The Single Axiomatization on CCRL-Fuzzy Rough Approximation Operators and Related Fuzzy Topology

: It is well known that lattice-valued rough sets are important branches of fuzzy rough sets. The axiomatic characterization and related topology are the main research directions of lattice-valued rough sets. For L = ( L , ⊛) , a complete co-residuated lattice (CCRL), Qiao recently deﬁned an L -fuzzy lower approximation operator ( LFLAO ) on the basis of the L -fuzzy relation. In this article, we give a further study on Qiao’s LFLAO around the axiomatic characterization and induced L -topology. Firstly, we investigate and discuss three new LFLAO generated by ⊛ -transitive, ⊛ - Euclidean and ⊛ -mediated L -fuzzy relations. Secondly, we utilize a single axiom to characterize the LFLAO generated by serial, symmetric, reﬂexive, ⊛ -transitive and ⊛ -mediate L -fuzzy relations and their compositions. Thirdly, we present a method to generate Alexandrov L -topology ( ALTPO ) from LFLAO and construct a bijection between ALTPO and ⊛ -preorder (i.e., reﬂexive and ⊛ -transitive L -fuzzy relation) on the same underlying set.


Introduction
Rough-set theory [1] was put forward by Pawlak.This theory plays a vital role in handling the uncertainty, granularity and incompleteness of knowledge in information systems.Classical rough sets with strict equivalence conditions restrict the development of rough sets, so people introduced generalized rough sets to avoid this situation.For the past few years, many experts and scholars have studied various types of generalized rough sets [2,3].It is well known that one pair of approximation operators is the basic concept of rough-set theory.Generally, we have two methods to study the approximation operators.One is the constructive method which constructs the approximation operators from relations, coverings, neighborhoods and so on [2].The other is the axiomatic method.An abstract operator is given firstly, and then we look for single axiom or axiom sets s.t. the operator happens to be the approximate operator from the construction method [3].In addition, the topologies induced by the approximation operators are also vital content of rough-set theory [2,4].The most well-known result may be the existence of bijection between Alexandrov topology (i.e., quasi-discrete topology) and a preorder relation based on the same underlying set.
Just like the classic rough sets, the construction and axiomatization of L-fuzzy approximation operators and the related topology are also important directions in L-fuzzy rough sets, where L = (L, * ) is a complete residuated lattice (CRL) and L = (L, ⊛) is a complete co-residuated lattice (CCRL).

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The work on L = (L, * ).For L-fuzzy relations (L-f r), Radzikowska introduced the pair of L-fuzzy upper and approximation operators.The basic properties of the Lfuzzy approximation operators generated by serial, reflexive, symmetric, transitive and Euclidean L-f r were also studied.Then, Wang [29] characterized Radzikowska's operators by using axiom set; She [18,27] improved Wang's work and characterized the related L-fuzzy approximation operators by a single axiom.Pang [23,24] defined and characterized the L-fuzzy approximation operators generated by mediated, Euclidean and adjoint L-f r.Hao [20] discussed the L-topological structure associated with L-f r and verified that there is a bijection between L-preorder (i.e., reflexive and transitive L-fuzzy relation) and Alexandrov L-topology.Ma further established the connections between L-closure and L-interior operators and L-fuzzy approximation operators.Zhao [35] introduced L-fuzzy variable-precision rough sets based on L-f r.Qiao [26] and Wang [28] further proposed the granular, variable-precision, L-fuzzy rough sets by the fuzzy granule associated with L-f r.Belohlavek built the connection between L-fuzzy rough sets and concept lattices.Han [19] discussed some categories of approximate-type systems generated by L-f r.In the above mentioned L-fuzzy rough set, the L-f r is based on the classical set.Quite recently, by considering the L-f r based on L-fuzzy sets, Wei [30] developed a general L-fuzzy rough set from both constructive and axiomatic methods.For L-fuzzy covering, Li [21] introduced and described several L-fuzzy approximation operators.Based on the L-f r generated by L-fuzzy covering, Jiang [37] proposed a covering-based variable-precision, L-fuzzy rough set and applied it in the study of multi-attribute decision-making problems when L = [0, 1].For L-fuzzifying neighborhood-systems and L-fuzzy neighborhoodsystems, Li [22] and Zhao [32,33] investigated two types of L-fuzzy rough sets and described them by one axiom each.Furthermore, Song [38] and Zhao [34] researched the lattice structure and L-topological structure associated with Zhao's L-fuzzy rough sets.For (L, M)-fuzzy neighborhood systems, El-Saady [39] established the (L, M)fuzzy rough sets, which unified Li's L-fuzzy rough sets [22] and Zhao's L-fuzzy rough sets [32,33] into one framework.

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The work on L = (L, ⊛).For L-f r, Qiao [25] defined and characterized a new L-fuzzy lower approximation operator on the basis of ⊛.He also proved that his reflexive L-fuzzy lower approximation operator induced an ALTPO in his sense.In [26], Qiao further proposed a variable-precision, L-fuzzy lower approximation operator.In [40], the author introduced an L-fuzzy upper approximation operator through ↝, the coimplication w.r.t.⊛.We verified that Qiao's L-fuzzy lower approximation operator is dual to our L-fuzzy upper approximation operator for some special L.

Motivations, Innovativeness and Contributions
From the above review, it is easy to find that compared with the CRL-fuzzy rough sets, the CCRL-fuzzy rough sets are still far from perfect.Therefore, from the perspective of theoretical development, the CCRL-fuzzy rough sets should continue to be studied in depth.In the paper, inspired by the following three aspects, we present further research on Qiao's L-fuzzy lower approximation operator (LFLAO) [25].

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The approximation operators, generated, respectively, by transitive (TR), Euclidean (EU) and mediate (ME) relations are important in the classical rough-set theory.In [25], Qiao defined TR and EU L-f r through * but not ⊛.Obviously, such indirect definition brings inconvenience to the research and limits the scope of theory.In addition, Qiao did not define ME L-f r.The first aims were to define directly ⊛-TR, ⊛-EU and ⊛-ME L-f rs and discuss the related LFLAO.

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The single-axiom description of the approximation operators is an amusive topic in various general rough sets [8,12,13,18,23,29,30].In the literature, Qiao did dot present the single axiomatic description of his lower approximation operator.The second aim was to use a single axiom to describe Qiao's LFLAO produced through serial (SR), symmetric (SY), reflexive (RF), ⊛-TR and ⊛-ME L-f rs.

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The construction of bijection between (fuzzy) Alexandrov topologies and (fuzzy) preorders is meaningful in (fuzzy) rough sets [20,30].The corresponding result has not been established on Qiao's L-fuzzy rough sets.The third aim was to redefine Alexandrov L-topologies and construct a bijection between them and ⊛-preorders.
In our opinion, the innovation of this paper includes the following two aspects.On the one hand, there is very little work on CCRL-fuzzy rough sets, and our results make a meaningful supplement in this respect.On the other hand, we characterize the proposed CCRL-fuzzy rough sets from an internal aspect (axiomatic characterization) and an external aspect (the bijection to ALTPO).
In Section 2, we give some basic concepts and Qiao's LFLAO as preliminaries.In Section 3, we investigate three new LFLAO produced through ⊛-TR, ⊛-EU and ⊛-ME L-f rs and characterize them through axiom sets.In Section 4, we use a single axiom each to describe five (including the three mentioned above) LFLAOs and their combinations.In Section 5, we redefine the concept of ALTPO and then construct a bijection between ALTPO and ⊛-preorder through LFLAO.In Section 6, we present conclusions.
In order to express conveniently, we give the following abbreviation table (Table 1).EQ.
As we all know, rough sets and their fuzzy generalizations are widely used in many fields [41][42][43].In recent years, the multi-attribute decision-making method based on fuzzy rough sets has been a hot topic in both rough sets and decision-making [44][45][46].When L = [0, 1], the CRL-fuzzy rough sets have already demonstrated their applications in medical diagnosis [47], attribute reduction [48] and decision analysis [49].Likewise, the CCRL-fuzzy rough sets have good application prospects in the above fields.However, the main purpose of this paper is to improve and expand the theoretical framework of fuzzy rough sets.As for the applications, we leave them for the future.

Preliminaries
In this section, we will recall some basic notions and notation used in this paper.

Basic Concepts
In this subsection, we review the basic concepts and properties of CCRL for later use.
(1) The standard max operator µ ⊛ SMO ν = max{µ, ν}; (2) The probabilistic sum µ Let us assume that W is a nonempty set.Every mapping f ∶ W → L is called an LF-set on W. All LF-sets on W are signed as L W .Take µ ∈ L. The symbol µ is also used to represent the constant LF-set valued as µ.Put B ⊆ W. The symbol ⊺ B represents an LF-set with ⊺ B (w) = 1 whenever w ∈ B and ⊺ B (w) = 0 whenever w ∈ B.
A mapping ¬ ∶ L → L is termed an involutive negation whenever it is non-increasing and ¬¬µ = µ for any µ ∈ L.
(4) S is called similar (SI) provided it is reflexive and symmetric.Definition 2. For an L-f r S ∶ W × Z → L, the triple (W, Z, S) refers to an L-fuzzy approximation space (LFASPC).Whenever W = Z, (W, Z, S) is simplified as (W, S).

Definition 4 ([25], Proposition 4.2).
A mapping Θ ∶ L Z → L W is also termed an LFLAO whenever Θ = S for some L-f r S.
, SY, RF and SI) LFLAO whenever Θ = S for some SR (resp., SY, RF and SI) L-f r S.

The Approximation Operators Produced through Three Special L-f r and Their Axiomatic Set Descriptions
In this section, we will define ⊛-TR, ⊛-EU and ⊛-ME conditions for L-f r.Then, we describe the associated LFLAO through axiom sets, respectively.Definition 6. Suppose that (W, S) is an LFASPC and ∀α, β, ρ ∈ W, (4) S is called ⊛-PR provided it is RF and ⊛-TR.
(5) S is called ⊛-EQ provided it is RF, SY and ⊛-TR.(2) Let ⊛ = ⊔.Then, it is easily noted: Both of them are well-known.
From Theorems 2 and 3, the next theorems can be deduced naturally: Theorem 4 (The combination of two L-frs).Let Θ ∶ L W → L W be a mapping.
Theorem 5 (The combination of three L-frs).Let Θ ∶ L W → L W be a mapping.

The Single-Axiom Description of LF LAO
In this section, we describe the mentioned LFLAO by single axiom.

One L-f r
In this subsection, we use a single axiom to describe the LFLAO generated by one L-f r.

Combination of Two L-f rs
In this subsection, we use a single axiom to describe the LFLAO generated by the combination of two L-f rs.

Combination of Three L-frs
In this subsection, we use a single axiom to describe the LFLAO generated by the combination of three L-f rs.By applying (L6) in (LSR-SY-TR), (LSR-SY) holds.
Remark 3. In [25], for a reflexive L-f r S on W, Qiao proved that the family T Q S = {B ∈ L W B = S(B)} forms an ALTPO in his sense.In this case, it is easily seen that T Q S = T S since ∀B ∈ L W and S(B) ≤ B by Proposition 3 (2).
Example 2. We assume W, L, ¬ and S are consistent with in Example 1. Obviously, S is reflexive.
Next, we will show that for any LTPO on W, we can define a ⊛-(PR) on W.
Definition 9. Suppose that T is an LTPO on W.Then, S T ∈ L W×W : is called the L-f r generated by T .
Theorems 26 and 27 illustrate that there is a bijection between ALTPO and ⊛-PR on W. We fix a lemma first.the single axiom characterizations on LFLAO associated with SR, SY, RF, ⊛-ME and ⊛-TR L-fuzzy relations and their combinations were presented, respectively.Thirdly, by the LFLAO, a one-to-one correspondence between ⊛-preorder and Alexandrov L-topology was obtained.
In the future, we shall further research CCRL-fuzzy rough sets from the following four angles.Firstly, notice that the CCRL-fuzzy rough sets based on L-fuzzy covering and L-fuzzy neighborhood systems are important branches of fuzzy-rough-set theory.Nowadays, both of them have not been studied.Hence, we will consider these two kinds of L-fuzzy rough sets in order to enrich and improve the theoretical framework of fuzzy rough sets.Secondly, it is known that the variable-precision fuzzy rough sets [51] and multi-granularity fuzzy rough sets [17] have attracted much attention because of their fault-tolerant ability.However, there is no research based on CCRL at present.Thus, we will discuss the variable-precision and multi-granularity fuzzy rough sets based on CCRL.Thirdly, as is known to all, category theory is an important tool for studying mathematical structure.With the help of category theory, people can understand the given structure at a higher level.Therefore, we will study the category properties and category relations of the proposed CCRL-fuzzy rough sets.Last but not least, when L = [0, 1], CRL-fuzzy rough sets have been widely used in medical diagnosis, rule extraction, decision analysis and many other fields [17,[45][46][47][48]. Hence, we will explore the application of CCRL-fuzzy rough sets in related fields.Especially, we will combine CCRL-fuzzy rough sets with three-way decisions.