Fuzzy formal concept analysis: approaches, applications and issues

ABSTRACT


INTRODUCTION
The mathematical foundation of formal concept analysis (FCA) is based on lattice theory [1]. The data analysis in FCA begins with a supplied cross table, wherein every row corresponds to the set of objects, every column refers to a collection of attributes, and the values of the cross-table indicates the relationship between them. The concept lattice is acknowledged as one of the primary outputs of formal concept analysis, reflecting generalization and specialization between the cross table's created formal concepts [2]. Formal concept is a fundamental unit of concept that play an essential role in knowledge processing containing the extent part (sets of objects) and the intents part (corresponding common attributes). FCA's traditional setting considers the context to be a table., where the rows in the context representing the domain objects, and the columns of the context refer to the attributes for each object in the domain under the study. The cross-table (context) inputs carrying the values ones and zeros (X symbol\ empty) based on whether or not an object possesses the attribute. As a result, the basic formal concept analysis is superior for the attributes that has a crisp value (0s or 1s). At the same time, features might be vague rather than precise(crisp). FCA has successfully been enhanced with a fuzzy setting to accommodate the ambiguity and vagueness in data. For example, if we ask if a man with a height of 170 cm is tall, we'll probably get an answer like "not totally tall but almost tall" or "to a high degree tall,". Lotfi Aliasker Zadeh introduced such a notion in fuzzy logic [3] to assign a truth level of belonging to an object based on the fuzzy attributes that the object contains. An L-scale of truth degrees (degree of belonging) is used to compute the degrees of belongings. One of the most popular  1]. To return to our earlier example, a guy of (170 cm) in height is tall to a level of 0.7. As a result, rather than 0 or 1, as in traditional FCA, the context values are becoming degrees from the interval [0,1]. FCA's immense potential is promoted by the concept lattice generated by the collection of formal concepts which retrieved from the formal context of a domain under the study. In reality, the lattice is used in the vast majority of applications., which is often depicted by a line diagram (Hass diagram). Large numbers of formal concepts might perhaps be produced from a tiny quantity of data [4]. In reality, FCA may lead to a great deal of computational complexity, and even with a small dataset, the resulting concept lattice can be huge [5]. For many applications, the computational cost is still too high. Furthermore, examining the final lattice might be challenging due to the enormous number of formal concepts and the intricacy of concept interactions [6]. In fact, obtaining a concept lattice of adequate complexity and scale is one of the most critical obstacles of employing formal concept analysis [4].
Concept lattice reduction strategies come in many different forms., each with its own set of advantages and disadvantages. Some of them purge the concept lattice from redundant information. Generally, their main goal is to seek the smallest number of objects or attributes which preserve the hierarchy order of the original concept lattice [7], [8]. Other techniques aim to create an abstraction of the concept lattice, or to achieve a high level of simplification that reveals the genuinely important aspects [9]. Lastly, some types of techniques employ a relevance criterion to select formal concepts, objects, or attributes [10]. The primary goal of this work is to clarify the relationship between the various approaches to Fuzzy formal concept analysis and to discuss the main issues associated with using it. The second direction in this work is to present a review of research articles on using fuzzy formal concept analysis in various applications such as knowledge discovery in databases and data mining, information retrieval, and ontology engineering.
The remainder of this paper is composed: In section 2 will give a theoretical background to the fuzzy formal concept analysis. In section 3, we will go over and compare the most important approaches for fuzzy formal concept analysis that have been proposed. Section 4 will provide a quick overview of the use of fuzzy formal concept analysis in several fields. In section 5, we will discuss the main techniques to reduce the size of concept lattice that considered as a main issue in several applications that used FCA as an analytical method.

FUZZY FORMAL CONCEPT ANALYSIS
Formal concept analysis is a theoretical framework that provides a foundation for conceptual data analysis and knowledge processing. It allows the representation of the relationships between objects and attributes in a specific domain [11]. Formal concept analysis offers a different graphical representation of tabular data that is easier to navigate and use [1]. A more detailed overview is provided in [1].

Formal concept analysis (FCA)
The ideas of a formal context are taken into account by FC), which describes the attributes of every object from the domain. Accordingly, a formal context may be conceptualized as a binary connection between both the object group and the attribute group, with values of 0 and 1. A cross-table (formal context) is given to FCA at the beginning, and it is described as a triple = ( , , ), where denotes a group of objects, denotes a set of attributes, and denotes a binary connection ( ⊆ × ). ( , ) ∈ I, can be read as an object has the attribute .
The usage of the words "object" and "attribute" is instructive because it may be advantageous in many situations to select things that resemble other objects as formal objects and then select their properties as formal attributes. In the field of information retrieval, documents, for instance, might be thought of as being object-like and phrases, as being attribute-like. As seen in Figure 1(a) from [11], the context is frequently represented as a cross table, with rows denoting formal objects, columns denoting formal attributes, and crosses denoting interactions between them. Definition 1. Formal Concept, a context = ( , , ), for ⊆ and for ⊆ applying a derivation operator: (1) The set of all objects having all of the attributes from is called A ′ , whereas the set of all attributes shared by all objects from A is called B ′ . As a result, a formal concept is defined as a pair (A, B) for a formal is referred to as the concept's extent and as its intent.
The Figure 1(a) depicts the line diagram for the concept lattice which constructed from the formal context that shows in the upper left corner of Figure 1(a). The concept lattice could be made up by the set of formal concepts which constructed from the formal context and the subconcept-superconcept relation between them [1]. The nodes in the line diagram refer to the formal concepts where the lower nodes noted in the diagram represented the formal objects and formal attributes are dipcted in the higher level of the diagram.
To obtain the extent of a formal concept, one must follow the descending path from the node to get the formal objects. In the Figure 1(a) we can notice that the formal objects for the node 2 are (URL_3, URL_2, URL_5). To obtain the intent of a formal concept, one must follow the ascending path from the node to get the formal attributes like the formal attribute "study" located at the top of the node 2 . Note that 2 is a formal concept with extent (URL_3, URL_2, URL_5) and intent (series, study). 2 is consider as a subconcept of 1 .

Fuzzy formal concept analysis
FCA has recently been used in several applications where the domain representation contains uncertain and ambiguous information. A generalized Wille's model was one of the first studies to incorporate fuzziness into FCA [12]. Specifically, the use of a residuated lattice [13]- [16] to extend the original formal concept analysis by determining the truth degree for the assertions "object x has attribute y" in fuzzy formal contexts. Degrees are calculated using an L-scale of truth degrees. Normally, real values in the range [0, 1] are used to value L. As a result, instead of values from 0 or 1 as in the basic setting of classical FCA, the entries of the cross-table describing objects and attributes become degrees from L. This is known as a fuzzy formal concept analysis. The fuzzy formal context in Figure 1(b) has a confidence threshold = 0.6. Where all objectsattributes relationships with membership values lower than 0.6 are hidden. Definition 6. Given two fuzzy formal concepts like ( ( 1 ), 1 ) and ( ( 2 ), 2 ) of a fuzzy formal context ( , , ). ( ( 1 ), 1 ) is the subconcept of ( ( 2 ), 2 ) denoted as ( ( 1 ), 1 )≤ ( ( 2 ), 2 ) if and inly if ( 1 ) ⊆ ( 2 ) ( 2 ⊆ 1 ). For example, in the Figure 1 129 The operators ∩,∪ indicate the intersection and union (respectively) operations on a fuzzy set. Tnorm and t-conorm are used to compute the fuzzy intersection and union. The minimum is the most widely used t-norm, whereas the maximum is the most widely used t-conorm. Assume two fuzzy sets with membership functions ( ), ( ), where ∈ (universe of discourse), the intersection and union operators are defined as ∩ ( ) = min ( ( ), ( )) and ∪ ( ) = max ( ( ), ( )). As an illustration, consider the similarity of the fuzzy formal concept that calculated between the concepts That given in the Figure 1

COMPARISON OF FCA APPROACHES 3.1. L-fuzzy concept lattice
The authors in their paper [16] were the first to indicate that FCA can be expanded to include fuzzy concepts. They proceed in the following manner: Let = ( , ≤, ′ , ⨁,0,1) be a structure such that ( , ≤, ′ , 0,1) indicates a complete lattice constrained by (0 and 1), ′ is refer to unary complementation operation, where ⨁ be a t-conorm on (a binary operation with the neutral element 0 that is associative, commutative, and associative).

FCA and related structures in a fuzzy setting
Belohlavek [19] and Pollandt [20] independently proposed the basic concepts of FCA in fuzzy environment. They developed the following approach, which proved to be a viable method for developing FCA and related structures in a fuzzy environment. Let ( , , ) be an -context, i.e., : × → . For a fuzzy set ∈ and ∈ , suppose fuzzy sets ↑ ∈ and ↓ = described by Using fundamental predicate fuzzy logic rules [15], [21], it is straightforward to understand that ↑ ( ) is the truth degree of is common for all objects in " and ↓ ( ) is the truth degree of possesses all attributes from ." As a result, we may claim that (5) and (6) are exact generalizations of (1) and (2). Putting ℬ( , , ) = {⟨A, B⟩ | ↑ = , ↓ = }. is refer to the set of all formal concepts ⟨A, B⟩, such that is refer to the set of all objects that share all the attributes of (the intent), and is refer to the set of all attributes that shared by all objects of (the extent). ℬ( , , ) refers to a collection of all formal concepts, where A denotes the collection of objects with all of 's features. known as the intent part, and is the collection of all features that all 's objects share. known as the extent part. ℬ( , , ) is regarded as as a fuzzy concept lattice of the formal context ( , , ). The extent part of a formal concept and the intent of a formal concept ( , ) are both in general fuzzy sets, as in the method of Burusco and Fuentes-Gonzalez [16]. This represents the reality that concepts, in general, apply to objects and attributes to varying degrees, rather than simply 0 and 1. Putting For ⟨A 1 , B 1 ⟩, ⟨A 2 , B 2 ⟩ ∈ ℬ( , , ), ≤ represented the subconcept-superconcept hierarchy in ℬ( , , )

Fuzzy concept lattice with non-commutative conjunction
The authors [22], defined the fuzzy concept lattice ( , ⋁, ⋀,⊗, →, ⟹ ,0,1) in their method, which combines fuzzy logic with a non-commutative conjunction ⊗ rather than a commutative conjunction. They claim that removing the commutativity condition is necessary in instances when the order of the terms of the conjunction concerns, in order to make the theory suitable for representing temporal data. In this case, the Galois connection will be made up of two pairs of functions, ↑, ⇑ : → and ↓, ⇓ : → , each in a symmetric position to his partner. The authors further demonstrate that any non-commutative fuzzy logic concept lattice may be understood using their framework of extended concept lattices with non-commutative conjunction.

One-sided fuzzy concept lattice
In [23] and [24] separately devised the "One-sided fuzzy concept lattice" approach. The definitions of the authors get identical outcomes for (X, Y, I −1 ), I −1 ∈ L X×Y defined by I −1 (x, y) = I(x, y), implying that the techniques are equivalent in terms of the function of objects and attributes.  The authors established that ℬ f,h (X, Y, I) could be fitted by the partial order ≤, as specified in (7), to form a complete lattice; the researchers have attributed for such a method as a "one-sided fuzzy concept

Crisply generated fuzzy concepts
Regarding the problem of a possibly big set of formal concepts , the authors in their work [25] recommended just using a portion of ℬ(X, Y, I), called ℬ c (X, Y, I), as opposed to using the complete ℬ(X, Y, I). ⟨A, B⟩ ∈ ℬ(X, Y, I) can be called crispy if there is a subset exist B c ⊆ Y (of attributes), such that A = B c ↓ (thus, B = B c ↓↑ ). Then, the complete lattice of crispy generated fuzzy concept represented by ℬ c (X, Y, I) = {⟨A, B⟩ ∈ ℬ(X, Y, I) |exists B c ⊆ Y ∶ A = B c ↓ }. Consider the ways in which intentions are crisp sets and extrentions are fuzzy sets. in Yahia and Krajci's "One-sided fuzzy concept lattice" approach, whereas both extention part and intention part are generally fuzzy sets in the "Crisply Generated Fuzzy Concepts" approach. B (f,h) (X,Y,I) equipped with the partial order (≤) in the "One-sided fuzzy concept lattice" given in (7) is a complete lattice that is isomorphic to ℬ c (X, Y, I) equipped with the partial order acquired from ℬ(X, Y, I). Additionally, there is an isomorphism for the equivalent notions, ⟨A, B⟩ ∈ ℬ f,h (X, Y, I) and (C, D) ∈ ℬ c (X, Y, I) such that A = C , B = D ↓↑ .

Generalized concept lattice
A "generalized concept lattice" is the objective of the author's investigation in [26]. In general, the author suggests that three sets of truth degrees (level of belonging) be taken into consideration: L X (refer to the objects set), L Y (attributes set), and L indicates the table entries (degree of attribute possession of objects). Assuming that X is objects set and Y is attributes set, the context that considered as a fuzzy context may be thought of as a triple (X, Y, I), where I denotes the L-relation between the objects set and the attributes set, i.e., I ∈ L X×Y . The author also asserts that is a partly ordered set and that L X and L Y are complete lattices. ≤ is used to represent all partial orders on (L X , L Y , and L). In order to define arrow-operators, the author makes the following assumption: It is satisfied by: If a j  b ≤ c for each j ∈ J then (⋁ a j j∈J ) b ≤ c, If a  b j ≤ c for each j ∈ J then a  (⋁ b j j∈J ) ≤ c, This is for each index set J and for all a, a j ∈ L X , b, b j ∈ L Y and c ∈ L. To put it another way, there are three levels of truth (L 1 , L 2 , L, , ≤, …). If it meets (8)- (11), this structure is referred to be Krajci's structure.
Then, Krajci moves on to mappings the arrow-operations ↑ : L X X → L Y Y and ↓: L Y Y → L X X by A ↑ (y) = ⋁{b ∈ L Y | ∀ x ∈ X: A(x)b ≤ I(x, y)} B ↓ (x) = ⋁{a ∈ L X | ∀ y ∈ Y: aB(y) ≤ I(x, y)} The formal concepts in (X, Y, I) are defined as pairs (A, B) ∈ L X X × L Y Y fulfilling A ↑ = B, B ↓ = A . ℬ = {⟨A, B⟩| A ↑ = B, B ↓ = A} (formal concepts) fitted with the partial order (≤) is a complete lattice (i.e., the generalized concept lattice for X, Y, I, ). In [27] establishes a fundamental theorem for an extended concept lattice.

APPLICATION DOMAINS
In numerous disciplines, formal concept analysis has been utilized in conjunction with fuzzy logic. In order to detect connections between demographic data and physical activity levels, Data from epidemiological surveys on physical exercise were examined using FCA by the authors of [28] [29]. The complicated structure of the dataset is formalized as a fuzzy many-valued context, which is then translated to a binary context using histogram scaling. The framework of the technique was based on the production and evaluation of formal concepts. The concepts were analyzed by a hydrobiologist, resulting in a collection of ecological features that were added to the initial context.
By fusing FCA and fuzzy characteristics, the authors of in [30] presented a framework that aids users in their discovery of semantic web resources. Lower and higher levels make up this structure. In the lowest layer, fuzzy multisets are created from the semantic representations of web services. The service's capabilities are represented in this representation (an OWL-S document). This representation, which is an OWL-S document, demonstrates the functionality of the service. Fuzzy C-Means clustering is used to group the web services into fuzzy clusters. Services that are near matches to the input request have been found using fuzzy matching. Using a fuzzy formal context, archetypes and ascribed ontological conceptions that are present or absent have been defined at the upper layer.
Ontology engineering is another research area that focuses on the relationships between individuals and classes. In [31], the authors use FCA in combined with fuzzy logic to automatically generate ontologies. The ontologies created will be used to support the Scholarly Semantic Web, that is used to share, reuse, and manage scholarly data. Quan et al. (2006) propose a "Fuzzy Ontology Generation System" for automatically creating an ontology that incorporates FCA and fuzzy logic. This approach is later utilized by the authors [32] to construct an ontology that might be used in "web-based help-desk applications." The authors of [11] described an ontology-based retrieval strategy that allows for data organizing and visualization while also offering a user-friendly navigation mechanism. To obtain conceptual frameworks from datasets and build a hierarchal structure representation of extracted information, it employs a fuzzy extension of Formal Concept Analysis theory. This approach contributes significantly to knowledge handling. It offers hierarchy exploration and query processing after performing knowledge extraction and structuring as well as ontologydriven discovery. The outcomes of the implementation are concentrated on hierarchical facet-based navigation.
Many papers have recommended combining fuzzy logic with formal concept analysis for information retrieval. In a citation database-based document retrieval system., the authors employed FCA with fuzzy features for conceptual grouping, according to their work in [33]. Using fuzzy logic and formal concept analysis, a fuzzy concept lattice is constructed on which "a fuzzy conceptual clustering approach" is conducted. the process of getting documentation will thereafter be accomplished through the use of fuzzy queries. The authors established a methodology for developing an ontology using formal concept analysis and fuzzy features in [34]. The initial collection of documents is broken into smaller groups of similar texts using the "Growing Hierarchical Self Organizing Map clustering method." "Agglomerative clustering" is used to combine the models into a hierarchy of concept lattices. To deal with empty responses for the queries based fuzzy, the authors employed formal concept analysis with fuzzy features, according to the work in [35]. Fuzzy querying processing based on Galois lattices helps discovering reasons for empty results by displaying the subqueries that are accountable for the mistake. In [36] proposed a query expansion technique based on FCA and fuzzy attributes.
Several researchers have recently demonstrated the use of formal concept analysis in reliability engineering. In [37], the authors' goal is to present the fundamentals of FCA and how it can be applied to reliability engineering problems in their paper. To accomplish this, four examples in reliability engineering were chosen for analysis from the literature as well as the authors' personal experience. The first example explains the FCA approach based on cut-sets in network-modeled systems. The second example analyzes which protection strategy could be used to prevent various types of attack scenarios in a given network using notions inferred from knowledge space theory. The last two examples show how binary formal contexts can be extended to analyse: i) failure events caused by different reasons (granularity levels); and ii) the significance of nodes in an electric power system based on several measures of significance (attributes with multiple values).
Another research direction to use FCA in crime prediction. In this paper [38] the authors provided a brief background on crime pattern analysis as well as available methods for resolving it. Simultaneously, some of the intriguing methods are empirically analyzed based on various parameters in order to understand their appropriate applicability. They also concentrated on the uncertainty analysis that exists in crime data sets with fuzzy attributes.

CURRENT ISSUES AND RESEARCH DIRECTIONS
FCA is a useful formalism for representing, extracting, and analyzing whatever information system, but it has a few issues that need to be settled. In general, contexts are large, complex, and contain a huge amount of redundant information. As a result, one of the main issues identified in practical FCA applications is that the computational cost of processing the information system with FCA is high and visualizing the lattice structure is difficult [39]. Because of FCA's scalability, this complexity issue arises. Considering that counting the formal concepts in the input context is #P-complete [40], and that the number of formal concepts in the input context might be exponential, all concepts can be constructed with polynomial latency. The sizes of implication bases, even the smallest implication base, can be exponential, with the size of the stem base being #P-hard [41] to calculate. A major challenge for FCA practical applications is the visualization of formal concepts in a hierarchy structure in the final outcome (concept lattice structure). One of the main issues with this technique is how large the concept lattice is when it is formed from a large formal context. The vast context concept lattice becomes difficult and unworkable. As a result, managing a large formal context and reducing the size of the concept lattice are highlighted as practical challenges in formal concept analysis applications. In general, procuring a concept lattice of sufficient complexity and size is one of the most fundamental challenges in formal concept analysis [39].
The literature describes a variety of techniques for controlling the complexity and size of formal contexts, formal concepts, concept lattices, and implications. To enhancing FCA scalability there are popular research techniques include iceberg concept lattices [10], matrix decompositions [42], conceptual scaling for many-valued contexts [43], the reduction of the concept lattices based on rough set theory [44], and other. In [39], the authors divided concept lattice reduction techniques into three categories. The first category of reduction techniques removes redundant information from the context, that means an object ∈ , ∈ (set of attributes) or incidence ∈ ( is a binary relation ( ⊆ × )) can be considered as redundant knowledge in the formal context if removing or transforming it results in a lattice isomorphic to the orginal. The techniques for removing redundant information aim to create a concept lattice that is isomorphic to the original. The authors in their paper [45] used the same teqnique of reduction on fuzzy formal context. This category of techniques is useful when there is a lot of redundant knowledge in the formal context.
The second category of the reduction techniques is simplification techniques. The concept lattice contains all relationships between concepts, including those between concepts that are very similar. For instance, the corresponding link is shown using formal concepts that differ only by a single characteristic. The lattice can be made simpler by omitting the property that separates these concepts in these situations if it is no longer relevant (as judged by some conditions). in order to emphasize only the important knowledge. This category of techniques is useful for identifying key aspects in formal context or concept lattices.
The third category of the reductions techniques is selection techniques. Several concepts, especially in a big concept lattice, may be deemed irrelevant in a given application. The "relevance" of a concept could be related to its cardinality, intention or extension, the relationship between some attributes, and so on. Selection techniques are those that select objects, attributes or concepts based on some relevance criterion. The authors [46] made a significant contribution in this direction by connecting frequent items and formal concepts. The terms "support" and "frequent sets" are discribed as: Let ⊆ , where M is a set of attributes and ( , ), is the counts of objects in that contains all the attributes of . We can say that a set of attributes ⊆ is frequent iff ( , ) ≥ (minimal support previously set). Iceberg concept lattices are concept lattices created by limiting the item sets to those that are frequently used. In this instance, just the most common formal concepts are employed, leading to a partial lattice. This occurs as a result of the support for the intent being a diminishing function. In other words, the gevin a two formal concept ( 1 , 1 ), ( 2 , 2 ), where ( 1 , 1 ) ≤ ( 2 , 2 ), sup ( 1 , ) ≤ sup ( 2 , ), where G is the sets of objects [10] describes how to create iceberg concept lattices using the Titanic method. The authors also show how to use these lattices for a number of tasks, such as large-scale database analysis, mining association rule extraction, implications extraction, and implications visualization. Following that, we summarize some of the advances in the literature on scalability issues in Table 1 (see in appendix) and briefly describe each work's contribution based on the categories that we mentioned.

CONCLUSION
In this work we presented an overview on the foundations of fuzzy formal concept analysis and its applications. In the literature, formal concept analysis with fuzzy attributes has gotten a lot of attention. The main focus of the researchers was on developing methods for extended FCA in a fuzzy environment like crisply generated, fuzzy concepts, generalized concept lattice, one-sided fuzzy concept lattice, and employing fuzzy formal concept analysis in domains such as reliability engineering, crime prediction, KDD, IR, and ontology engineering, An major challenge for FCA practical applications is the display of formal concepts in hierarchy structure in the concept lattice structure. One of the major issues in this process is the the large structure of the concept lattice (big line diagram) constructed from a large formal context. The concept lattice constructed from the big context becoming difficult and unworkable. Therefore, it is emphasized as a real difficulty in FCA applications to deal with a big formal context and minimize the size of the concept lattice.
To deal with such problems several techniques to control the complexity and size of a concept lattice have Work's contribution Category [1] By removing reducible objects and attributes, the authors were able to obtain a clarified context, and the resulting concept lattice maintains the isomorphism with the original.
Redundant information removal [47] The authors demonstrated how to factorize concept lattices based on concept similarity. It's also demonstrated how to speed up the computation of similarity relations. They defined and examined the similarity relations at three levels: similarity of objects and attributes, similarity of concepts, and similarity of concept lattices.
Redundant information removal [48] This paper investigates the granular structure of concept lattices and how it can be used to reduce knowledge in formal concept analysis. The properties of information granules are first discussed in a formal context. In the formal context, the concepts of a granular consistent set and granular reducts are introduced.
Redundant information removal [49] The attributes in a decision formal context have been reduced using a homomorphism consistent set from the concept lattice.
Redundant information removal [50] The author proposes a framework for knowledge reduction from a decision formal context that employs rule acquisition to discover a new set of non-redundant decision rules.
Redundant information removal [51] Ciobanu later used the reduction described in [47] to introduce a new reduction in the property-oriented and object-oriented concept lattice frameworks.
Redundant information removal [52] The authors in their article covered two important research topics in FCA: attribute reduction and size reduction in concept lattices. The authors present a procedure that uses an irreducible -cut concept lattice to simultaneously reduce attributes and concept lattice size.
Redundant information removal [53] JBOS (Junction based on object similarity) uses background knowledge to replace similar objects with representative elements that are similar to a certain degree.
Simplification [54] Using fuzzy k means (FKM) clustering, the size of the concept lattices was reduced. The context matrix is reduced, and quotient lattices are obtained using FKM Clustering-derived equivalence relations. Each element is associated with a set of membership levels, and each record can belong to more than one cluster.

Simplification
The authors extended previous work on the correspondence of block relations of formal contexts and complete tolerances on concept lattices to a fuzzy setting and provided an example of how to use block relations to reduce a concept lattice.
Simplification [55] The authors investigated concept lattices in uncertain environments. They investigated the fuzziness in a multivalued context, which is then transformed into fuzzy formal contexts and fuzzy formal concepts. By simplifying the corresponding fuzzy concept lattice structure, they were able to reduce the number of fuzzy formal concepts.
Simplification [56] Based on their characteristics, fuzzy formal contexts are reduced using attribute reduction. The term "one-sided fuzzy concept" is used for the first time. The attributes are divided into three categories: core attributes, relatively important attributes, and unimportant attributes. By virtue of attribute characteristics, an attribute reduction method is presented.
Simplification [57] They've implemented a mechanism to reduce attributes in fuzzy FCA, considering the reduction procedure and tolerance relations introduced in RST. This new method for reducing attributes reduces the original concept lattice significantly. The most important feature of this method is that it partially preserves the structure of the original concept lattice when using this new mechanism, i.e., no new join-irreducible elements appear after the reduction procedure.
Simplification [10] The authors made a significant contribution by connecting frequent items and formal concepts as described in [57].
Selection [58] The authors introduced a type of concept lattice, which are like iceberg concept lattices. Some class restraints are built with attributes in a formal context. The authors named the resulting concept lattice alpha concept lattice. An iceberg concept lattice is formed by an unrestricted lattice, which contains only frequent formal concepts.
Selection [59] The choice of formal concepts in the proposed method is based on the concept of distance or similarity. In the process of selecting important concepts, the concepts of equivalence classes and object or attribute similarity are used.
Selection [4] In this work, each attribute is given a weight to demonstrate its relevance, and thereafter formal concepts that are relevant are chosen. Equal weights are assigned to attributes derived from multivalued attributes to facilitate the application of weights. The sum of the weights of a formal concept's attributes intention divided by the cardinality of its intention determines its importance.
Selection [45] The authors introduce the Titanic algorithm for generating iceberg concept lattices and demonstrate the utility of these lattices in a variety of applications, including large-scale database analysis, extraction of implications, visualization of implications and mining association rules.
Selection [60] For fuzzy formal concepts, the authors presented a similarity metric. In order to choose a subset of formal concepts that are related to one another, the similarity measure is utilized. This subset of formal concepts may be much smaller than the initial set of formal concepts. A measurement of similarities between formal concept extensions is defined as follows. Given two forma concepts ( 1 , 1 ), ( 2 , 2 ), the similarity between the extentions 1 and 2 given by ( 1 , 2 )=1-| 1 ′ ∩ 2 ′ |∕ | 1 ′ ∪ 2 ′ |.| Selection [61] The authors focused on using entropy to reduce the number of formal concepts in formal concept analysis with fuzzy attributes. Furthermore, at a given granulation of the entropy-based attribute intent weight, the number of fuzzy formal concepts is reduced.