ B( f(x))) ≥

inf

(sup

′

B( f(x

′

) ∗ R( f(x), f(x

′

)) 7→ B( f(x)) ≥

inf

(sup

′

B(y

′

) ∗ R(y, y

′

) 7→ B(y)) = K

(B)

and

( f

−1

(B)) = f

−1

(B) ֒→ l

( f

−1

(B)) =

inf

( f

−1

(B)(x) 7→ l

( f

−1

(B))(x)) =

inf

(B( f(x)) 7→ inf

′

(R(x, x

′

) 7→ f

−1

′

)) =

inf

′

(B( f(x)) 7→ (R(x, x

′

) 7→ B( f(x

′

))) ≥

inf

′

(B( f(x)) 7→ (R( f(x), f(x

′

)) 7→ B( f(x)) ≥

inf

′

(B(y) 7→ (R(y, y

′

) 7→ B(y)) =

B ֒→ l

(B) = T

(B).

Example 5.5. Let ∗

be the Łukasiewicz t-norm on

the unit interval L = [0, 1], that is

α∗ β = min(α+ β− 1, 1)

and 7→

: L × L → L be the corresponding residium,

that is

α 7→

β = max{1− α+ β, 0}.

Then, given an equivalence L-relation R on a set X

and A ∈ L

we have:

K (A) = inf

inf

′

(2−A(x)+ A(x

′

)−R(x, x

′

));

T (A) = inf

inf

′

(2−A(x)+A(x

′

)−R(x

′

, x)).

In particular, if R : X × X → [0, 1] is the discrete rela-

tion, that is

R(x, x

′

) =



1 if x = x

′

0 otherwise,

we have

R A(A) = 1 for every A ∈ L

On the other hand for the indiscrete relation (that is

R(x, x

′

) = 1 for all x, x

′

∈ X)

R A(A) = 1 − inf

x,x

′

| A(x) − A(x

′

) | for all A ∈ L

Example 5.6. Let ∗ = ∧ be the minimum t-norm on

the unit interval L = [0, 1], and 7→: L× L be the cor-

responding residium, that is

α 7→ β =



1 if α ≤ β

β otherwise

Then given a reﬂexive transitive L-relation R on a

set X and A ∈ L

we have:

K (A) = inf

inf

′

(A(x

′

) ∧ R(x, x

′

) 7→ A(x));

T (A) = inf

inf

′

(A(x

′

) ∧ R(x

′

, x) 7→ A(x)).

In particular, R A(A) = 1 for every A ∈ L

in case the

relation R is symmetric.

MeasureofRoughnessforRoughApproximationofFuzzySetsandItsTopologicalInterpretation

Example 5.7. Let ∗ = · be the product t-norm on the

unit interval [0,1], and 7→: L×L be the corresponding

residium, that is

α 7→ β =



1 if α ≤ β

if α > β

Then, for a reﬂexive transitive L-relation we have:

K (A) = inf

inf

′

(A(x

′

) · R(x, x

′

) 7→ A(x));

T (A) = inf

inf

′

(A(x

′

) · R(x

′

, x) 7→ A(x)).

In particular

R A(A) = inf

inf

′

(A(x

′

) · R(x, x

′

) 7→ A(x))

in case R is symmetric.

6 MEASURE OF ROUGHNESS OF

A FUZZY SET:

DITOPOLOGICAL

INTERPRETATION

Notice that conditions (2), (4), and (6) of Theorem

5.3 actually mean that the mapping T : L

→ L is an

L-fuzzy topology on the set X, (see e.g. Sostak 1989,

Sostak 1996), while conditions (1), (3), and (5) of this

Theorem mean that the mapping K : L

→ L is an L-

co-topology on this set (see e.g. (Sostak 1985, Kubiak

1985, Sostak 1989, Sostak 1996). Since the mappings

T and K are not mutually related via complementa-

tion on the lattice L (which even need not exist on

the lattice) we may interpret the pair (T , K ) as an

L-fuzzy ditopology on the set X (Brown, Ert¨urk and

Dost 2000).

Let α ∈ L be ﬁxed and let

= {A | K (A) ≥ α} and T

= {A | T (A) ≥ α}.

Then, applying again Theorem 5.3, we easily con-

clude that T

satisﬁes the axioms of a Chang-Goguen

L-topology, see (Chang 1968, Goguen 1973) and

satisﬁes the axioms of a Chang-Goguen L-co-

topology. Hence for each α ∈ L the pair (T

, K

)

can be realized as a a Chang-Goguen L-topology on

X (Brown, Ert¨urk and Dost 2000).

From Theorem 5.4 we conclude that if f :

(X, R

) → (Y, R

) is a morphism in the category

REL(L) of sets endowed with reﬂexive transitive L-

relations, then

f : (X, T

, K

) → (Y, T

, K

)

In this section we give an alternative view on the con-

cepts studied in the work. A reader not interested in the

topological aspects of approximation, may omit this sec-

tion.

is continuous mapping of the corresponding L-fuzzy

ditopological spaces. Thus we come to the following

Theorem 6.1. By assigning to every object (X, R

)

from the category REL(L) (see subsection 2.2) an L-

fuzzy ditopological space (X, T

, K

), and interpret-

ing a morphism f : (X, R

) → (Y, R

) of REL(L) as

a mapping f : (X, T

, K

) → (Y, T

, K

) we obtain a

functor

Φ : REL(L) → L-DiTop(L),

where L-DiTop(L), is the category of L-fuzzy ditopo-

logical spaces and their continuous mappings.

Corollary 6.2. Let α ∈ L be ﬁxed. By as-

signing to every object (X, R

) from the cate-

gory REL(L) a Chang-Goguen L-ditopological space

(X, T

, K

), and realizing a morphism f :

(X, R

) → (Y, R

) from REL(L) as a mapping f :

(X, T

, K

) → (Y, T

, K

) we obtain a functor

: REL(L) → L-DiTop,

where L-DiTop is the category of Chang-Goguen L-

ditopological spaces and their continuous mappings.

7 CONCLUSION

In this paper we proposed an approach allowing to

measure the roughness of lower and upper rough ap-

proximation for fuzzy subsets of a set endowed with

a reﬂexive transitive L-relation. The basics of the the-

ory of roughness measure were developed here. Be-

sides, a natural interpretation of the operator of mea-

sure of rough approximation as a fuzzy ditopology

was sketched here. However, several crucial issues

concerning this theory remain untouched in this work.

As one of the ﬁrst goals for the further work we see

the development of a consistent categorical viewpoint

on the measure of rough approximation. In particular,

it is important to study the behavior of the measure

of approximation under operations of products, direct

sums, quotients, etc, and to research the behavior of

the measure of roughness under images and preim-

ages of special mappings between sets endowed with

reﬂexive transitive fuzzy relations..

Another interesting, in our opinion, direction of

the research is to developthe topologicalmodel of this

theory sketched in Section 6. The restricted volume of

this work does not allow us to linger on this subject.

However, in our opinion the topological interpretation

of the theory could be helpful for further studies.

Besides we hope that the concept of measure of

rough approximation will be helpful also in some

problems of practical nature, since since it allows in a

certain sense to measure the quality of the rough ap-

proximation.

FCTA2014-InternationalConferenceonFuzzyComputationTheoryandApplications

ACKNOWLEDGEMENTS

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

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

REFERENCES

Birkhoff, G . Lattice Theory, AMS Providence, RI, (1995)

Brown, L.M., Ert¨urk, R. Dost S¸. Ditopological texture

spaces and fuzzy topology, I. Basic concepts, Fuzzy

Sets and Syst. 110, 227–236 (2000).

Ciucci, D. Approximation algebra and framework, Fund.

Inform. 94(2) (2009) 147–161.

Chang, C.L. Fuzzy topological spaces, J. Math. Anal. Appl.

24, 182–190 (1968)

Chen, P., Zhang, D. Alexandroff L-cotopological spaces,

Fuzzy Sets and Systems 161 (2010) 2505 – 2514.

Ciucci, D. Approximation algebra and framework, Fund.

Inform. 94(2) (2009) 147–161.

Gierz, G., Hoffman, K.H., Keimel, K., Lawson, J.D., Mis-

love, M.W., Scott, D.S. Continuous Lattices and Do-

mains, Cambridge Univ. Press, Cambridge, 2003

Goguen, J.A. The fuzzy Tychonoff theorem, J. Math. Anal.

Appl. 43 (1973) 734–742.

Dubois, D., Prade, H. Rough fuzzy sets and fuzzy rough

sets, International J. General Systems 17 (2–3) (1990)

191–209.

El¸kins, A.,

Sostak, A. On some categories of approximate

systems generated by L-relations. In: 3

Rough Sets

Theory Workshop, pp. 14–19 Milan, Italy (2011)

Hao J., Li Q. The relation between L-fuzzy rough sets and

L-topology, Fuzzy Sets and Systems, 178 (2011)

H¨ohle U. Commutative residuated l-monoids, in: U. H¨ohle

abd E.P. Klement eds., Nonclassical Logics and their

Appl. to Fuzzy Subsets, 53–106, Kluwer Acad. Publ.,

Docrecht, Boston, 1995.

J¨arvinen J. On the structure of rough approximations, Fund.

Inform. 53 (2002) 135–153.

J¨arvinen J., Kortelainen J. A uniﬁed study between modal-

like operators, topologies and Fuzzy Sets, Fuzzy Sets

and Systems 158 (2007) 1217–1225.

Kaufman A. Theory of Fuzzy Subsets, Academic Press,

New York, 1975.

Klement, E.P., Mesiar, R., Pap, E. Triangular Norms.

Kluwer Acad. Publ. (2000)

Kortelainen J. On relationship between modiﬁed sets, topo-

logical spaces and rough sets, Fuzzy Sets and Systems

61 (1994) 91-95.

Kubiak T. On fuzzy topologies, PhD Thesis, Adam Mick-

iewicz University Poznan, Poland (1985)

Menger K. Geometry and positivism - a probabilistic mi-

crogeometry, Selected Papers in Logic, Foundations,

Didactics, Economics, Reidel, Dodrecht, 1979.

Mi J.S., Hu B.Q. Topological and lattice structure of L-

fuzzy rough stes determined by upper and lower sets,

Information Sciences 218 (2013) 194–204.

Pawlak, Z. Rough sets, Intern. J. of Computer and Inform.

Sci. 11, 341–356 (1982)

Qin K., Pei Z. On the topological properties of fuzzy rough

sets, Fuzzy Sets and Systems 151 (2005) 601–613.

Qin K., Pei Z. Generalized rough sets based on reﬂexive and

transitive relations, Information Sciences 178 (2008)

4138–4141.

Radzikowska, A.M., Kerre, E.E. A comparative study of

fuzzy rough sets. Fuzzy Sets and Syst. 126, 137–155

(2002)

Rosenthal, K.I. Quantales and Their Applications. Pirman

Research Notes in Mathematics 234. Longman Scien-

tiﬁc & Technical (1990)

Schweitzer, B., Sklar, A. Probabilistic Metric Spaces. North

Holland, New York (1983).

Sostak, A. On a fuzzy topological structure, Suppl. Rend.

Circ. Matem. Palermo, Ser II 11, 125–186 (1980).

Russian Math. Surveys 44 (1989) 125–186

Sostak, A. Two decades of fuzzy topology: Basic ideas,

notions and results, Russian Math. Surveys 44 (1989)

125–186

Sostak, A. Basic structures of fuzzy topology, J. Math. Sci.

78 (1996) 662–701.

Sostak A. Towards the theory of M-approximate systems:

Fundamentals and examples. Fuzzy Sets and Syst.

161, 2440–2461 (2010)

Sostak, A. Towards the theory of approximate sys-

tems: variable range categories. In: Proceedings of

ICTA2011, Islamabad, Pakistan. pp. 265–284. Cam-

bridge University Publ. (2012)

Tiwari, S.P., Srivastava, A.K. Fuzzy rough sets. fuzzy pre-

oders and fuzzy topoloiges, Fuzzy Sets and Syst., 210

(2013), 63–68.

Valverde, L. On the structure of F-indistinguishability op-

erators, Fuzzy Sets and Syst. 17, 313–328 (1985)

Yao, Y.Y. A comparative study of fuzzy sets and rough sets.

Inf. Sci. 109, 227–242 (1998)

Yao, Y.Y. On generalizing Pawlak approximation opera-

tors. In: Proc. First Intern. Conf. Rough Sets and Cur-

rent Trends in Computing. pp. 298–307 (1998)

Yu H., Zhan W.R. On the topological properties of gener-

alized rough sets, Information Sciences 263 (2014),

141–152.

Zadeh L. Similarity relations and fuzzy orderings, Inf. Sci.

3 (1971) 177–200.

MeasureofRoughnessforRoughApproximationofFuzzySetsandItsTopologicalInterpretation