Approximations of New MV -Valued Types of Fuzzy Sets
Ji
ˇ
r
´
ı Mo
ˇ
cko
ˇ
r
a
University of Ostrava, Institute for Research and Applications of Fuzzy Modeling,
30. dubna 22, 701 03 Ostrava 1, Czech Republic
Keywords:
Semirings, Dual Pair of Semirings, Transformation of New Fuzzy Sets, Approximation of New Fuzzy Sets.
Abstract:
Many of the new types of fuzzy sets, such as intuitionistic, neutrosophic, multi-level or fuzzy soft sets and
their combinations, can be transformed into one common type of fuzzy sets, called (R, R
∗
)-fuzzy sets, with
values in a set R that is a common underlying set of complete commutative idempotent semirings R and R
∗
.
For (R, R
∗
)-fuzzy sets, the theory of lower and upper approximations by (R, R
∗
)-relations is defined and
the basic properties of these approximations are presented. Using examples of the transformation of some
new types of MV -valued fuzzy sets and corresponding fuzzy relations into R-fuzzy sets and R-fuzzy relations,
examples of approximation of these new types of fuzzy sets through their fuzzy relations are presented, without
having to define these operators separately for each new type of fuzzy set.
1 INTRODUCTION
The paper deals with the application of approximation
methods to new types of MV -valued fuzzy sets. The
development of fuzzy mathematics and their theory
and applications leads to an expansion of new types
of fuzzy set with values in various ordered algebraic
structures. Let us mention, for example, intuitionistic
fuzzy sets (Aggarwal et al., 2019), (Atanassov, 1986),
(Atanassov, 1984), (Kozae and et al., 2020), fuzzy
soft sets (Aktas and Cagman, 2007), (Maji and et al,
2001), (Maji et al., 2003), (Maji and et al., 2002),
(Molodtsov, 1999), (Mushrif et al., 2006), or neutro-
sophic fuzzy sets (Hu and Zhang, 2019), (James and
Mathew, 2021), (Zhang et al., 2018) and their mutual
combinations, such as intuitionistic fuzzy soft sets
(Agarwal et al., 1013), (Garg and Arora, 2018) and
many others. Although all of these fuzzy structures
are different from each other, they still use many of
the analogous methods and tools, typical for classical
fuzzy sets.
The importance of this topic lies mainly in the
fact that nowadays the use of these new fuzzy struc-
tures for solving specific applications is expanding
very quickly, on the one hand, but on the other hand,
the theoretical part of these methods is often not fully
solved using the tools of these new fuzzy sets. This
often leads to the creation of own theories (some-
times not fully correct), mainly motivated by classic
a
https://orcid.org/0000-0002-5464-9521
fuzzy sets, within individual methods using new fuzzy
sets. And this regardless of the fact that a large part
of these new fuzzy sets can be isomorphically trans-
formed into one type of parametric fuzzy sets, where
parameters are represented as special examples of op-
erations in some semirings. The transformed struc-
tures then correspond to standard MV -valued fuzzy
sets.
This transformation then makes it possible to use
the entire range of theoretical methods of classic MV -
valued fuzzy sets and, using the inverse aforemen-
tioned isomorphic transformation, to convert them to
the appropriate methods in new types of fuzzy sets.
And without any need to prove the properties of these
transformed operations for individual types of new
fuzzy sets.
This possible approach to the unification of new
types of fuzzy sets was published in (Mo
ˇ
cko
ˇ
r, 2021),
where some of these new MV -valued fuzzy sets were
transformed into the so-called (R, R
∗
)-fuzzy sets,
that is, the mappings X → R, where R is the (com-
mon) underlying set of complete commutative idem-
potent semirings R and R
∗
with involutive isomor-
phisms ¬ : R → R
∗
. The advantage of this construc-
tion is, among other things, that the pair (R, R
∗
) ex-
plicitly defines the pair of dual constructions, com-
monly used in the theory of fuzzy sets, such as, for
example, upper and lower approximations, upper and
lower F-transforms or closure and interior operators.
In this paper, we focus on building the theory of
Mo
ˇ
cko
ˇ
r, J.
Approximations of New MV-Valued Types of Fuzzy Sets.
DOI: 10.5220/0012157300003595
In Proceedings of the 15th International Joint Conference on Computational Intelligence (IJCCI 2023), pages 327-337
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)
327

approximations of (R, R
∗
)-fuzzy sets and on illus-
trative examples of how different types of approxima-
tions for new types of MV -valued fuzzy sets can be
defined using this theory. For this purpose, we also
use the elementary theory of monads in categories, as
introduced in (Manes, 1976).
2 PRELIMINARIES
In this preliminary section, we introduce the basic
definitions and properties of dual pairs of semirings
(R, R
∗
) and (R, R
∗
)-fuzzy sets that were presented
(in a modified form) in (Mo
ˇ
cko
ˇ
r, 2021). The main
motivation for the introduction of dual pairs of semir-
ings (R, R
∗
) and (R, R
∗
)-fuzzy sets is that this type
of fuzzy sets can be used to transform many of the
new MV -valued fuzzy sets in a set X into mappings
X → R. And even in cases where the given new type
fuzzy set is not a mapping X → L, although tradition-
ally it is called L-fuzzy set. A typical example of this
obstacle in the application of classic L-fuzzy sets tools
to new types of fuzzy sets is represented by L-fuzzy
soft sets. A L-fuzzy soft sets are defined in a space
(K, X) where K is the (fixed) set of criteria and X is
a set. L-fuzzy soft set is then defined as a pair (E, f ),
where E ⊆ K and f : E → L
X
. This form of fuzzy
set requires completely new definitions of operations
with these new fuzzy sets.
In this introductory section, we present the defi-
nition of (R, R
∗
)-fuzzy sets, and we also show how
some new types of fuzzy sets can be isomorphically
transformed into (R, R
∗
)-fuzzy sets, with the basic
operations defined formally in the same way as for
L-fuzzy sets. Construction of (R, R
∗
)-fuzzy sets is
based on the notion of a complete commutative idem-
potent semiring.
Definition 1. ((Berstel and Perrin, 1985)) A com-
plete commutative idempotent semiring (or a semir-
ing, shortly) R = (R, +, ×, 0, 1) is an algebraic struc-
ture with the following properties:
1. (R, +, 0) is a complete idempotent commutative
monoid,
2. (R, ×, 1) is a commutative monoid,
3. x ×
∑
R
y∈I
y =
∑
R
y∈I
(x × y) holds for all x ∈ R, I ⊆ R,
where
∑
R
=
∑
is the complete operation + in R,
4. 0 × x = 0 holds for all x ∈ R.
The notion of a semiring homomorphism is de-
fined as a standard homomorphism between algebraic
structures. We introduce the dual pair of semirings
(R, R
∗
) that was originally introduced in modified
form in (Mo
ˇ
cko
ˇ
r, 2021). The following definition is a
simplification of the original definition.
Definition 2. Let R = (R, +, ×, 0, 1) and R
∗
=
(R, +
∗
, ×
∗
, 0
∗
, 1
∗
) be complete idempotent commu-
tative semirings with the same underlying set R.
The pair (R, R
∗
) is called the dual pair of semirings
if there exists a semiring isomorphism ¬ : R → R
∗
and the following axioms hold:
1. ¬ : R → R
∗
is the involutive isomorphism;
2. ∀a ∈ R, S
0
⊆ R a ×
∗
(
∑
b∈S
0
b) =
∑
b∈S
0
(a ×
∗
b);
3. ∀a ∈ R, S
0
⊆ R a + (
∑
∗
b∈S
0
b) =
∑
∗
b∈S
0
(a + b),
where
∑
∗
is the complete operation +
∗
in R
∗
;
4. ∀a, b ∈ R, a + b = a ⇔ a +
∗
b = b.
Using the isomorphism ¬, it is easy to see that the
following dual statements also hold for arbitrary dual
pairs of semirings (R, R
∗
):
2’. ∀a, ∈ R, S ⊆ R, a ×
∑
∗
b∈S
b =
∑
∗
b∈S
(a × b),
3’. ∀a, ∈ R, S ⊆ R, a +
∗
∑
b∈S
b =
∑
b∈S
(a +
∗
b).
The basic properties of the dual pair of semirings
(R, R
∗
), including the orderings defined in the un-
derlying set R, are described in the following lemma.
Lemma 1. Let (R, R
∗
) be the dual pair of semirings.
1. Let the relations ≤ and ≤
∗
be defined by
x, y ∈ R, x ≤ y ⇔ x+y = y, x ≤
∗
y ⇔ x+
∗
y = y.
The following statements hold:
(a) ≤ and ≤
∗
are the order relations on R,
(b) x ≤
∗
y ⇔ x ≥ y ⇔ ¬x ≤ ¬y,
(c) x ≤ y ⇒ x + z ≤ y + z, x +
∗
z ≤ y +
∗
z,
(d) x ≤ y ⇒ x × z ≤ y × z, x ×
∗
z ≤ y ×
∗
z.
2. (R, +, ×, ≤) and (R, +
∗
, ×
∗
, ≤
∗
) are lattice-
ordered semirings, where, for arbitrary S ⊆ R,
sup S =
∑
x∈S
x, inf S =
∗
∑
x∈S
x, in (R, ≤),
sup S =
∗
∑
x∈S
x, inf S =
∑
x∈S
x, in (R, ≤
∗
),
where
∑
∗
x∈S
x is the sum of elements with respect
to +
∗
.
Proof is only a simple application of Definition 2
and will be omitted. 
The dual pairs of semirings (R, R
∗
) with a com-
mon underlying set R represent a special value set
structure for the so-called (R, R
∗
)-fuzzy sets, which
can be effectively used to transform some of the new
fuzzy structures into the shape of classic fuzzy sets in
the set X, i.e. in the mappings X → R.
The following definition was introduced in
(Mo
ˇ
cko
ˇ
r, 2021).
Definition 3. (Mo
ˇ
cko
ˇ
r, 2021) Let (R, R
∗
) be a dual
pair of semirings and X be a set.
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
328

1. A mapping s : X → R is called a (R, R
∗
)-fuzzy set
in a set X.
2. Operations with (R, R
∗
)-fuzzy sets are defined by
(a) The intersection s u t is defined by (s ut)(x) =
s(x) +
∗
t(x), x ∈ X,
(b) The union s tt is defined by (s t t)(x) = s(x) +
t(x), x ∈ X,
(c) Complement ¬s is defined by (¬s)(x) =
¬(s(x)),
(d) The external multiplication  of (R, R
∗
)-fuzzy
set s by an element a ∈ R is defined by (a 
s)(x) = a × s(x), x ∈ X,
(e) The order relation ≤ between s, t is defined by
s ≤ t ⇔ (∀x ∈ X)s(x) ≤ t(x) where ≤ is the or-
der relation defined in Lemma 1.
The set of all (R, R
∗
)-fuzzy sets in a set X will be
denoted by (R, R
∗
)
X
. In the following proposition,
some basic properties of operations with (R, R
∗
)-
fuzzy sets are summarized. It follows that the al-
gebraic structure ((R, R
∗
)
X
, u, t, ¬, ) has analogi-
cal properties as the algebraic structure of classical
L-fuzzy sets.
Proposition 1. (Mo
ˇ
cko
ˇ
r, 2021) Let (R, R
∗
) be a
dual pair of semirings. Let X be a set, and let s, t, w
be (R, R
∗
)-fuzzy sets. Then the following statements
are valid.
1. s u s = s, s t s = s,
2. s u t ≤ s, s ≤ s tt,
3. s u t ≤ s t t,
4. s u (t t w) = (s u t) t (s uw),
5. s t (t u w) = (s t t) u (s tw),
6. a  (t t w) = (a  t) t (a  w),
7. a  (t u w) = (a  t) u (a  w),
8. ¬(s t t) = ¬s u ¬t, ¬(s ut) = ¬s t ¬t,
9. s ≤ t ⇒ s tw ≤ t t w, s u w ≤ t u w.
As we mentioned in the Introduction, in the paper
we will use some basic constructions from the theory
of monads, which will be used in the next parts of
the paper. Elements of category theory can be found
in many publications, such as (Herrlich and Strecker,
2007). By Set we denote the category of sets with
mappings as morphisms. For more information on
monads, see, for example, (Manes, 1976; Herrlich
and Strecker, 2007).
Definition 4. (Manes, 1976) The structure T =
(T, ♦, η) is a monad in the category Set, if
1. T : Set → Set is a mapping of objects,
2. η is a system of mappings {η
X
: X → T (X)|X ∈
Set},
3. For each X,Y, Z ∈ Set and any pair of mappings
f : X → T (Y ), g : Y → T (Z), there exists a com-
position (called Kleisli composition) g♦ f : X →
T (Z), which is associative,
4. For every mapping f : X → T (Y ), η
Y
♦ f = f and
f ♦η
X
= f hold,
5. For arbitrary mappings f : X → Y , g : Y → T (Z),
we have g♦(η
Y
. f ) = g. f : X → T (Z), where . is
the composition of morphisms in the category Set.
Remark 1. Morphisms X → T (Y ) will be denoted by
X Y and will be called T-relations (or Kleisli mor-
phisms) from X to Y . The composition of T-relations
f : X Y and g : Y Z is defined by g♦ f : X Z.
The following proposition presents an example of
a monad that is defined by a complete commutative
idempotent semiring. This monad will be the key the-
oretical tool for the approximation of (R, R
∗
)-fuzzy
sets.
Proposition 2. Let R = (R, +, ×, 0, 1) be a com-
plete commutative idempotent semiring. The struc-
ture T
R
= (T, ♦, η) is defined by
1. The mapping T : Set → Set of objects is defined
by T (X) = R
X
,
2. For T
R
-relations f : X Y and g : Y Z their
composition g♦ f : X Z is defined by
(g♦ f )(x)(z) =
R
∑
y∈Y
f (x)(y) × g(y)(z).
3. η = {η
X
: X ∈ Set}, where η
X
is the T
R
-relation
X X defined by
η
X
(x)(y) =
(
1
R
, x = y,
0
R
, x 6= y,
.
Then T
R
is the monad in the category Set.
Remark 2. 1. If (R, R
∗
) is a dual pair of semir-
ings, according to Proposition 2, there also exists
the monad T
R
∗
= (T, ♦
∗
, η
∗
), where for f : X Y
and g : Y Z,
g♦
∗
f (x)(z) =
R
∗
∑
y∈Y
f (x)(y) ×
∗
g(y)(z),
η
∗
X
(x)(y) =
(
1
∗
= 0, x = y,
0
∗
= 1, x 6= y.
2. T
R
-relations will be simply called R-relations
and T
R
∗
-relations will be called R
∗
-relations.
Both relations represent the same morphisms X →
R
Y
, the difference is only related to their composi-
tions ♦ and ♦
∗
. Both relations will sometimes be
called (R, R
∗
)-relations.
Approximations of New MV-Valued Types of Fuzzy Sets
329

3. Instead of category Set we can consider the
Kleisli category Set
R
of the monad T
R
, where ob-
jects are the same as in Set and morphisms are
T
R
-relations X Y , with ♦ as the composition
of morphisms.
3 EXAMPLES OF
TRANSFORMATION OF NEW
TYPES OF MV -VALUED FUZZY
SETS INTO (R, R
∗
)-FUZZY
SETS
In this section we present examples of transforma-
tions of MV -valued fuzzy soft sets, neutrosophic MV -
valued fuzzy sets and Ω-level MV -valued fuzzy sets
into (R, R
∗
)-fuzzy sets which extend some results
presented in (Mo
ˇ
cko
ˇ
r, 2021). We also show that alge-
bras of MV -valued fuzzy soft sets, neutrosophic fuzzy
sets or Ω-level MV -valued fuzzy sets are isomorphic
to the corresponding algebras of (R, R
∗
)-fuzzy sets.
These isomorphisms enable us to define basic oper-
ations with these new MV-valued fuzzy sets in an
analogous way to classical L-fuzzy sets. In this sec-
tion, L = (L, ∨, ∧, ⊕, ⊗, ¬, 0
L
, 1
L
) is a complete MV -
algebra.
3.1 MV -valued Fuzzy Soft Sets
Let K be a fixed set of criteria. L-fuzzy soft set in a set
X is a pair (E, s), where E ⊆ K and s : K → L
X
such
that s(k) = 0
L
for k ∈ K \ E. Therefore, (E, s) can be
interpreted as a mapping (E, s) : K → L
X
such that
(E, s)(k)(x) =
(
s(k), k ∈ E,
0
L
, k 6∈ E.
By FSS(X) we denote the set of all L-fuzzy soft sets in
a set X (Maji and et al, 2001). For the operations with
fuzzy soft sets, see (Ayg
¨
uno
˘
glu and Ayg
¨
un, 2009).
We define the dual pair of semirings (R, R
∗
) that
transform L-fuzzy soft sets into (R, R
∗
)-fuzzy sets.
We use the following notation. For arbitrarily s ∈ L
K
and E ⊆ K, the object s
E
is defined by
s
E
: K → L, ∀e ∈ K, s
E
(e) =
(
s(e), e ∈ E,
0
L
, e ∈ K \ E.
For an element α ∈ L, by α we denote the constant
function K → L with the only value α. It should be
mentioned that because s
E
are functions K → L, s
E
=
t
F
iff s
E
(e) = t
F
(e) for arbitrary e ∈ K. We set
R = {s
E
: s ∈ L
K
, E ⊆ K},
and in the set R we define two structures:
1. Let R = (R, +, ×, 0, 1) be defined by
(a) s
E
+t
F
:= (s
E
∨t
F
)
E∪F
∈ R,
(b) s
E
× t
F
= (s
E
⊗ t
F
)
E∪F
. It should be observed
that in that case (s
E
⊗ t
F
)
E∪F
= (s
E
⊗ t
F
)
E∩F
holds,
(c) 0 = 0
L
K
, 1 = 1
L
K
,
2. Let R
∗
= (R, +
∗
, ×
∗
, 0
∗
, 1
∗
) be defined by
(a) s
E
+
∗
t
F
= (s
E
∧t
F
)
E∪F
. It should be observed
that in that case (s
E
∧ t
F
)
E∪F
= (s
E
∧ t
F
)
E∩F
holds,
(b) s
E
×
∗
t
F
= (s
E
⊕t
F
)
E∪F
∈ R,
(c) 0
∗
= 1, 1
∗
= 0.
3. The mapping ¬
R
: R → R
∗
is defined by
s
E
∈ R, ¬
R
(s
E
) := (¬(s
E
))
K
∈ R,
where ¬ is the negation in L.
It is easy to see that the operations +, +
∗
can be
extended to the complete operations
∑
,
∑
∗
by
(
∑
s
E
∈S
s
E
) = (
_
s
E
∈S
s
E
)
S
s
E
∈S
E
,
(
∗
∑
s
E
∈S
s
E
) = (
^
s
E
∈S
s
E
)
S
s
E
∈S
E
= (
^
s
E
∈S
s
E
)
T
s
E
∈S
E
,
where S ⊆ R.
Lemma 2. (R, R
∗
) is the dual pair of semirings with
involutive isomorphism ¬
R
.
Proof is only a technical verification of properties
of (R, R
∗
) from Definition 2 and due to the limited
scope of the paper it will be omitted. 
Proposition 3 (Transformation of L-fuzzy soft sets
onto (R, R
∗
)-fuzzy sets). Let L be a complete MV -
algebra. In the set FSS(X ) of all L-fuzzy soft sets, the
union ∪, intersection ∩ and the complement ¬ can be
defined so that there exist homomorphisms
Λ : (FSS(X), ∪, ∩, ¬) → ((R, R
∗
)
X
, t, u, ¬),
Λ
−1
: ((R, R
∗
)
X
, t, u, ¬) → (FSS(X), ∪, ∩, ¬),
such that Λ.Λ
−1
.Λ = Λ.
Sketch of the proof. We define the mapping Λ :
FSS(X) → (R, R
∗
)
X
by
(E, s) ∈ FSS(X ), Λ(E, s) : X → R,
x ∈ X , Λ(E, s)(x) = s
x
E
∈ R, where
s
x
∈ L
K
, s
x
(k) := s(k)(x), ∀k ∈ K. (1)
To define the mapping Λ
−1
: (R, R
∗
)
X
→ FSS(X), let
f ∈ (R, R
∗
)
X
. According to the definition of R, for
x ∈ X we have f (x) = f
x
E
x
∈ R, where f
x
∈ L
K
, E
x
⊆ K.
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
330

The mapping Λ
−1
: (R, R
∗
)
X
→ FSS(X) is defined
by
Λ
−1
( f ) = (E, s) ∈ FSS(X),
E =
\
x∈X
E
x
, s : K → L
X
,
k ∈ K, x ∈ X, s(k)(x) =
(
f
x
(k), k ∈ E ⊆ E
x
,
0
L
, k 6∈ E.
(2)
It is easy to see that Λ
−1
.Λ = id
FSS(X)
and it follows
that Λ : FSS(X) → R
X
in the injective map. The map-
ping Λ represents the transformation of MV -valued
fuzzy soft sets onto (R, R
∗
)-fuzzy sets and, con-
versely, Λ
−1
is the inverse transformation of (R, R
∗
)-
fuzzy sets onto MV -valued fuzzy soft sets. Using
these transformation and inverse transformation we
can define basic operations with MV -valued fuzzy
soft sets, such as union, intersection or complement in
such a way that the resulting algebras of MV -valued
fuzzy soft sets and (R, R
∗
)-fuzzy sets are isomor-
phic.
For example, we present the constructions of the
operations ∪ and ∩ in FSS(X). Let (E, s), (F, t) ∈
FSS(X) and let f = Λ(E, s), g = Λ(F,t) ∈ R
X
. There-
fore, according to (1), for x ∈ X ,
f (x) = Λ(E, s)(x) = s
x
E
, g(x) = Λ(F,t)(x) = t
x
F
.
Using the operation t in (R, R
∗
)-fuzzy sets from
Definition 3, we can define the operation ∪ by
(E, s) ∪(F, t) := Λ
−1
(Λ(E, s) tΛ(F, t)) =
Λ
−1
( f t g), (3)
where according to Definition 3, for x ∈ X we have
( f t g)(x) = f (x)+ g(x) = s
x
E
+t
x
F
=
(s
x
E
∨t
x
F
)
E∪F
: K → L.
Using the definition (2) of the inverse transformation
Λ
−1
, we obtain
Λ
−1
( f t g) = (E ∪ F, q), q : K → L
X
is such that
q(k)(x) = (s
x
E
∨t
x
F
)
E∪F
(k) =









s(k) ∨ t(k). k ∈ E ∩ F,
s(k), k ∈ E \ F,
t(k), k ∈ F \ E,
0
L
, k 6∈ E ∪ F.
The operation ∩ can be calculated in a similar way,
that is,
(E, s) ∩(F, t) = Λ
−1
(Λ(E, s) uΛ(F, t)) =
Λ
−1
( f u g) = (G, h),
x ∈ X , ( f u g)(x) = f (x)+
∗
g(x) = s
x
E
+
∗
t
x
F
=
(s
x
E
∧t
x
F
)
E∪F
: K → L.
Therefore, we obtain
(E, s) ∩(F, t) = (E ∪F, h), h : K → L
X
is such that
h(k)(x) = (s
x
E
∧t
x
F
)
E∪F
(k) =
(
s(s) ∧t(k), k ∈ E ∩ F
0
L
, s 6∈ E ∩ F.
Moreover, Λ is the homomorphism of algebras. In
fact, it should be observed that for an arbitrary f ∈ R
X
such that it satisfies the following condition (*)
(∀x ∈ X)( f (x) = s
x
E
x
=⇒ E
x
= E), (4)
the identity Λ.Λ
−1
( f ) = f also holds. It is easy to
see that for arbitrary (E, s) ∈ FSS(X) this condition
(4) is satisfied for f = Λ(E, s) ∈ R
X
and it follows
that Λ.Λ
−1
.Λ(E, s) = Λ(E, s). Then, for example, for
(E, s), (F,t) ∈ FSS(X), using identity (3), we have
Λ((E, s) ∪(F, t)) = Λ(Λ
−1
(Λ(E, s) tΛ(F, t))) =
ΛΛ
−1
(Λ(E, s) tΛ(F, t)) = Λ(E, s) t Λ(F, t).
Analogously, it can be proven that Λ is a homomor-
phism for other operations ∩, ¬. 
It should be mentioned that the basic operations on
the set FSS(X) defined in Proposition 3 correspond
to the operations defined in (Ayg
¨
uno
˘
glu and Ayg
¨
un,
2009).
3.2 MV -valued Neutrosophic Fuzzy Sets
Recall (James and Mathew, 2021) that a neutrosophic
L-fuzzy set is mapping X → L
3
. By NFS(X) we de-
note the set of all neutrosophic L-fuzzy sets in a set
X. Operations with neutrosophic L-fuzzy sets f , g,
where f (x) = (α
x,1
, α
x,2
, α
x,3
), g(x) = (β
x,1
, β
x,2
, β
x,3
)
for x ∈ X , are defined by
f ∪ g(x) = (α
x,1
∨ β
x,1
, α
x,2
∨ β
x,2
, α
x,3
∧ β
x,3
),
(¬ f )(x) = (α
x,3
, ¬α
x,2
, α
x,1
),
f ∩ g(x) = (α
x,1
∧ β
x,1
, α
x,2
∧ β
x,2
, α
x,3
∨ β
x,3
).
We define the dual pair of semirings (R, R
∗
) that
transforms neutrosophic L-fuzzy sets into (R, R
∗
)-
fuzzy sets. Let R = L
3
.
1. The semiring R = (R, +, ×, 0, 1) is defined by
(a) (α, β , γ) + (α
1
, β
1
, γ
1
) := (α ∨ α
1
, β ∨ β
1
, γ ∧
γ
1
),
(b) (α, β , γ) ×
1
(α
1
, β
1
, γ
1
) := (α ⊗ α
1
, β ⊗ β
1
, γ ⊕
γ
1
),
(c) 0 = (0
L
, 0
L
, 1
L
), 1 = (1
L
, 1
L
, 0
L
),
2. The semiring R
∗
= (R, +
∗
, ×
∗
, 0
∗
, 1
∗
) is de-
fined by
(a) (α, β , γ) +
∗
(α
1
, β
1
, γ
1
) := (α ∧ α
1
, β ∧ β
1
, γ ∨
γ
1
),
Approximations of New MV-Valued Types of Fuzzy Sets
331

(b) (α, β , γ) ×
∗
(α
1
, β
1
, γ
1
) := (α ⊕ α
1
, β ⊕ β
1
, γ ⊗
γ
1
),
(c) 0
∗
= (1
L
, 1
L
, 0
L
), 1
∗
= (0
L
, 0
L
, 1
L
).
Let ¬
R
: R → R
∗
be defined by
(α, β , γ) ∈ R, ¬
R
(α, β , γ) = (γ, ¬β , α).
Proposition 4 (Transformation of neutrosophic
L-fuzzy sets into (R, R
∗
)-fuzzy sets). (R, R
∗
) is the
dual pair of semirings. For arbitrary set X there exists
an identity isomorphism Λ = 1
NSF
Λ : (NFS(X), ∪, ∩, ¬) → ((R, R
∗
)
X
, t, u, ¬)
between the algebra of (R, R
∗
)-fuzzy sets and the al-
gebra of neutrosophic L-fuzzy sets in a set X.
Proof. The proof is straightforward and will be
omitted. 
3.3 Ω-Level L-Fuzzy Sets
Let L be the complete MV -algebra. If we want to
specify how an element x ∈ X corresponds to the
fuzzy set s ∈ L
X
, sometimes this value depends on
the ”observation points” α ∈ Ω, the points x ∈ X are
observed from. Therefore, instead of a L-fuzzy set s :
X → L we should consider the function s : X ×Ω → L.
For the correct determination of the value s(x, α), it
should be assumed that if the positions of two obser-
vation points are similar, the observed values should
also be close. A similar approach was first discussed
by A.
ˇ
Sostak in (
ˇ
Sostak A. et al., 2019), where he
introduced the concept of many-level L-fuzzy rela-
tions. We show that these fuzzy structures can be
transformed into (R, R
∗
)-fuzzy sets.
Let ρ be a L-fuzzy equivalence relation in Ω. Re-
call that a L-fuzzy set f ∈ L
Ω
is called ρ-extensional,
if
∀α, β ∈ Ω, f (α) ⊗ ρ(α, β ) ≤ f (β ).
For arbitrary g ∈ L
Ω
, the ρ-extensional hull g of g is
defined by g(α) =
W
β ∈Ω
g(β )⊗ρ(β, α). The Ω-level
L-fuzzy set is defined in the following definition.
Definition 5. Let X be a set and let (Ω, ρ) be a set
with the L-fuzzy equivalence relation ρ. The Ω-level
L-fuzzy set s in a set X is a mapping s : X × Ω → L,
such that for arbitrary x ∈ X, the mapping s(x, −) ∈
L
Ω
is ρ-extensional.
The set of all Ω-level L-fuzzy sets in a set X is
denoted by Ω(X, ρ).
We show that if L is the MV -algebra, Ω-level
L-fuzzy sets can also be transformed into (R, R
∗
)-
fuzzy set. In fact, let
R = { f ∈ L
Ω
: f is ρ-extensional} ⊆ L
Ω
.
1. Let R = (R, +, ×, 0, 1) be defined by
(a) f + g = f ∨ g, where ∨ is the supremum in L
Ω
,
(b) f ×g = f ⊗ g, where f ⊗ g is defined pointwise
in L
Ω
,
(c) 0(α) = 0
L
, 1(α) = 1
L
, for arbitrary α ∈ Ω,
2. Let R
∗
= (R, +
∗
, ×
∗
, 0
∗
, 1
∗
) be defined by
(a) f +
∗
g = f ∧ g,
(b) f ×
∗
g = ¬(¬( f ⊕ g)), where f ⊕ g is defined
point-wise in L
Ω
,
(c) 0
∗
= 1, 1
∗
= 0,
Let ¬ : R → R
∗
be defined by
f ∈ R, (¬ f )(α) = ¬( f (α)).
Proposition 5 (Transformation of Ω-level L-fuzzy
sets into (R, R
∗
)-fuzzy sets). (R, R
∗
) is the dual
pair of semirings. In the set Ω(X , ρ) the operations
∪, ∩, ¬ can be defined such that there exists the iso-
morphism of algebras
Λ : (Ω(X, ρ), ∪, ∩, ¬) → ((R, R
∗
)
X
, t, u, ¬).
Sketch of the proof. We use the fact that in the
MV -algebra the negation of ρ-extensional mappings
is also ρ-extensional. Using the properties of opera-
tions in MV -algebra it is easy to prove that the defini-
tion of (R, R
∗
) is correct.
We define two mappings
Λ : Ω(X, ρ) → (R, R
∗
)
X
,
Λ
−1
: (R, R
∗
)
X
→ Ω(X, ρ),
where for x ∈ X , α ∈ Ω,
s ∈ Ω(X, ρ), Λ(s)(x) := s(x, −) : Ω → L,
f ∈ R
X
, Λ
−1
( f )(x, α) := f (x)(α).
Λ and Λ
−1
are mutually inverse mappings, as follows
from the following:
Λ.Λ
−1
( f )(x) = Λ(Λ
−1
( f ))(x) =
Λ
−1
( f )(x, −) = f (x),
Λ
−1
.Λ(s)(x, α) = Λ
−1
(Λ(s))(x, α) =
Λ(s)(x)(α) = s(x, α).
The mapping Λ represents the transformation of Ω-
level L-fuzzy sets into (R, R
∗
)-fuzzy sets and Λ
−1
is
the inverse transformation. Using these transforma-
tions, we can define basic operation ∩,∪ and .
For example, if x ∈ X, α ∈ Ω, s, t ∈ Ω(X, δ ), the
operation ∪ on Ω(X, δ ) can be defined by
(s ∪t)(x, α) := Λ
−1
(Λ(s) t Λ(t))(x, α) =
(Λ(s) t Λ(t))(x, −)(α) =
(Λ(s)(x) + Λ(t)(x))(α) = s(x, α) ∨ t(x, α).
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
332

Furthermore, if r ∈ R, s ∈ R
X
, r  s also represents a
Ω-level L-fuzzy set t ∈ Ω(X, ρ) defined by
t := Λ
−1
(r  s)(x, −)(α) = (r  s)(x)(α) =
(r ×s(x))(α) = (r ⊗s(x))(α) =
_
β ∈Ω
r(β ) ⊗ s(x, β ) ⊗ ρ(β , α).
If we want to calculate the external multiplication λ .s,
where λ ∈ L, s ∈ Ω(X, ρ), we can take the constant
function λ ∈ R with the only value λ. In that case,
λ .s(x, α) =
Λ
−1
(λ  Λ(s))(x, α) = Λ
−1
(λ ⊗ s(x, −))(α) =
_
β ∈Ω
λ (β ) ⊗ s(x, β) ⊗ ρ(β , α) =
λ ⊗
_
β ∈Ω
s(x, β ) ⊗ ρ(β , α) = λ ⊗ s(x, α).
It is easy to see that Λ
−1
and Λ are isomorphisms of
the corresponding structures. 
4 EXAMPLES OF
TRANSFORMATION OF NEW
TYPES OF MV -VALUED FUZZY
RELATIONS INTO
(R, R
∗
)-FUZZY RELATIONS
Fuzzy type relations are basic structures for approx-
imation of fuzzy sets, including new types of fuzzy
sets. In this section we show two examples of trans-
formation of fuzzy relation in new MV -valued types
of fuzzy sets into (R, R
∗
)-relations. In this sec-
tion L = (L, ∨, ∧, ⊕, ⊗, ¬, 0
L
, 1
L
) is a complete MV -
algebra.
The notion of the L-fuzzy soft relation was intro-
duced, for example, in (Sut, 2012). We repeat this
definition.
Definition 6. (Sut, 2012) Let X ,Y, Z be sets.
1. A L-fuzzy soft relation from X to Y (denoted X (
Y ) is the L-fuzzy soft set (E, r) in the Cartesian
product X ×Y .
2. If (E, r) and (F, s), respectively, are L-fuzzy soft
relations in X × Y and Y × Z, respectively, their
composition (F, s)◦(E, r) = (E ∩F, sr) : X ( Z,
where s  r : E ∩ F → L
X×Z
is defined by
(s  r)(k)(x, z) =
_
y∈Y
r(k)(x, y) ⊗ s(k)(y, z).
for k ∈ E ∩ F, x ∈ X, y ∈ Z.
3. For a set X, 1
X
: X ( X is defined by 1
X
= (K, ∆),
where ∆ : K → L
X×X
is defined by ∆(k)(x, y) =
(
1
L
, x = y,
0
L
, x 6= y.
The following proposition holds.
Proposition 6 (Transformation of L-fuzzy soft re-
lations into (R, R
∗
)-fuzzy selations). Let (R, R
∗
)
be the dual pair of semirings defined in Section 3.1.
There exists the embedding homomorphism
H : FSS → Set
R
of the category FSS of sets with L-fuzzy soft relations
as morphisms into the Kleisli category Set
R
defined
in Remark 2.
Sketch of the proof. Let (E, r) : X ( Y be a mor-
phism in the category FSS. The embedding functor
H : FSS → Set
R
is defined by
H(X) = X, H(E, r) : X Y,
(∀x ∈ X, y ∈ Y )H(E, r)(x)(y) = r
xy
E
∈ R,
r
xy
: K → L, r
xy
(k) = r(k)(x, y), k ∈ K.
For x, y ∈ X, we have
H(1
X
)(x)(y) = H(K, ∆)(x)(y) = ∆
xy
K
= η
X
(x)(y) = 1
H(X)
(x)(y).
For illustration, we show that H respects the composi-
tion of the morphisms. In fact, let (E, r) : X ( Y and
(F, s) : Y ( Z be morphisms in FSS. For x ∈ X, z ∈ Z
we obtain the following:
H((F, s) ◦ (E, r))(x)(z) = H(F ∩E, s  r)(x)(z) =
(s  r)
xz
E∩F
: K → L,
k ∈ E ∩ F, (s  r)
xz
E∩F
(k) =
_
y∈Y
r
xy
E
(k) ⊗ s
yz
F
(k) =
(
_
y∈Y
(r
xy
E
⊗ s
yz
F
)
E∩F
)
E∩F
= (
R
∑
y∈Y
r
xy
E
× s
yz
F
)(k) =
R
∑
y∈Y
H(E, r)(x)(y) × H(F, s)(y)(z) =
(H(F, s)♦H(E, r))(x)(z).
Therefore, H is the functor and it is easy to see that H
is the embedding.

Neutrosophic fuzzy relations were introduced, for
example, in (Salama et al., 2014).
Definition 7. 1. Neutrosophic L-fuzzy relation from
X to Y (also denoted by X ( Y ) are neutrosophic
L-fuzzy sets in X ×Y .
Approximations of New MV-Valued Types of Fuzzy Sets
333

2. For neutrosophic L-fuzzy relations r : X ( Y and
s : Y ( Z, their composition s ◦ r : X ( Z is de-
fined by
r(x, y) = (α
1
x,y
, α
2
x,y
, α
3
x,y
),
s(y, z) = (β
1
y,z
, β
2
y,z
, β
3
y,z
),
(s ◦ r)(x, z) =
_
y∈Y
α
1
x,y
⊗ β
1
y,z
,
_
y∈Y
α
2
x,y
⊗ β
2
y,z
,
^
y∈Y
α
3
x,y
⊕ β
3
y,z
!
.
3. For a set X, the unit morphisms 1
X
: X ( X are
defined by
1
X
(x, y) =
(
(1
L
, 1
L
, 0
L
), x = y,
(0
L
, 0
L
, 1
L
), x 6= y.
Proposition 7 (Transformation of neutrosophic
L-fuzzy relations into (R, R
∗
)-fuzzy relations). Let
(R, R
∗
) be the dual pair of semirings defined in Sec-
tion 3.2. There exists the identity isomorphic functor
H = 1
NSF
,
H : NFS → Set
R
between the category NFS of sets with neutrosophic
L-fuzzy relations as morphisms and the Kleisli cate-
gory Set
R
defined in Remark 2.
The proof of this proposition is straightforward
and will be omitted.

An analogous situation concerns Ω-level L-fuzzy
sets. Ω-level L-fuzzy relations are defined in the fol-
lowing definition.
Definition 8. 1. Ω-level L-fuzzy relations from X to
Y (denoted by X ( Y ) are Ω-level L-fuzzy sets in
a set X ×Y .
2. For Ω-level L-fuzzy relations r : X ( Y and s :
Y ( Z, their composition s ◦r : X ( Z is defined
by
(s ◦ r)(x, y, α) =
_
b∈Ω
_
y∈Y
r(x, y, β) ⊗ s(y, z, β ) ⊗ρ(β , α).
3. For a set X, the unit morphism 1
X
: X 7→ X is de-
fined by 1
X
(x, y, α) = η
X
(x)(y).
The following proposition holds, where L is again
the complete MV -algebra.
Proposition 8 (Transformation of Ω-level L-fuzzy re-
lations into (R, R
∗
)-fuzzy relations). Let (R, R
∗
)
be the dual pair of semirings defined in Section 3.3.
There exists the isomorphic functor
H : Ω(ρ) → Set
R
between the category Ω(δ ) of sets with Ω-level L-
fuzzy relations as morphisms and the Kleisli category
Set
R
defined in Remark 2
The technical proof will be omitted.
5 EXAMPLES OF
APPROXIMATIONS OF NEW
TYPES OF MV -VALUED FUZZY
SETS
Basic operations with classic L-fuzzy sets certainly
include approximation of fuzzy sets by fuzzy rela-
tions. This operation makes it possible to transform
fuzzy sets into a simpler form, and thus facilitate the
work with fuzzy sets modified in this way. The ad-
vantage of this operation is also that in some cases a
reverse process can also be defined, which restores the
original fuzzy set from the simplified fuzzy set. Typi-
cal examples of approximations using fuzzy relations
are, for example, rough fuzzy sets or F-transform op-
erations. It is therefore natural to deal with the is-
sue of approximations for new types of fuzzy set. For
these purposes, we can again use the transformation
of new fuzzy structures into (R, R
∗
)-fuzzy sets and
the inverse transformation back to the original types
of fuzzy sets. This can be done in the following steps:
Step 1 Using the transformation morphisms Λ from
Section 3, we transform new types of MV -
valued fuzzy sets f into (R, R
∗
)-fuzzy sets,
that is, we need to calculate Λ( f ) ∈ (R, R
∗
)
X
,
Step 2 Using the transformation functors H from
Section 4, we transform new types of MV -
valued fuzzy relations Q into morphisms
H(Q) : X Y (that is, H(Q) : X → R
Y
) in
Kleisli category Set
R
(i.e., (R, R
∗
)-fuzzy re-
lations).
Step 3 Calculate the upper approximation
H(Q)
↑
(Λ( f )) of Λ( f ) by the (R, R
∗
)-
relation H(Q), where H(Q)
↑
(and similarly
H(Q)
↓
) is defined in a way similar to the
upper and lower approximations defined for
classical L-fuzzy sets (see Definition 9.
Step 4 Using the inverse transformations Λ
−1
from
Section 3, for arbitrary new MV-valued fuzzy
set f in a set X we obtain new MV -valued
fuzzy sets Q
↑
( f ) and Q
↓
( f ) in the set Y that
are approximations of f , that is,
Q
↑
( f ) := Λ
−1
(H(Q)
↑
(Λ( f ))),
Q
↓
( f ) := Λ
−1
(H(Q)
↓
(Λ( f ))).
Therefore, to apply the above-mentioned proce-
dure, we need to define the notion of the upper
and lower approximations of (R, R
∗
) fuzzy sets by
(R, R
∗
)-fuzzy relations. This is done in the follow-
ing definition, and it represents, in fact, an analogy of
these notions defined for classical L-fuzzy sets.
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
334

Because this construction uses the composition of
(R, R
∗
)-relations, we need to distinguish between
two types of these approximations, depending on the
type of Kleisli composition ♦ or ♦
∗
we use.
Definition 9. Let (R, R
∗
) be the dual pair of semir-
ings with the involutive isomorphism ¬ and let S :
X Y be a (R, R
∗
)-relation.
1. The upper approximation defined by the (R, R
∗
)-
relation S is the mapping S
↑
: R
X
→ R
Y
, such that
S
↑
= S♦1
R
X
.
2. The lower approximation defined by the (R, R
∗
)-
relation S is the mapping S
↓
: R
X
→ R
Y
, such that
S
↓
= (¬S)♦
∗
1
R
X
.
For illustration, the following basic properties of
these approximation operators can be simply proven.
Lemma 3. Let (R, R
∗
) be the dual pair of semirigs
with the involutive isomorphism ¬ and let S : X Y
be a (R, R
∗
)-relation. Let a ∈ R and s, t, s
i
∈ R
X
,
i ∈ I.
1. S
↑
(
F
i∈I
s
i
) =
F
i∈I
S
↑
(s
i
), S
↓
(
F
i∈I
s
i
) =
F
i∈I
S
↓
(s
i
),
2. S
↑
(a  s) = a  S
↑
(s), S
↓
(a 
∗
s) = a 
∗
S
↓
(s),
3. s ≤ t ⇒ S
↓
(s) ≤ S
↓
(t), S
↑
(s) ≤ S
↑
(t),
4. S
↓
(s) = ¬(S
↑
(¬s)), S
↑
(s) = ¬(S
↓
(¬s)).
5. Let R : X Y and S : Y Z be (R, R
∗
)-relations.
We have
S
↑
.R
↑
= (S♦R)
↑
, S
↓
.R
↓
= (S♦R)
↓
.
In the next three subsections we present examples
of Steps 1-4 for neutrosophic, fuzzy soft sets, and Ω-
level MV -valued fuzzy sets, respectively. These ex-
amples illustrate the possibilities of applying the clas-
sical theory of L-fuzzy sets to new types of fuzzy sets,
without having to derive this theory or prove its prop-
erties for individual types of new fuzzy sets.
5.1 Approximations of Neutrosophic
L-Fuzzy Sets
In Sections 3 and 4 we show that for L= complete
MV -algebra, neutrosophic L-fuzzy sets and neutro-
sophic L-fuzzy relation can be equivalently expressed
as (R, R
∗
)-fuzzy sets and (R, R
∗
)-fuzzy relations,
that is, according to Proposition 4, the algebra of neu-
trosophic L-fuzzy sets in a set X is isomorphic to
the algebra of (R, R
∗
)-fuzzy sets in a set X, where
(R, R
∗
) was introduced in Section 3. According to
Proposition 7, the neutrosophic L-fuzzy relation Q :
X ( Y can be equivalently defined as the (R, R
∗
)-
relation Q : X → R
Y
.
Let us consider a neutrosophic MV -fuzzy set f
and neutrosophic MV -fuzzy relation Q : X ( Y from
X to Y . According to the definition of neutrosophic
fuzzy sets, for x, y ∈ X we set f (x) = (α
x
, β
x
, γ
x
) and
Q(x)(y) = (α
xy
, β
xy
, γ
xy
). We show how Steps 1-4 are
realized.
Step 1, Step 2
These are the trivial steps because according to Propo-
sition 4 and Proposition 7, Λ is the identity mapping
and H is the identity functor. Therefore, Λ( f ) = f ,
H(Q) = Q.
Step 3
According to the definition of operations in (R, R
∗
)
in Section 3, the upper and lower approximations
Q
↑
(s), Q
↓
(s) of s in a set Y are defined by
∀y ∈ Y, Q
↑
(s)(y) =
R
∑
x∈X
s(x) × Q(x)(y) =
_
x∈X
α
x
⊗ α
xy
.
_
x∈X
β
x
⊗ β
xy
,
^
x∈X
γ
x
⊕ γ
xy
!
,
∀y ∈ Y, Q
↓
(s)(y) =
R
∗
∑
x∈X
s(x) ×
∗
¬Q(x)(y) =
^
x∈X
α
x
⊕ γ
xy
,
^
x∈X
β
x
⊕ ¬β
xy
,
_
x∈X
γ
x
⊗ α
xy
!
.
Step 4
This is also trivial step, because Λ
−1
is the identity
map. Hence, the results of this step are idetical to the
results from Step 3.
5.2 Approximations of L-Fuzzy Soft
Sets
We show how the Steps 1-4 can be realized for L-
fuzzy soft sets, where L is the complete MV -algebra.
In that case the situation is more complicated, because
Λ is not the identity map and H is not the identity
functor.
Let K be the set of criteria and let X be a set. We
use the notation form Section 3.1. and Section 4. Let
us consider a L-fuzzy soft set f = (E, s) and L-fuzzy
soft relation Q = (F, q) : X ( Y .
Step 1
According to Proposition 3, the transformation of f
into (R, R
∗
)-fuzzy set is defined by Λ( f ) = Λ(E, s) :
X → R, where for x ∈ X we have
Λ(E, s)(x) = s
x
E
∈ R,
k ∈ K, s
x
E
(k) =
(
s(k)(x), k ∈ E,
0
L
, k ∈ K \ E.
Approximations of New MV-Valued Types of Fuzzy Sets
335

Step 2
Recall that a L-fuzzy soft relation Q = (F, q) is a L-
fuzzy soft set in the set X ×Y . That is, q is a mapping
K → L
X×Y
, such that q(k)(x, y) = 0
L
, if k ∈ K \ E.
According to Proposition 6 it can be transformed into
the (R, R
∗
)-relation H(Q) = H(F, q) : X → R
Y
, such
that for x ∈ X , y ∈ Y ,
H(F, q)(x)(y) = q
xy
F
∈ R,
q
xy
(k) :=

q(k)(x, y) ∈ L, k ∈ F,
0
L
, k ∈ K \ F.
Step 3
We need to calculate the upper and lower approxi-
mations H(F, q)
↑
(Λ(E, s)) and H(F, q)
↓
(Λ(E, s)) of
Λ(E, s) by the (R, R
∗
)-relation H(F, q) according to
Definition 9. We have
H(F, q)
↑
(Λ(E, s))(x)(k) =
R
∑
z∈X
(Λ(E, s)(z) × H(F, q)(z)(x))(k) =
R
∑
z∈X
(s
z
E
× q
zx
F
)
E∩F
(k) =

W
z∈X
(s
z
E
(k) ⊗ q
zx
F
(k)), k ∈ E ∩F
0
L
, k 6∈ E ∩ F,
=

W
z∈X
(s(k)(z) ⊗ q(k)(z)(x)), k ∈ E ∩ F
0
L
, k 6∈ E ∩ F,
Analogically, we obtain the lower approximation:
H(F, q)
↓
(Λ(E, s))(x)(k) =
((¬
R
H(F, q))♦
∗
1
R
X
)(Λ(E, s))(x)(k) =
R
∗
∑
y∈X
(Λ(E, s))(z) ×
∗
¬
R
H(F, q)(z)(x))(k) =
R
∗
∑
z∈X
(s
z
E
×
∗
¬
R
q
zx
F
)
E∩F
(k) =
R
∗
∑
z∈X
(s
z
E
⊕ (¬(q
zx
F
))
E∪F
(k) =

V
z∈X
(s
z
E
(k) ⊕ (¬q
zx
F
(k)), k ∈ E ∪F,
0
L
, k 6∈ E ∪ F
=

V
z∈X
(s(k)(z) ⊕ ¬q(k)(z, x)), k ∈ E ∪F,
0
L
, k 6∈ E ∪ F.
Step 4
We use the inverse transformation Λ
−1
from
Proposition 3 to obtain the L-fuzzy soft sets
(F, q)
↑
(E, s) := Λ
−1
(H(F, q)
↑
(Λ(E, s))) and
(F, q)
↓
(E, s) := Λ
−1
(H(F, q)
↓
(Λ(E, s))) We ob-
tain the following formulas for upper and lower
approximations of L-fuzzy soft sets by L-fuzzy soft
relations.
(F, q)
↑
(E, s)(k)(x) =
(
W
z∈X
s(k)(z) ⊗ q(k)(z)(x), k ∈ E ∩F,
0
L
, k 6∈ E ∩F
(F, q)
↓
(E, s)(k)(s) =
(
V
z∈X
s(k)(z) ⊕ ¬q(k)(z, x), k ∈ E ∪F,
0
L
, k 6∈ E ∪F
.
5.3 Approximations of Ω-Level L-Fuzzy
Soft Sets
We show how the steps 1-4 can be realizded for Ω-valued
L-fuzzy soft sets. We use the notation of Sections 3.3 and
4. Let ρ be the L-fuzzy equivalence relation in the set Ω
and let f ∈ Ω(X, ρ), f : X ×Ω → L be an Ω-level L-fuzzy
sets and let Q : X ( Y be a Ω-level L fuzzy relation, that
is, Q : X ×Y × Ω → L.
Step 1
According to Proposition 5, the transformation Λ( f ) of
f into the (R, R
∗
)-fuzzy set X → R is defined by
x ∈ X, Λ( f )(x) = f (x, −) ∈ R.
Step 2
The transformation of Ω-level L-fuzzy relation Q is de-
fined by Proposition 8 by
H(Q) : X → R
Y
,
x ∈ X, y ∈ Y, H(Q)(x)(y) := Q(x, y, −) ∈ R.
Step 3
According to definition of operations in (R, R
∗
) in Sec-
tion 3.3, the upper and lower approximations of Λ( f ) by
H(Q) are defined for y ∈ Y, α ∈ Ω by
H(Q)
↑
(Λ( f ))(y)(α) =
R
∑
x∈X
(Λ( f )(x) × H(Q)(x, y)
!
(α) =
_
β ∈Ω
_
x∈X
f (x, β ) ⊗ Q(x, y, β) ⊗ ρ(β , α), (5)
H(Q)
↓
(Λ( f ))(y)(α) =
R
∗
∑
x∈X
(Λ( f )(x) ×
∗
¬H(Q)(x, y)
!
(α) =
^
x∈X
¬(¬(s(x, −) ⊕ ¬Q(x, y, −))(α) =
^
x∈X
¬


_
β ∈Ω
¬s(x, β ) ⊗ Q(x, y, β) ⊗ ρ(α, β )


=
^
x∈X
^
β ∈Ω
s(x, β ) ⊕ ¬Q(x, y, β) ⊕ ¬ρ(α, η). (6)
FCTA 2023 - 15th International Conference on Fuzzy Computation Theory and Applications
336

Step 4
According to Proposition 5, the inverse transforma-
tion Λ
−1
is define such that
Q
↑
( f )(y, α) = Λ
−1
(H(Q)
↑
(Λ( f )))(y, α) =
H(Q)
↑
(Λ( f ))(y)(α) = (5),
Q
↓
( f )(y, α) = Λ
−1
(H(Q)
↓
(Λ( f )))(y, α) =
H(Q)
↓
(Λ( f ))(y)(α) = (6).
6 CONCLUSIONS
In this short paper, we show that some types of new
L-fuzzy sets with values in complete MV -algebras
L (such as neutrosophic L -fuzzy sets, L -fuzzy soft
sets, intuitionistic L-fuzzy sets or multi-level L-fuzzy
sets and some others) can be transformed into the so-
called (R, R
∗
)-fuzzy sets, where (R, R
∗
) is a pair
of commutative complete idempotent semirings with
involutive isomorphisms between them.
Using this value structure (R, R
∗
), the theories
of these new L-fuzzy sets can be defined in a uni-
fied way, without having to prove the properties of
this theory for individual types of these new L-fuzzy
sets. For illustration, we have shown how the theory
of upper and lower approximations can be defined us-
ing the corresponding types of new L-fuzzy relations
for neutrosophic MV -valued fuzzy sets, MV -valued
fuzzy soft sets, and for Ω-level MV -valued fuzzy sets.
In a completely analogous way, this theory can also
be defined, for example, for intuitionistic MV -valued
fuzzy sets or for combinations of these new types of
fuzzy sets. Moreover, if we consider some new types
of pairs (R, R
∗
), we can obtain completely new types
of L-fuzzy sets.
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