On the Categories of Coalgebras, Dialgebras and Powerset Theory over
L-Fuzzy Approximation Spaces
Sutapa Mahato
1
and S. P. Tiwari
2
1
Centre for Data Science, ITER, Siksha ’O’ Anusandhan (Deemed to be University), Bhubaneswar-751030, India
2
Department of Mathematics and Computing, Indian Institute of Technology(ISM), Dhanbad-826004, India
Keywords:
L-Fuzzy Approximation Space, Category Theory, Functor, Coalgebra, Dialgebra, Powerset Theory.
Abstract:
This paper is to establish a relationship between powerset theories and the category of dialgebras over the
category of L-fuzzy approximation space, where L is a residuated lattice. Also, we show that the category
FAS of L-fuzzy approximation spaces is a category of F-coalgebras. Interestingly, we introduce a functor
having both the left/right adjoint from the category FAS to the category UAS of upper approximation sets.
Further, the category J-Coal of J-coalgebras and the category (J,K)-Dial of (J,K)-dialgebras are introduced
over the category FAS and it is shown that they are topological categories.
1 INTRODUCTION
The notion of rough sets was introduced by Pawlak
(Pawlak, 1982) in his seminal paper in the early eight-
ies. This idea applies to data systems that contain in-
consistencies. Rough set theory is a database min-
ing or knowledge discovery approach for relational
databases. Equivalence relations played an important
part in the rough set theory presented by Pawlak. Sev-
eral generalizations of rough sets have been made in
the literature (Kondo, 2006; Qin et al., 2008; Yao,
1998) by using an arbitrary relation in the place of
equivalence relation. By using the ideas of fuzzy set
theory, Dubois and Prade (Dubois and Prade, 1990)
introduced the concept of fuzzy rough set, in which
fuzzy relations play the key role instead of crisp rela-
tions, which turned into a powerful tool in analyzing
inconsistent and vague data. Further, the combina-
tions of fuzzy sets and rough sets were studied using
binary fuzzy relations and different fuzzy logic oper-
ations in (D’eer et al., 2015; Mi et al., 2008; Morsi
and Yakout, 1998; Radzikowska and Kerre, 2002;
Radzikowska and Kerre, 2005; Tiwari et al., 2018;
Wang and Hu, 2015; Wu et al., 2013; Yao et al.,
2014); and the topological properties of fuzzy rough
sets were discussed in (Perfilieva et al., 2017; Srivas-
tava and Tiwari, 2003; Tiwari et al., 2014; Tiwari and
Srivastava, 2013; Wang, 2023).
Eilenberg and Mac Lane (Eilenberg and MacLane,
1945) developed the concept of category theory,
which is well-known. A number of researchers fur-
ther developed this theory (Freyd, 1964; Lawvere,
1963; Lawvere, 1966) and demonstrated it to help
develop many aspects of theoretical computer sci-
ence. Coalgebra is an abstract theory that emerged
as a relatively new theory within or closely related
to category theory. Coalgebras have the advantage
of naturally dealing with nondeterminism and unde-
finedness concepts that are difficult or impossible to
deal with algebraically. Categories of dialgebras were
initially investigated as extended algebraic categories
(Ad
´
amek, 1976; Trnkov
´
a and Goral
ˇ
c
´
ık, 1969). In
computer science, these structures have been used to
specify data types (Hagino, 1987; Poll and Zwanen-
burg, 2001) Powerset structures are commonly used
in algebra, logic, topology, and computer science. Al-
most all areas of mathematics and its applications,
including computer science, use the primary exam-
ple of a powerset structure P(X ) = {A : A ⊆ X} and
the related extension of a mapping f : X → Y to the
map f
→
P
: P(X) → P(Y ). Classical set theory can be
thought of as a subset of the fuzzy set theory; natu-
rally, powerset objects associated with fuzzy sets were
examined as generalizations of classical powerset ob-
jects. The first approach was made by Zadeh (Zadeh,
1965), who defined a new powerset object instead of
P(X) and introduced new powerset operators named
Zadeh’s forward and backward operator. Many pa-
pers (Rodabaugh, 1999a; Rodabaugh, 1999b) have
been published about Zadeh’s extension and general-
izations. Rodabaugh investigated Zadeh’s extension
for lattice-valued fuzzy sets for the first time in (Rod-
348
Mahato, S. and Tiwari, S.
On the Categories of Coalgebras, Dialgebras and Powerset Theory over L-Fuzzy Approximation Spaces.
DOI: 10.5220/0012182100003595
In Proceedings of the 15th International Joint Conference on Computational Intelligence (IJCCI 2023), pages 348-358
ISBN: 978-989-758-674-3; ISSN: 2184-3236
Copyright © 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

abaugh, 1999a). Rodabaugh’s work laid a solid foun-
dation for future research into powerset objects and
operators. The relationship of algebraic theories to
powerset theories and fuzzy topological theories have
been studied in (Rodabaugh et al., 2007). In (Mo
ˇ
cko
ˇ
r,
2020), it is proved that fuzzy soft sets also give rise to
a powerset theory, which is defined by a monad. Fur-
ther, in (Mo
ˇ
cko
ˇ
r, 2023), two basic types of relational
powerset theory have been introduced for semiring-
valued fuzzy structures and examine the basic rela-
tionships between these theories. The relationships
between powerset theories and F-transforms are in-
vestigated in (Mo
ˇ
cko
ˇ
r, 2018). Recently, in (Perfilieva
et al., 2017), it has been shown that F-transform is a
realization of an abstract fuzzy rough set theory. So,
it is obvious to think about the relationship between
fuzzy approximation spaces and powerset theories.
We use category and powerset theories to expand this
study’s idea of L-fuzzy approximation spaces. Specif-
ically,
• we show that the category FAS is a category of F-
coalgebras where F is a functor from the category
SET to SET;
• The categories J-Coal and (J, K)-Dial are intro-
duced over FAS and it is shown that they are topo-
logical categories;
• The relationship between powerset theories and
the category of dialgebras is established over the
category EFAS of L-fuzzy equivalence approxi-
mation space.
The paper is organized as follows. In Section 2,
we recall some basic properties of residuated lattice,
L-fuzzy sets and category theory. In Section 3, it
is shown that the category FAS is a category of F-
coalgebras. The categories J-Coal and (J,K)-Dial are
introduced and it is shown that they are topological
categories. Next, the relationship between powerset
theories and the category of dialgebras is established.
At end, we conclude our work.
2 PRELIMINARIES
This section recalls the ideas respective to the resid-
uated lattice, L-fuzzy sets, MV -algebra, and category
theory (cf., (Barr and Wells, 1990; Belohlavek, 2012;
Blount and Tsinakis, 2003;
´
Ciri
´
c et al., 2007; Goguen,
1967; Mac Lane, 2013; Pei, 2004; She and Wang,
2009; Zadeh, 1975)). We begin with the following
definition.
Definition 2.1. A complete residuated lattice
is a structure L = (L, ∧, ∨, ⊗, →, 0, 1) such that
(L, ∧, ∨, 0, 1) is a complete lattice with lower bound
0 and upper bound 1, (L, ⊗, 1) is a commutative
monoid and (→, ⊗) is an adjoint pair, i.e. a ⊗ b ≤ c
⇐⇒ a ≤ b → c, ∀a, b, c ∈ L.
A complete residuated lattice L is said to be di-
visible if for b, c ∈ L and b ≤ c, ∃ d ∈ L such that
b = c ⊙ d. A negation in L is a map ¬ : L → L such
that ¬b = b → 0, ∀ b ∈ L. If ¬(¬b) = b, ∀ b ∈ L, then
L is called a complete regular residuated lattice. An
MV -algebra is a complete regular residuated lattice
satisfies divisibility property.
Proposition 2.1. Let L be a complete residuated lat-
tice. Then for a, b, c ∈ L,
• 1 ⊗ a = a ⊗ 1 = a, a ⊗ 0 = 0 ⊗ a = 0;
• a → (b → c) = b → (a → c);
• a ⊗ (∨
i∈I
b
i
) = ∨
i∈I
(a ⊗ b
i
);
• a → (∧
i∈I
b
i
) = ∧
i∈I
(a → b
i
).
If L is a complete regular residuated lattice, Then
• ¬a → ¬b = b → a;
• ¬ ∨
i∈I
a
i
= ∧
i∈I
¬a
i
.
Throughout this paper, an L-fuzzy set is identified
with its membership function and takes values from a
fixed complete residuated lattice L. For a nonempty
set X, the collection of all L-fuzzy sets of X is
denoted by L
X
. Also, for all a ∈ L; a(x) = a denotes a
constant L-fuzzy set. Further, for A ∈ L
X
, the core(A)
is a set of all elements x ∈ X such that A(x) = 1 and
1
x
is the characteristic function of {x} in X.
For A
1
, A
2
∈ L
X
and x ∈ X, new L-fuzzy sets
are defined as follows: (A
1
∨ A
2
)(x) = A
1
(x) ∨ A
2
(x);
(A
1
∧ A
2
)(x) = A
1
(x) ∧ A
2
(x); (A
1
⊙ A
2
)(x) =
A
1
(x) ⊙ A
2
(x); (A
1
→ A
2
)(x) = A
1
(x) → A
2
(x).
We recall the following concept of an L-fuzzy
relation and Zadeh’s L-fuzzy operators from (Mahato
and Tiwari, 2020).
Definition 2.2. An L-fuzzy relation R on X is an L-
fuzzy set of X × X. An L-fuzzy relation R is called
reflexive if R(x, x) = 1, symmetric if R(x, y) = R(y, x)
and transitive if R(x, y) ⊗ R(y, z) ≤ R(x, z), for all
x, y, z ∈ X.
A reflexive, symmetric and transitive L-fuzzy re-
lation on X is called an L-fuzzy equivalence relation
on X.
Definition 2.3. Let φ : X → Y be a map, then Zadeh’s
L-fuzzy forward operators φ
→
Z
: L
X
→ L
Y
is defined as
φ
→
Z
(A)(x
′
) = ∨
φ(x)=x
′
A(x), ∀A ∈ L
X
, ∀x
′
∈ Y.
Next, we recall the ideas related to category the-
ory. For details, we refer to the work done in (Barr
and Wells, 1990; Mac Lane, 2013). Some standard
On the Categories of Coalgebras, Dialgebras and Powerset Theory over L-Fuzzy Approximation Spaces
349

categories used in this paper are: category SET of
sets as objects and maps as morphisms; and the cat-
egory CSLAT(∨) of complete ∨-semilattices as ob-
jects with ∨-preserving maps as morphisms. Sim-
ilarly, we can define the category CSLAT(∧). We
shall write CSLAT if there is no need to differentiate
between ∨ and ∧.
Definition 2.4. Let F : C → D be a functor and M be
a D-object. Then the functor F has left adjoint if for
some C-object N, there exists a pair (N, τ), where τ :
M → F(N) is a D-morphism such that for all C-object
N
′
and D-morphism f : M → F(N
′
), there exists a
unique C-morphism g : N → N
′
such that the diagram
in Figure 1 commutes.
N F(N) M
N
′
F(N
′
)
f
F(g)
g
τ
Figure 1: Diagram for Definition 2.4.
Definition 2.5. Let G : C → D be a functor and M be
a D-object. Then the functor G has right adjoint if for
some C-object N, there exists a pair (N, η), where η :
G(N) → M is a D-morphism such that for all C-object
N
′
and D-morphism f : G(N
′
) → M, there exists a
unique C-morphism g : N
′
→ N such that the diagram
in Figure 2 commutes.
N G(N) M
N
′
G(N
′
)
f
G(g)
g
η
Figure 2: Diagram for Definition 2.5.
We close this section by recalling the concepts of
F-coalgebras and (F, G)-dialgebras.
Definition 2.6. For a functor F : C → C, F-coalgebra
is a pair (X, α), where X ∈ ob j(C) and α : X → F(X)
is a map of the coalgebra. A morphism between two
F-coalgebras (X
1
, α
1
) and (X
2
, α
2
) is a map f : X
1
→
X
2
such that F( f ) ◦ α
1
≤ α
2
◦ f . The class of F-
coalgebras alongwith their morphisms form a cate-
gory.
Definition 2.7. For functors F, G : C → D, a (F, G)-
dialgebra is a pair (X, γ), where X ∈ ob j(C) and
γ : F(X) → G(X) is a map . A morphism between
two (F, G)-dialgebra (X
1
, γ
1
) and (X
2
, γ
2
) is a map
f : X
1
→ X
2
such that G( f ) ◦ γ
1
≤ γ
2
◦F( f ). The class
of all (F, G)-dialgebras alongwith their morphisms
form a category.
3 COALGEBRAS AND
DIALGEBRAS OVER L-FUZZY
APPROXIMATION SPACES
In this section, we show that the category of L-
fuzzy approximation spaces FAS is a category of
F-coalgebra, where F is a functor from SET to SET.
The concept of the category of upper approximation
set UAS is introduced, and we show that there exists
a functor G from the category FAS to the category
UAS, which has both the left and right adjoint.
Further, the category J-Coal of J-coalgebras and
the category (J,K)-Dial of (J, K)-dialgebras are
introduced over the category FAS and it is shown that
they are topological categories.
Now, we recall the following from (Mahato and
Tiwari, 2020).
Definition 3.1. A pair (X, R) is called an L-fuzzy ap-
proximation space, where X is a nonempty set and R is
an L-fuzzy relation on X. The operators R, R : L
X
→
L
X
are respectively called the L-fuzzy upper and L-
fuzzy lower approximation operators of (X, R), where
for A ∈ L
X
R(A)(x) = ∨
y∈X
(R(x, y) ⊗ A(y))
R(x) = ∧
y∈X
(R(x, y) → A(y)).
The pair (R(A), R(A)) is called an L-fuzzy rough set
of an L-fuzzy set of A ∈ L
X
in (X, R). Further, let
(X
1
, R
1
) and (X
2
, R
2
) be two L-fuzzy approximation
spaces. Then a map f : X
1
→ X
2
is relation preserv-
ing if R
1
(x
1
, y
1
) ≤ R
2
( f (x
1
), f (y
1
)) , ∀x
1
, y
1
∈ X
1
.
L-fuzzy approximation spaces along with relation
preserving maps form a category, say, FAS.
Proposition 3.1. Let (X, R) be an L-fuzzy approxima-
tion space. Then
(1) If R is an L-fuzzy equivalence relation on X, then
R
◦ R = R and R ◦ R = R.
(2) If L is a regular residuated lattice, then ¬R(A) =
R(¬A).
Now, we introduce a functor F : SET → SET such
that for X ∈ ob j(SET), F(X) = L
X
and for a map f ,
F( f ) = f
→
Z
. The following is towards the category
FAS, which is a category of F-coalgebras. Let (X , R)
be an L-fuzzy approximation space. Then the L-fuzzy
relation R : X ×X → L may also be interpreted as a F-
coalgebra map, R : X → L
X
such that for all x, y ∈ X ,
R(x)(y) = R(x, y).
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
350

Theorem 3.1. The category FAS is a category of F-
coalgebra.
Proof: Let (X
1
, R
1
) and (X
2
, R
2
) ∈ ob j(FAS).
Then X
1
, X
2
∈ ob j(SET) and the maps R
1
: X
1
→
L
X
1
and R
2
: X
2
→ L
X
2
are defined as R
1
(x
1
)(y
1
) =
R
1
(x
1
, y
1
) and R
2
(x
2
)(y
2
) = R
2
(x
2
, y
2
). Thus (X
1
, R
1
)
and (X
2
, R
2
) are F-coalgebras. Now, let f : X
1
→ X
2
be a map in SET such that f is relation preseving map
and F( f ) = f
→
Z
. Then the diagram in Figure 3 com-
mutes, i.e., for x
1
∈ X
1
and y
2
∈ X
2
,
X
1
X
2
≤
L
X
1
L
X
2
R
2
f
→
Z
R
1
f
Figure 3: Diagram for Theorem 3.1.
(F( f )◦ R
1
(x
1
))(y
2
) = ( f
→
Z
(R
1
(x
1
)))(y
2
)
= ∨
f (y
1
)=y
2
R
1
(x
1
)(y
1
)
= ∨
f (y
1
)=y
2
R
1
(x
1
, y
1
)
≤ ∨
f (y
1
)=y
2
R
2
( f (x
1
), f (y
1
))
≤ R
2
( f (x
1
), y
2
)
≤ (R
2
◦ f )(x
1
)(y
2
).
Thus F( f ) ◦ R
1
≤ R
2
◦ f and f is a morphism of the
category of F-coalgebras, whereby the category FAS
is a category of F-coalgebras.
Now, we introduce the following.
Definition 3.2. Upper approximation set R
X
of an L-
fuzzy approximation space (X , R) is defined as
R
X
= {R(A) : A ∈ L
X
} ⊆ L
X
.
Let (X
1
, R
1
) and (X
1
, R
2
) be two L-fuzzy approxima-
tion spaces and φ : X
1
→ X
2
be their relation preserv-
ing map. Again, let R
1
X
1
and R
2
X
2
be corresponding
upper approximation sets of (X
1
, R
1
) and (X
2
, R
2
) re-
spectively. Then the morphism φ
↑
: R
1
X
1
→ R
2
X
2
is
defined as φ
↑
(R
1
(A
1
) = R
2
(φ
→
z
(A
1
)), ∀A
1
∈ L
X
1
.
The class of all upper approximation sets along-
with their morphism form a category, say, UAS. Next
are to introduce a functor G from the category FAS to
the category UAS having left and right adjoint.
Proposition 3.2. Let G : FAS → UAS be a map such
that for all (X, R) ∈ ob j(FAS), G(X, R) = R
X
and for
every FAS-morphism φ : (X
1
, R
1
) → (X
2
, R
2
), G(φ) =
φ
↑
. Then G is a functor.
Theorem 3.2. The functor G : FAS → UAS has a left
adjoint.
Proof: To prove that G has left adjoint, it is suf-
ficient to show that for each UAS-object R
X
corre-
sponding to an L-fuzzy approximation space (X, R),
there exist a pair (M, τ), where M is FAS-object and
τ : R
X
→ G(M) is an UAS-morphism alongwith for all
FAS-object M
′
and UAS-morphism g : R
X
→ G(M
′
),
there exists a unique FAS-morphism f : M → M
′
such
that the diagram in Figure 4 commutes.
M G(M) R
X
M
′
G(M
′
)
g
G( f )
f
τ
Figure 4: Diagram for Theorem 3.2.
Now, let M = (X, R) be an FAS-object and
τ : R
X
→ G(M) = R
X
be a map such that for all
R(A) ∈ R
X
, τ(R(A)) = R(A). To show that τ is an
UAS-morphism, let us construct an identity map I :
(X, R) → (X, R), then τ = G(I) is an identity mor-
phism of UAS. Let M
′
= (X
′
, R
′
) be an FAS-object
and g : R
X
→ G(M
′
) = R
′
X
′
. Then there exists an
FAS-morphism f
′
: (X, R) → (X
′
, R
′
) such that for
R(A) ∈ R
X
, g(R(A)) = R
′
( f
′
Z
→
(A)). Next, we de-
fine a map f : M → M
′
such that f = f
′
. Then for
R(A) ∈ R
X
,
(G( f ) ◦ τ)(R(A)) = G( f )(τR(A))
= G( f )(R(A))
= G( f
′
)(R(A))
= R
′
( f
′
Z
→
(A))
= g(R(A)).
Thus G( f ) ◦ τ = g. Further, the uniqueness of f is
trivial. Hence G has a left adjoint.
Theorem 3.3. The functor G : FAS → UAS has a
right adjoint.
Proof: To prove that G has a right adjoint, it is
sufficient to show that for each UAS-object R
X
corre-
sponding to an L-fuzzy approximation space (X, R),
there exists a pair (M, η), where M is an FAS-object
and η : G(M) → R
X
is an UAS-morphism along-
with for all FAS-object M
′
and UAS-morphism g :
G(M
′
) → R
X
, there exists an FAS-morphism f : M
′
→
M such that the diagram in Figure 5 commutes.
On the Categories of Coalgebras, Dialgebras and Powerset Theory over L-Fuzzy Approximation Spaces
351

M G(M) R
X
M
′
G(M
′
)
g
G( f )
f
η
Figure 5: Diagram for Theorem 3.3.
Let M = (X, R) be an FAS-object and η : G(M) =
R
X
→ R
X
be a map such that for all R(A) ∈ R
X
,
η(R(A)) = R(A). To show that η is an UAS-
morphism, let us construct an identity map I :
(X, R) → (X, R), then η = G(I) is an identity mor-
phism of UAS. Let M
′
= (X
′
, R
′
) be an FAS-object
and g : G(M
′
) = R
′
X
′
→ R
X
. Then there exists an
FAS-morphism f
′
: (X
′
, R
′
) → (X, R) such that for
R
′
(A
′
) ∈ R
′
X
′
, g(R
′
(A
′
)) = R( f
′
Z
→
(A
′
)). Next, we de-
fine a map f : M
′
→ M such that f = f
′
. Then for
R
′
(A
′
) ∈ R
′
X
′
,
(η ◦ G( f ))(R
′
(A
′
)) = η(G( f )(R
′
(A
′
)))
= η(R( f
→
Z
(A
′
)))
= R( f
′
Z
→
(A
′
))
= g(R
′
(A
′
)).
Thus η ◦ G( f ) = g. Further, the uniqueness of f is
trivial. Hence G has a right adjoint.
Let R
X
be an upper approximation set and
S : R
X
× R
X
→ L be the L-fuzzy relation on R
X
. Then
the pair (R
X
, S) is called upper approximation space,
where S(R(A), R(B)) = ∨
x∈X
{R(A)(x) ⊗ R(B)(x)},
for R(A), R(B) ∈ R
X
. Now, we have the following.
Proposition 3.3. Let (X
1
, R
1
) and (X
2
, R
2
) be two
L-fuzzy approximation spaces and φ : X
1
→ X
2
be
their relation preserving map. Then (R
1
X
1
, S
1
) and
(R
2
X
2
, S
2
) are the object of FAS. Again let φ
↑
:
(R
1
X
1
, S
1
) → (R
2
X
2
, S
2
) be a morphism such that
φ
↑
(R
1
(A
1
)) = R
2
(φ
→
Z
(A
1
)). Then φ
↑
is a relation pre-
serving map, i.e., φ
↑
∈ FAS-morphism.
Proof: By the definition, R
X
is a set and S is an
L-fuzzy relation on R
X
. Thus (R
X
, S) is an L-fuzzy
approximation space. Now R
1
(A
1
), R
1
(B
1
) ∈ R
X
,
S
2
(φ
↑
(R
1
(A
1
)), = ∨
x
2
∈X
2
φ
↑
R
1
(A
1
)(x
2
) ⊗
φ
↑
(R
1
(B
1
))) φ
↑
(R
1
(B
1
)(x
2
)
= ∨
x
2
∈X
2
{φ
↑
R
1
(A
1
)(x
2
) ⊗
φ
↑
(R
1
(B
1
)(x
2
)}
= ∨
x
2
∈X
2
R
2
(φ
→
z
(A
1
))(x
2
) ⊗
R
2
(φ
→
z
(B
1
))(x
2
)
≥ ∨
x
1
∈X
1
R
1
(A
1
)(x
1
) ⊗ R
1
(B
1
)(x
1
)
≥ S
1
(R
1
(A
1
), R
1
(B
1
).
Thus φ
↑
is a relation preserving map. This completes
the proof.
The following is towards to a category of J-coalgebras
based on the category FAS. Let J : FAS → FAS be a
map such that J(X, R) = (R
X
, S) and J(φ) = φ
↑
. Then
J is also a functor. Before stating the next theorem,
let define a morphism α
R
X
: (X, R) → (R
X
, S), such
that α
R
X
: X → R
X
is a map and α
R
X
(x) = R(1
x
).
Theorem 3.4. Let J : FAS → FAS be a functor,
(X, R) ∈ ob j(FAS), and φ be an FAS-morphism. Then
the set of the pairs ((X , R), α
R
X
) with their morphisms
φ form a category of J-coalgebras.
Proof: Let (X
1
, R
1
), (X
2
, R
2
) ∈ ob j(FAS) and
φ : (X
1
, R
1
) → (X
2
, R
2
) ∈ FAS-morphism such that
J(X
1
, R
1
) = (R
1
X
1
, S
1
), J(X
2
, R
2
) = (R
2
X
2
, S
2
) and
J(φ) = φ
↑
. Then by the definition, it is clear that
the pairs ((X
1
, R
1
), α
R
1
X
1
) and ((X
2
, R
2
), α
R
2
X
2
) are J-
coalgebras. Now, we show that φ is a morphism of
the category of J-coalgebras, i.e., the diagram in Fig-
ure 6 commutes.
(X
1
, R
1
) (X
2
, R
2
)
(R
1
X
1
, S
1
) (R
2
X
2
, S
2
)
α
R
2
X
2
φ
↑
α
R
1
X
1
φ
Figure 6: Diagram for Theorem 3.4.
For x
1
∈ X
1
and x
2
∈ X
2
,
(J(φ) ◦ α
R
1
X
1
(x
1
))(x
2
) = J(φ)(α
R
1
X
1
(x
1
)(x
2
))
= φ
↑
◦ R
1
(1
x
1
)(x
2
)
= R
2
◦ φ
→
z
(1
x
1
)(x
2
)
= R
2
(1
φ(x
1
)
)(x
2
)
= α
R
2
X
2
(φ(x
1
))(x
2
)
= (α
R
2
X
2
◦ φ)(x
1
)(x
2
).
Thus J(φ) ◦ α
R
1
X
1
= α
R
2
X
2
◦ φ. Hence φ is a morphism of
the category of J-coalgebra.
We denote the above category of J-coalgebras
as J-Coal. Before stating the next theorem, we recall
the definition of topological category from (Mo
ˇ
cko
ˇ
r,
2019).
Definition 3.3. A concrete category C with a forgetful
functor U : C → SET is a topological category if let
a system of objects A
i
∈ ob j(C) and X ∈ SET. Then
for any system of maps g
i
: X → U(A
i
), there exists an
initial lift, which is to say
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
352

(i) an object A ∈ ob j(C), such that U(A) = X,
(ii) a system of K-morphism f
i
: A → A
i
, such that
U( f
i
) = g
i
,
(iii) for each B ∈ ob j(C), a map w : U(B) → X, a
system of C-morphism t
i
: B → A
i
such that g
i
◦ w =
U(t
i
), there exists the unique K-morphism h : B → A
such that U(h) = w and f
i
◦ h = t
i
.
Theorem 3.5. The category J-Coal is a topological
category.
Proof: Let U : J-Coal → SET be a functor.
For ((X , R), α
R
X
) ∈ ob j − J-Coal, U((X , R), α
R
X
) =
X. Again, let f be a morphism of J-Coal. Then
U( f ) = f . Now, let ((X
i
, R
i
), α
R
i
X
i
) be a sys-
tem of J-Coal-objects and f
i
: X → X
i
be maps,
where X ∈ SET is set. Then define an L-fuzzy
relation R : X × X → L such that for x, y ∈ X,
R(x, y) = ∧
i∈I
R
i
( f
i
(x), f
i
(y)) and α
R
X
(x) = R(1
x
).
Thus f
i
: ((X, R), α
R
X
) → ((X
i
, R
i
), α
R
i
X
i
) are the J-Coal-
morphisms, where U( f
i
) = f
i
. Again, let ((Y, S), η)
be an object of J-Coal. Then U((Y, S), η) = Y and for
a map w : Y → X and a morphism t
i
: ((Y, S), η) →
((X
i
, R
i
), α
R
i
X
i
), the diagram in Figure 7 commutes,
i.e., f
i
◦ w = U(t
i
) = t
i
. Now, we have to prove
h : ((Y, S), η) → ((X , R), α
R
X
) is a J-Coal-morphism
such that U(h) = w.
((X, R), α
R
X
) (X
i
, R
i
), α
R
i
X
i
)
((Y, S), η)
t
i
h
f
i
X X
i
Y
t
i
w
f
i
Figure 7: Diagram for Theorem 3.5.
F(h) ◦ η(y)(x) = F(h) ◦ S(1
y
)(x)
= R(h
→
Z
(1
y
)(x)
= R(1
h(y)
)(x)
= α
R
X
(h(y)(x)
= (α
R
X
◦ h)(y)(x).
Thus F(h) ◦ η = α
R
X
◦ h. Since U(h) = w implies
h = w. Thus h is unique morphism. This completes
the proof.
Similar to the concept upper approximation space, we
next introduce the concept of lower approximation
space.
Definition 3.4. Lower approximation space of an L-
fuzzy approximation space (X, R) is defined as a pair
(R
X
, T ), where R
X
= {R(A) : A ∈ L
X
} ⊆ L
X
and T :
R
X
× R
X
→ L is an L-fuzzy relation on R
X
, such that
for R(A), R(B) ∈ R
X
,
T (R(A), R(B)) = ∧
x∈X
{R(A)(x) → ¬R(B)(x)}.
Now, we have the following.
Proposition 3.4. Let L be a regular complete residu-
ated lattice, (X
1
, R
1
) and (X
2
, R
2
) be two L-fuzzy ap-
proximation spaces, and φ : X
1
→ X
2
be their rela-
tion preserving map. Then (R
1
X
1
, T
1
) and (R
2
X
2
, T
2
)
are the object of FAS. Again let φ
↓
: (R
1
X
1
, T
1
) →
(R
2
X
2
, T
2
) be a morphism such that φ
↓
(R
1
(A
1
)) =
R
2
(¬φ
→
Z
(¬A
1
)). Then φ
↓
is a relation preserving
map, i.e., φ
↓
∈ FAS-morphism.
Proof: By the definition, R
X
is a set and T is an
L-fuzzy relation on R
X
. Thus (R
X
, T ) is an L-fuzzy
approximation space. Now R
1
(A
1
), R
1
(B
1
) ∈ R
X
,
T
2
(φ
↓
(R
1
(A
1
)), = ∧
x
2
∈X
2
{φ
↓
R
1
(A
1
)
φ
↓
(R
1
(B
1
))) (x
2
) → ¬φ
↓
(R
1
(B
1
)(x
2
))}
= ∧
x
2
∈X
2
{φ
↓
R
1
(A
1
)(x
2
)
→ ¬φ
↓
(R
1
(B
1
)(x
2
)}
= ∧
x
2
∈X
2
{R
2
(¬φ
→
z
(¬A
1
)(x
2
) →
¬R
2
(¬φ
→
z
(¬B
1
)(x
2
)}
≥ ∧
x
1
∈X
1
{R
1
(A
1
)(x
1
) →
¬R
2
(¬φ
→
z
(¬B
1
)(x
2
)
≥ ∧
x
1
∈X
1
{R
1
(A
1
)(x
1
) →
¬R
1
(B
1
)(x
1
)}
≥ T
1
(R
1
(A
1
), R
1
(B
1
)).
Thus φ
↓
is a relation preserving map. This completes
the proof.
Now, we introduce a functor K from the cate-
gory FAS to the category FAS.
Proposition 3.5. Let K : FAS → FAS be a map such
that for all (X, R) ∈ ob j(FAS), K(X , R) = (R
X
, T )
and for every FAS-morphism φ : (X
1
, R
1
) → (X
2
, R
2
),
K(φ) : (R
1
X
1
, T
1
) → (R
2
X
2
, T
2
) is a map such that
K(φ) = φ
↓
. Then K is a functor.
Theorem 3.6. Let L be a complete regular resid-
uated lattice, J, K : FAS → FAS be two functors
such that for all (X, R) ∈ ob j(FAS), and φ ∈ FAS-
morphism such that J(X , R) = (R
X
, S), K(X, R) =
(R
X
, T ), J(φ) = φ
↑
, and K(φ) = φ
↓
. Again, let γ :
On the Categories of Coalgebras, Dialgebras and Powerset Theory over L-Fuzzy Approximation Spaces
353

J(X , R) → K(X, R) be a map such that γ(R(A)) =
R(¬A), for all R(A) ∈ R
X
. Then the set of all pair
((X, R), γ) with their morphisms φ form a category of
(J, K)-dialgebras.
Proof: Let J, K : FAS → FAS be two func-
tors. Now, for all (X, R) ∈ FAS, J(X, R) = (R
X
, S),
K(X, R) = (R
X
, T ) and γ : J(X, R) → K(X, R) is a map
such that γ(R(A)) = R(¬A). Also, by the definition of
dialgebra, the pair ((X , R), γ) is an (J, K)-dialgebra.
Again, let φ : (X
1
, R
1
) → (X
2
, R
2
) be a morphism of
the category FAS. Then we show that φ is a morphism
of the category of (J, K)-dialgebras, i.e., the diagram
in Figure 8 commutes.
(R
1
X
1
, S
1
) (R
2
X
2
, S
2
)
(R
1
X
1
, T
1
) (R
2
X
2
, T
2
)
γ
2
K(φ)
γ
1
J(φ)
Figure 8: Diagram for Theorem 3.6.
K(φ) ◦ γ
1
(R
1
(A
1
)) = K(φ)(R
1
(¬A
1
))
= R
2
(¬φ
→
Z
(¬¬A
1
))
= R
2
(¬φ
→
Z
(A
1
))
= γ
2
(R
2
(φ
→
Z
(A
1
))
= γ
2
J(φ)(R
1
(A
1
)).
Thus K(φ) ◦ γ
X
1
= γ
X
2
◦ J(φ). Hence φ is a homomor-
phism of the category of (J, K)-dialgebras.
We shall denote this category by (J,K)-Dial.
Theorem 3.7. The category (J,K)-Dial is a topologi-
cal category.
Proof: Let V :(J,K)-Dial→ SET be a func-
tor. Then for ((X, R), γ) ∈ ob j − (J,K)-Dial,
V ((X, R), γ) = X. Again, let f be a morphism of
(J,K)-Dial. Then V ( f ) = f . Now, let ((X
i
, R
i
), γ
i
)
be a system of (J,K)-Dial-objects and f
i
: X → X
i
be maps, where X ∈ SET is set. Then define an L-
fuzzy relation R : X × X → L such that for x, y ∈ X,
R(x, y) = ∧
i∈I
R
i
( f
i
(x), f
i
(y)) and γ(R(A)) = R(¬A),
for all R(A) ∈ J(X, R). Since f
i
are relation pre-
serving map then f
i
: ((X, R), γ) → (X
i
, R
i
), γ
i
) are
the (J,K)-Dial-morphisms, where V ( f
i
) = f
i
. Now,
let ((Y, S), δ) be an object of (J,K)-Dial. Then
V ((Y, S), δ) = Y and there exist a map w : Y → X
and a morphism t
i
: ((Y, S), η) → ((X
i
, R
i
), γ
i
) such
that the diagram in Figure 9 commutes, i.e., f
i
◦ w =
V (t
i
) = t
i
. Now, we prove h : ((Y, S), η) → ((X, R), γ)
is a (J,K)-Dial-morphism such that V (h) = w implies
h = w. Firstly, we show that h : (Y, S) → (X, R) is a
relation preserving map.
((X, R), γ) ((X
i
, R
i
), γ
i
)
((Y, S), δ)
t
i
h
f
i
X X
i
Y
t
i
w
f
i
Figure 9: Diagram for Theorem 3.7.
For y
1
, y
2
∈ Y,
R(h(y
1
), h(y
2
)) = ∧
i∈I
R
i
( f
i
(h(y
1
)), f
i
(h(y
2
)))
= ∧
i∈I
R
i
( f
i
(w(y
1
)), f
i
(w(y
2
)))
= ∧
i∈I
R
i
(t
i
(y
1
),t
i
(y
2
))
≥ ∧
i∈I
S(y
1
, y
2
)
≥ S(y
1
, y
2
).
Thus h is a relation preserving map. Now, for B ∈ L
Y
,
K(h) ◦ δ(S(B)) = K(h)(S(¬B))
= R(¬h
→
Z
(¬¬B))
= R(¬h
→
Z
((B))
= γ(R(h
→
Z
((B))
= γ(J(h)(S(B))).
Thus K(h) ◦ δ = γ ◦ J(h). Also, U(h) = w implies h =
w, whereby h is unique morphism. This completes the
proof.
It is advisable to keep all the given values because
any text or material outside the aforementioned mar-
gins will not be printed.
4 POWERSETS VS DIALGEBRAS
In this section, the relationship between powerset the-
ories and the category of dialgebras is established.
We recall the following definition of powerset theory
from (Mo
ˇ
cko
ˇ
r, 2018).
Definition 4.1. Let C be a category with a forget-
ful functor |.| : C → SET. Then (T, →, η) is called
a CSLAT-powerset theory in C, if
(1) T : ob j(C) → ob j(CSLAT) is a map.
(2) For each C-morphism f : X → Y , there exist f
T
→
:
T (X) → T (Y ) in CSLAT.
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
354

(3) η determines for each X ∈ C a mapping η
X
: |X| →
T (X) in SET.
(4) For each f : X → Y in C, f
T
→
◦ η
X
= η
Y
◦ | f |.
In general T may not be a functor. If T is a functor
then it is called a functor type powerset theory.
Definition 4.2. A structure (T, →, η) is called L
∨
-
powerset in a category C, if
(1) (T, →, η) is a CSLAT(∨)-powerset theory in the
category C.
(2) For each X ∈ ob j(C),
(a) there exist ∨-preserving embedding i
X
: T (X) →
L
X
,
(b) for each x ∈ |X |, core(i
X
(η
X
)(x)) ̸=
/
0,
(c) there exist an external operation  : L × T (X) →
T (X), for a ∈ L and A ∈ T (X), i
X
(a  A) = a ⊗ i
X
(A)
and f
→
T
(a  A) = a  f
→
T
(A).
Definition 4.3. Let (X, R) be an L-fuzzy approxima-
tion space and R be an L-fuzzy equivalence relation.
Then the L-fuzzy approximation space is called L-
fuzzy equivalence approximation space.
L-fuzzy equivalence approximation spaces with
their relation preserving maps form a category, say,
EFAS.
Example 4.1. Let (X
1
, R
1
), (X
2
, R
2
) ∈ ob j(EFAS)
and T(X
i
, R
i
) = R
i
X
i
, i = 1, 2. Then f
T
→
:
T (X
1
, R
1
) → T (X
2
, R
2
) such that for all R
1
(A) ∈
R
1
X
1
, f
T
→
(R
1
(A
1
)) = R
2
( f
Z
→
(A
1
)). Also, for each
(X, R) ∈ EFAS, η
X
: X → R
X
such that η
X
(x) = R(1
x
)
and i
X
: R
X
→ L
X
is ∨-preserving map such that
i
X
(R(A)) = R(A) ∈ L
X
. Since R is reflexive then
core(i
X
(R(1
x
))) ̸=
/
0. For x ∈ X
1
and y
′
∈ X
2
,
f
→
T
◦ η
X
1
(x)(y
′
) = f
→
T
(
R
1
(1
x
))(y
′
)
= R
2
( f
→
Z
(1
x
))(y
′
)
= ∨
z
′
∈X
2
{R
2
(y
′
, z
′
) ⊗ f
→
Z
(1
x
)(z
′
)}
= ∨
z
′
∈X
2
{R
2
(y
′
, z
′
) ⊗ (1
f (x)
)(z
′
)}
= R
2
(1
f (x)
)(y
′
)
= η
X
2
( f (x))(y
′
)
= (η
X
2
◦ f )(x)(y
′
).
Thus f
→
T
◦ η
X
1
= η
X
2
◦ f .
Again, for a ∈ L and R(A) ∈ R
X
, we define exter-
nal operation a  R(A) = a ⊗ R(A). Then i
X
1
(a 
R
1
(A
1
)) = i
X
(a ⊗ R
1
(A
1
)) = a ⊗ R
1
(A
1
) = R
1
(a ⊗
A
1
) ∈ T (X
1
). Also, f
→
T
(a ⊗ R
1
(A
1
)) = f
→
T
(R
1
(a ⊗
A
1
)) = R
2
( f
Z
→
(a ⊗ A
1
)) = a ⊗ R
2
( f
Z
→
(A
1
)) = a ⊗
f
→
T
(R
1
(A
1
)). Hence (T, →, η) is L
∨
-powerset theory
of EFAS.
Definition 4.4. The structure (P, →, ζ) is called L
∧
-
powerset theory in the category C, if
(1) (P, →, ζ) is a CSLAT(∧)-powerset theory in cate-
gory C.
(2) For each X ∈ C,
(a) there exist ∧-preserving embedding j
X
: P(X) →
L
X
,
(b) for each x ∈ |X |, such that core(¬ j
X
(ζ(x))) ̸=
/
0,
(c) there exist an external operation + : L × P(X) →
P(X), for a ∈ L and A ∈ P(X ), j
X
(a + A) = ¬a →
j
X
(A) and f
→
P
(a + A) = a + f
→
P
(A).
Example 4.2. Let L be a complete regular residu-
ated lattice and (X
1
, R
1
), (X
2
, R
2
) ∈ ob j(EFAS) such
that P(X
i
, R
i
) = R
i
X
i
, for i = 1, 2. Also, f
P
→
:
P(X
1
, R
1
) → P(X
2
, R
2
) is defined as f
P
→
(R
1
(A
1
)) =
R
2
(¬ f
Z
→
(¬A
1
)). For each (X, R) ∈ EFAS, ζ
X
:
X → P(X, R) is defined as ζ
X
(x) = R(¬1
x
) and
j
X
(R(A)) = R(A). Then for x, y ∈ X , ¬ j
X
(ζ
X
(x))(y) =
¬R(¬1
x
)(y) = R(x, y). Since R is reflexive relation
then core(¬ j
X
(ζ
X
(x)) ̸=
/
0. Now for x ∈ X
1
and y
′
∈
X
2
,
f
P
→
◦ ζ
X
1
(x)(y
′
) = f
P
→
(R
1
(¬1
x
))(y
′
)
= R
2
(¬ f
Z
→
(¬¬1
x
))(y
′
)
= R
2
(¬ f
Z
→
(1
x
))(y
′
)
= ∧
z
′
∈X
2
{R
2
(y
′
, z
′
) →
¬ f
Z
→
(1
x
)(z
′
)}
= ∧
z
′
{R
2
(y
′
, z
′
) →
¬ ∨
z: f (z)=z
′
(1
x
)(z)}
= ∧
z
′
{R
2
(y
′
, z
′
) →
∧
z: f (z)=z
′
(¬1
x
)(z)}
= ∧
z
′
∧
z: f (z)=z
′
{R
2
(y
′
, z
′
) →
(¬1
x
)(z)}
= ∧
z
′
R
2
(y
′
, z
′
) → (¬1
f (x)
)(z
′
)
= R
2
(¬1
f (x)
)(y
′
)
= ζ
X
2
( f (x))(y
′
).
Thus f
P
→
◦ ζ
X
1
= ζ
X
2
◦ f .
Now, for a ∈ L and R(A) ∈ R
X
, we define external op-
eration a +R(A) = ¬a → R(A). Then j
X
(a+ R(A)) =
j
X
(R(a + A)) = R(¬a → A) = ¬a → R(A) = ¬a →
j
X
(R(A)). For R
1
(A
1
) ∈ R
1
X
1
, x
′
∈ X
2
,
f
→
P
(a + R
1
(A
1
))(x
′
) = f
→
P
(R
1
(a + A
1
))(x
′
)
= R
2
(¬ f
→
Z
(¬(a + A
1
)))(x
′
)
= ∧
y
′
∈X
2
R
2
(x
′
, y
′
) →
¬ f
→
Z
(¬(a + A
1
)))(y
′
)
= ∧
y
′
∈X
2
{R
2
(x
′
, y
′
) →
∧
f (y)=y
′
(a + A
1
)(y)}
= ∧
y
′
∈X
2
{R
2
(x
′
, y
′
) →
(¬a → ∧
f (y)=y
′
(A
1
)(y))}
= ¬a → ∧
y
′
∈X
2
{R
2
(x
′
, y
′
) →
¬ f
→
Z
(¬A
1
)(y
′
)}
= a + R
2
(¬ f
→
Z
(¬A
1
))(x
′
)
= a + f
→
P
(R
1
(A
1
))(x
′
).
On the Categories of Coalgebras, Dialgebras and Powerset Theory over L-Fuzzy Approximation Spaces
355

Thus f
→
P
(a+R(A)) = a+ f
→
P
(R(A)). Hence (P, →, ζ)
is L
∧
-powerset theory.
Let (T, →, η) be a L
∨
-powerset theory of a cate-
gory C. Then define a map T : L
|X|
→ T (X), such
that T (A) = ∨
x∈|X|
{η
X
(x) A(x)}. Again let (P, →, ζ)
be L
∧
-powerset theory in C. Then define a map
P : L
|X|
→ T (X) such that P = ∧
x∈|X|
{¬ζ
X
(x)+A(x)}.
Now, we have the following.
Theorem 4.1. Let (T, →, η) be an L
∨
-powerset the-
ory of EFAS. Then (T, →, η) is also L
∧
-powerset the-
ory of EFAS such that R = T and R = T , provided L
is a complete MV -algebra.
Proof: Let f : (X
1
, R
1
) → (X
2
, R
2
) be a mor-
phism in the category EFAS. We define T :
EFAS → CSLAT(∨) such that T (X
1
, R
1
) = L
X
1
and
|(X
1
, R
1
)| = X
1
. Also, define f
→
T
: T (X
1
, R
1
) →
T (X
2
, R
2
) such that f
→
T
= f
→
Z
. It is clear that T (X , R)
is a complete ∨-semilattice and f
→
T
is ∨-preserving
map. Next, we define η
X
: X → T (X , R) such that for
x, y ∈ (X, R), η
X
(x)(y) = R(x, y). Then we show that
the following diagram commute.
X
1
X
2
T (X
1
, R
1
) T (X
2
, R
2
)
η
X
2
f
→
T
η
X
1
f
Figure 10: Diagram for Theorem 4.1.
f
→
T
(η
X
1
(x))(y
′
) = f
→
Z
η
X
1
(x)(y
′
)
= ∨
y: f (y)=y
′
η
X
1
(x)(y)
= ∨
y: f (y)=y
′
R
1
(x, y)
= ∨
y: f (y)=y
′
R
2
( f (x), f (y))
= ∨
y: f (y)=y
′
R
2
(y
′
, f (x))
= R
2
(y
′
, f (x))
= η
X
2
( f (x))(y).
Thus f
→
T
◦ η
X
1
= η
X
2
◦ f . Hence (T, →, η) is
CSLAT(∨) powerset theory. For a ∈ L, x ∈ X and
A ∈ L
X
, (a  A)(x) = a ⊗ A(x). For A ∈ L
X
, x ∈ X, we
have
T (A)(x) = ∨
y∈X
(η
X
(y)  A(y))(x)
= ∨
y∈X
η
X
(y)(x) ⊗ A(y)(x)
= ∨
y∈X
R(x, y) ⊗ A(y)(x)
= R(A)(x)
Hence T = R.
Let L be the complete MV -algebra. For arbitrary
morphism f : (X
1
, R
1
) → (X
2
, R
2
), T (X, R) = L
X
is
also complete ∧-semilattice. Since any complete
MV -algebra is completely distributive and f
→
T
= f
→
Z
is ∧-preserving. Thus (T, →, η) can also consider
as CSLAT(∧) powerset theory. To prove T is
also L
∧
-powerset theory, we need to change only
definition of the external operation + as follows. For
A ∈ L
X
, a ∈ L and x ∈ X, (a + A)(x) = ¬a → A(x).
Then for A ∈ L
X
and x ∈ X,
T (A)(x) = (∧
y∈X
¬η
X
(y) + A(y))(x)
= ∧
y∈X
{¬¬η
X
(y)(x) → A(y)}
= ∧
y∈X
{η
X
(y)(x) → A(y)}
= ∧
y∈X
{R(x, y) → A(y)}
= R(A)(x).
Thus T = R. This completes the proof.
Proposition 4.1. Let (T, →, η) be a L
∨
-powerset the-
ory of a category C. Then for any A ∈ T (X) can
be written as A = ∨
y∈X
{i
X
(A)(y)  η
X
(y)}, provided
i
X
(η
X
(x)) = 1
x
.
Proof: Let A ∈ T (X ). Then i
X
(A) ∈ L
X
. Thus
i
X
(A) can be written as i
X
(A) = ∨
x∈X
{i
X
(A)(x)  1
x
}.
Now,
i
X
(A) = ∨
x∈X
{i
X
(A)(x)  1
x
}
= ∨
x∈X
{i
X
(A)(x)  i
X
(η
X
(x))}
= ∨
x∈X
i
X
{i
X
(A)(x)  η
X
(x)}
= i
X
{∨
x∈X
{i
X
(A)(x)  η
X
(x)}}.
Since i
X
is one-one, whereby A = ∨
x∈X
{i
X
(A)(x) 
η
X
(x)}.
Theorem 4.2. Let (T
1
, →, η
1
) and (T
2
, →, η
2
) be two
functor type L
∨
and L
∧
-powerset theories of a cate-
gory C, respectively. Again, for each X ∈ ob j(C), φ
X
:
T
1
(X) → T
2
(X) is a map such that φ
X
◦ η
1
X
= η
2
X
.
Then the set of pair (X, φ
X
) with their morphisms
as the morphisms in C form a category of (T
1
, T
2
)-
dialgebras, provided A = ∨
x∈X
{i
1
X
(A)(x)  η
1
X
(x)},
for all A ∈ T
1
(X).
Proof: Let (T
1
, →, η
1
) and (T
2
, →, η
2
) be two
functor type L
∨
and L
∧
-powerset theories of a cate-
gory C, respectively. Then for a C-morphism f : X →
Y , f
→
T
1
◦ η
1
X
= η
1
Y
◦ | f | and f
→
T
2
◦ η
2
X
= η
2
Y
◦ | f |.
Now, T
1
, T
2
: C → CSLAT are two functors such
that T
1
( f ) = f
→
T
1
and T
2
( f ) = f
→
T
2
. For each X ∈ C,
define a map φ
X
: T
1
(X) → T
2
(X) such that for all
A ∈ T
1
(X), φ
X
(A) = φ
X
(∨
x∈X
(i
1
X
(A)(x)  η
1
X
(x))) =
∧
x∈X
(¬i
1
X
(A)(x) + φ
X
(η
1
X
(x))). Then by the def-
inition of dialgebra, the pair (X, φ
X
) is an (T
1
, T
2
)-
dialgebra. Now, we show that the map f : X → Y is
an (T
1
, T
2
)-morphism, i.e., the diagram in Figure 11
commutes. Now, for all A ∈ T
1
(X),
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
356

X Y
T
1
(X) T
1
(Y )
T
2
(X) T
2
(Y )
η
1
Y
F
→
T
1
η
1
X
f
F
→
T
2
φ
X
φ
Y
Figure 11: Diagram for Theorem 4.2.
f
→
T
2
◦ φ
X
(A) = f
→
T
2
◦ φ
X
(∨
x∈X
(i
1
X
(A)(x)  η
1
X
(x)))
= f
→
T
2
(∧
x∈X
(¬i
1
X
(A)(x) + φ
X
(η
1
X
)(x)))
= f
→
T
2
(∧
x∈X
(¬i
1
X
(A)(x) + η
2
X
(x)))
= ∧
x∈X
{¬i
1
X
(A)(x) + f
→
T
2
(η
2
X
(x))}
= ∧
x∈X
{¬i
1
X
(A)(y) + (η
2
Y
◦ | f |)(x)}
= ∧
x∈X
{¬i
1
X
(A)(x) + φ
Y
◦ η
1
Y
◦ | f |(x)}
= φ
Y
{∨
x∈X
(i
1
X
(A)(x)  f
→
T
1
◦ η
1
X
(x))}
= φ
Y
◦ f
→
T
1
{∨
x∈X
(i
1
X
(A)(x)  η
1
X
(x))}
= φ
Y
◦ f
→
T
1
(A).
Thus f
→
T
2
◦ φ
X
1
= φ
Y
◦ f
→
T
1
. Hence f is a morphism of
the category of (T
1
, T
2
)-Dialgebras.
Example 4.3. Let (T, →, η) and (P, →, ζ) be the L
∨
,
L
∧
-powerset theories of EFAS as defined in Exam-
ple 4.1 and Example 4.2, respectively. Then T, P :
EFAS → CSLAT are functor type powerset theories.
Since R is an equivalence L-fuzzy relation, R(A) ∈ R
X
can be written as R(A) = ∨
x∈X
(i
X
(R(A))(x)⊗η
X
(x)).
For (X, R) ∈ EFAS, φ
X
: R
X
→ R
X
is defined as fol-
lows.
φ
X
(R(A)) = φ
X
{∨
x∈X
(i
X
(R(A))(x) ⊗ η
X
(x))}
= ∧
x∈X
(¬i
X
(R(A))(x) + φ
X
(η
X
(x)))
= ∧
x∈X
(¬R(A)(x) + ζ
X
(x))
= ∧
x∈X
(R(¬A)(x) + R(¬1
x
)(y))
= ∧
x∈X
(¬R(¬A)(x) → ¬R(x, y))
= ∧
x∈X
(R(x, y) → R(¬A)(x))
= R(R(¬A))(x)
= R(¬A)(x).
Thus the pair ((X, R), φ
X
) is a (T, P)-dialgebra. Now,
let f : (X
1
, R
1
) → (X
2
, R
2
) be a EFAS-morphism.
Then for R
1
(A
1
) ∈ R
1
X
1
,
f
→
P
(φ
X
1
(R
1
(A
1
))) = f
→
P
(R
1
(¬A
1
))
= R
2
(¬ f
→
Z
(¬¬A
1
))
= R
2
(¬ f
→
Z
(A
1
))
= φ
X
2
(R
2
( f
→
Z
(A
1
)))
= φ
X
2
( f
→
T
(R
1
(A
1
))).
Thus f
→
P
◦ φ
X
1
= φ
X
2
◦ f
→
T
. So, f is a morphism of the
category of (T, P)-dialgebras.
5 CONCLUSIONS
In this paper, we have studied L-fuzzy approxima-
tion space from the categorical point of view. It is
shown that the category FAS is the category of F-
coalgebras. Also, we have introduced two categories
J-Coal and (J,K)-Dial, which are topological cate-
gories. In fuzzy set theory and applications, fuzzy ap-
proximation operators and powerset theories in fuzzy
structures are extensively used concepts. Despite the
fact that these concepts appear to be independent in
terms of methodology, there are significant connec-
tions between them. Interestingly, it is shown that
there is a special type of powerset theories in which
maps defined by these powerset theories are fuzzy
approximation operators. Furthermore, we have es-
tablished a bijective correspondence between power-
set theories and the category of dialgebras. Because
of the close link of L-fuzzy approximation operators
with powerset theories, in the future, it will be inter-
esting to derive or discuss the results in fuzzy rough
set theory via the properties of powerset theories.
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