Conceptual Identification Within the Decomposition of Fuzzy Homogeneous Classes of Objects Dmytro O. Terletskyi, and Sergey V. Yershov V. M. Glushkov Institute of Cybernetics of NAS of Ukraine, Academician Glushkov avenue, 40, 03187 Kyiv, Ukraine Abstract Conceptual identification of fuzzy knowledge is one of the important knowledge-processing methods, which can be used for such tasks as concept matching, computation of concept similarity, re-engineering of conceptual hierarchies, etc. Since wildly used approaches to conceptual identification, which are based on the formal concept analysis and fuzzy formal concept analysis, do not consider the internal semantic dependencies among the attributes, it may lead to the construction of semantically inconsistent concepts. Therefore, in this paper, we propose a new approach to the conceptual identification of fuzzy knowledge within the decomposition of nodes of fuzzy object-oriented dynamic networks. The decomposition of fuzzy homogeneous classes of objects is considered the space for the identifying their fuzzy sub- concepts within the corresponding identification lattice. To implement the proposed approach, we developed the algorithm for identifying semantically consistent subclasses of fuzzy homogeneous classes of objects. The algorithm constructs a semantically consistent lattice of fuzzy class subclasses and discovers all subclasses and superclasses for a selected fuzzy class subclass, creating a corresponding identification lattice. In addition, we introduce a notion of a subclass neighborhood within its identification lattice, which allows the consideration of a conceptual locus of the subclass instead of the subclass itself. It makes it possible to operate with subclasses of a fuzzy class in a broader sense, calculating their similarities and differences. To explain the proposed approach, we have provided a detailed example of the conceptual identification of a particular fuzzy homogeneous class of objects, demonstrating the application of the developed algorithm. Keywords Fuzzy knowledge identification, fuzzy class identification, fuzzy concept identification, fuzzy class decomposition 1 1. Introduction Nowadays, conceptual (class) hierarchies are the most common complex knowledge representation structure within modern object-oriented knowledge-based systems and programming languages. It provides an opportunity to formalize a particular domain via constructing a corresponding hierarchy of concepts that encapsulates the representation of concepts themselves and the relations among them. The knowledge-based systems can use such hierarchies for conceptual knowledge processing, including representation, analysis, COLINS-2024: 8th International Conference on Computational Linguistics and Intelligent Systems, April 12-13, 2024, Lviv, Ukraine Dmytro.terletskyi@nas.gov.ua (D. O. Terletskyi); ErshovSV@nas.gov.ua (S. V. Yershov) 0000-0003-7393-1426 (D. O. Terletskyi); 0000-0002-9895-777X (S. V. Yershov) © 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). CEUR ceur-ws.org Workshop ISSN 1613-0073 Proceedings classification, integration, identification, retrieval, inferring, and transferring. Concept identification is one of the important tasks related to knowledge analysis, integration, retrieval, and inferring since it allows the system to detect a place of particular concepts and how they are connected with other concepts in the hierarchy. Consequently, a concept can be considered not only as a single node from the hierarchy but also as its neighborhood or sub-hierarchy, which includes some number of adjacent nodes and relations among them. Using such a representation, the system can operate by concepts in the broader meaning, for example, for computation of similarity or difference of certain concepts from the same hierarchy or a few different hierarchies. In addition, a hierarchical neighborhood of a concept can be used to reduce a hierarchy representation and consequently the search space during the concept retrieval or inferring, as well as for detecting the best place for integrating a new concept into a hierarchy. Conceptual identification has a few interpretations depending on the specifics of a certain hierarchy, the nature of the concepts, and the relations among them. Since sometimes concepts themselves, as well as the relations among them, can be vague and imprecise, knowledge-based systems should be able to perform the identification of fuzzy concepts. Therefore, in this paper, we study the identification of fuzzy concepts in fuzzy object-oriented dynamic networks, considering the decomposition of fuzzy homogeneous classes of objects, which are nodes of the networks, as spaces for the identification of fuzzy sub-concepts. As a result, we propose a new approach to identifying semantically consistent fuzzy sub-concepts of fuzzy homogeneous classes of objects. The paper has the following structure. Section 2 contains the analysis of the main approaches to the conceptual identification of fuzzy knowledge. Section 3 presents a morphological analysis of a particular fuzzy homogeneous class of objects. Section 4 provides an approach to reducing the space for identifying fuzzy subclasses via the semantically consistent decomposition of the fuzzy homogeneous class of objects. Section 5 presents the algorithm for identifying fuzzy sub-concepts and an example of its application to identifying semantically consistent subclasses of the fuzzy homogeneous class of objects. The conclusions and acknowledgments sections finish the paper. 2. Conceptual Identification Nowadays, one of the most common approaches to formal representation of concept hierarchies is a formal concept analysis (FCA) proposed by Ganter and Wille in [5]. It is a powerful framework that focuses on representing concept hierarchies in terms of so-called, concept lattices. Such representation of the essences from a chosen domain consists of three main stages. Firstly, to define a formal context for the domain by constructing the cross- table with a set of attributes of domain entities as columns and the set of objects, which model their particular instances, as rows. The cells of such a table usually contain a Boolean value meaning that a particular object has or does not have a corresponding attribute. Secondly, to define formal concepts within the formal context by constructing a collection of pairs of appropriate extents and intents, where the extent is a set of common attributes for the particular set of objects, while the intent is a set of objects with common attributes. And finally, to build a corresponding concept lattice by constructing a complete lattice of objects and a complete lattice of attributes, which are isomorphic to each other. The lattice structure assumes that a set of formal concepts is a partially ordered set with defined sub- concept-super-concept relations. Later, the FCA framework was extended for the representation of fuzzy domains (FFCA), consequently, notions of fuzzy formal context, fuzzy formal concepts, and fuzzy concept lattice were introduced [3, 11]. In contrast to the crisp formal concepts, the fuzzy ones are defined using the confidence threshold, which allows modeling the membership measure of a particular attribute for a certain object, and a certain object for the set of objects. Constructing a concept lattice or fuzzy concept lattice for a particular domain creates its lattice-based formal model that can be used for conceptual identification. According to [20], conceptual identification is the detection of the taxonomic position of a particular object within a certain classification. In the case of FCA/FFCA, the concept lattice is used as such classification, therefore identification of a concept transforms into the detection of sub-concepts and super-concepts within the lattice. One of the commonly used approaches to conceptual identification is rule-based identification. The main idea of the approach is to define a system of implication rules extracting them from the defined formal context and corresponding concept lattice. In general, an implication rule can be defined in the form P → Q , where P and Q are subsets of attributes of the set of all tributes used to determine a formal context or a fuzzy formal context. In the case of FCA, an implication rule can be interpreted in the following way – if an object has all attributes from the set P , it also has all attributes from the set Q . This approach was used to identify: a set of professional competencies that can help people successfully take a new position when professional retraining or changing jobs [15]; conservative access patterns, minimum behavior patterns, and canonical access patterns in two-mode social networks [13]. In the case of FFCA, an implication rule can be interpreted as follows – if a fuzzy object has all fuzzy attributes from the set P to the corresponding degree, then it also has all fuzzy attributes from the set Q to the corresponding degree. This version of the approach was used to identify: differential diagnoses for patients by a conversational recommender system [2]; causes and consequences of customer complaints within customer relationship management in financial services helping managers to accommodate the required dynamic changes according to customer expectations [14]; exceptional or suspicious cases specific to the event logs, NTFS file system, the Windows operating system, or a type of anomaly, to provide warnings for the security analysts [16-17]. However, the approach assumes the rules extraction from the formal context and corresponding concept lattice analyzing sub- concept and super-concept relations. In the case of big formal contexts, this task becomes more complicated from the computational perspective. Moreover, to identify specific concepts within a concept lattice, the system must discover in the set of rules those rules that are associated with these concepts, including all transitive rules. Another approach to conceptual identification is the multi-stage intersection identification of formal concepts. The approach involves extracting new concepts via the sequence-based intersection of formal concepts within a constructed formal context. The discovered hidden concepts are identified and then integrated into the classification, where identification of the concepts means retrieval of their sub-concepts and super-concepts. Such integration extends the initial formal context enriching it with previously non-obvious or hidden concepts. The approach was used to detect missing or hidden concepts and improve the completeness of concept coverage in biomedical terminologies NCI Thesaurus and SNOMED CT [23-24]. However, the larger the size of the formal context, the more difficult identification becomes due to the increasing number of intersections being calculated. One more approach to conceptual identification is the criterion-based identification of a group of concepts. The main idea of the approach is to identify a group of concepts within a formal context or fuzzy formal context, which satisfies particular identification criteria. The FCA-version of the approach was used to identify: key nodes in massive networks using cross-face scalable centrality measure [9]; key nodes in a two-mode network, using bi-face bipartite centrality measure [10]; diversified top- k maximal clique in a social Internet of things [8]; dynamic maximal clique in online social networks [21]; user-friendly communities in signed social networks and  -quasi-cliques for closely related users within them [22]; key structures from social networks [6]. The FFCA-version of the approach was used for the identification of location-based and content-based communities of users in social networks [4]; skyline (  , k ) -cliques in a fuzzy attributed social network [7]; cloud services in collaborative filtering-based recommendation system [12]. Each of the considered approaches implements a specific strategy for solving the problem of conceptual identification based on FCA/FFCA. In all the mentioned FCA/FFCA applications, the formal context and the fuzzy formal context were constructed using a set of objects and a set of attributes. However, this approach has one major drawback: if we consider objects as instances of a particular class of objects, this means, that they are encapsulated containers for storing data, and we do not see how their attributes are defined relative to each other. It is known that the attributes of all class objects are defined at the class level, and, as was shown in [18-19], some attributes (properties and methods) of a class may have internal semantic dependencies on other attributes. It is crucial for the semantic consistency of formal concepts within the constructed concept lattice or fuzzy concept lattice because the construction of new formal concepts is based on the set- theoretical intersection of extents ignoring internal semantic dependencies among the attributes [19]. Consequently, some constructed formal concepts may be semantically inconsistent and, therefore, physically impossible or unrealistic in a modeled domain. Another feature of FCA/FFCA is the fact that attributes are treated only as properties of objects, not as methods defined within an object class, and can be executed on all objects to change their state and attribute values. Therefore, we propose an alternative lattice-based approach to the conceptual identification of fuzzy knowledge, based on the analysis of internal semantic dependencies between the attributes (fuzzy properties and fuzzy methods) of fuzzy homogeneous classes of objects. In addition, we introduce the notion of concept' neighborhood, which allows the consideration of some subclass and superclass locus within a concept lattice instead of the single concept. 3. Fuzzy Concepts Morphology To explain our approach to conceptual identification of fuzzy knowledge, we use the fuzzy homogeneous classes of objects which are nodes of fuzzy object-oriented dynamic networks. The formalization of internal semantic dependencies among the attributes (properties and methods) of fuzzy homogeneous classes of objects was introduced in [18]. For this purpose, the abstract model of chemical atoms and molecules was used, according to which, atoms are indivisible particles and molecules are the union of atoms and (or) smaller molecules. This model can be interpreted by attributes defined independently of other fuzzy class attributes (fuzzy atoms) and attributes defined based on them (fuzzy molecules). Definition 1. A fuzzy atom of a fuzzy homogeneous class of objects T / M (T ) is a singleton collection Ai (T / M (T ) ) = T .xi /  (T .xi ) , ( ) ( ) where T .xi /  (T .xi )  P (T ) / M P (T )  F (T ) / M F (T ) is a crisp or fuzzy property or a method defined without using any other properties and (or) methods of the fuzzy class T / M (T ) , where P (T ) / M ( P (T ) ) and F (T ) / M ( F (T ) ) are collections of its properties and methods, respectively. Definition 2. A fuzzy molecule of a fuzzy homogeneous class of objects T / M (T ) is a collection  ( ) ( M i (T / M (T ) ) = T .xi /  (T .xi ) , T . y j1 /  T . y j1 ,..., T . y jn /  T . y jn ) , ( ) ( ) where T .xi /  (T .xi )  P (T ) / M P (T )  F (T ) / M F (T ) , and 1  i  T / M (T ) is a crisp or fuzzy property or a method defined based on the other methods and (or) properties ( ) ( ) T . y j1 /  T . y j1 ,..., T . y jn /  T . y jn  P (T ) / M ( P (T ) )  F (T ) / M ( F (T ) ) which are crisp or fuzzy atoms and (or) parts of smaller fuzzy molecules of the fuzzy class T / M (T ) , where 1  j1  ...  jn  T / M (T ) , where P (T ) / M ( P (T ) ) and F (T ) / M ( F (T ) ) are collections of properties and methods of the fuzzy class T / M (T ) , respectively. Fuzzy atoms and fuzzy molecules of a fuzzy homogeneous class of objects, together determine its internal semantic dependencies. Definition 3. Internal semantic dependencies of a fuzzy homogeneous class of objects T / M (T ) is a set of its atoms and molecules, i.e.  ISD (T / M (T ) ) = A1 (T / M (T ) ) ,..., An (T / M (T ) ) , M1 (T / M (T ) ) ,..., M m (T / M (T ) ) ( ) where Ai T / M (T ) , i = 1, n are fuzzy atoms of the fuzzy class T / M (T ) , while M j (T / M (T ) ) , j = 1, m are its fuzzy molecules, respectively. Now, let us consider an example of a fuzzy homogeneous class of objects Gp / 0.9 , which defines the concept of a geographic place and has the following structure: ( Gp p1 = ( latitude, ( , ) ) / 0.91, p2 = ( longitude, (  , ) ) / 0.91, p3 = ( name, ( n  Vn , str ) ) / 0.71, p4 = ( region, ( r Vr , str ) ) / 0.78, f1 = get _ latitude ( gp, ) / 0.95, f 2 = get _ longitude ( gp, ) / 0.95, f3 = get _ name ( gp, str ) /1, f 4 = get _ region ( gp, str ) /1) / 0.9 where Gp. p1 / 0.91 and Gp. p2 / 0.91 are fuzzy quantitative properties, which mean the latitude and longitude of the geographic place gp /  ( gp ) defined in degrees, i.e. gp.latitude =  /  ( ) , gp.longitude =  /  (  ) ; Gp. p3 / 0.71 is a fuzzy quantitative property which means the name of the geographic place gp /  ( gp ) , and is defined by the following fuzzy set Vn = official _ name / 0.95, historical _ name / 0.74, regional _ name / 0.62, local _ name / 0.55 , where elements of the fuzzy set Vn are corresponding names of the geographic place gp /  ( gp ) , i.e. gp.name = n /  ( n ) Vn ; Gp. p4 / 0.78 is a fuzzy quantitative property, which means the region name where the geographic place gp /  ( gp ) is located, and is defined by the following fuzzy set Vr = official _ name / 0.95, historical _ name / 0.73, local _ name / 0.56 , where elements of the fuzzy set Vr are corresponding names of the region where the geographic place gp /  ( gp ) is located, i.e. gp.region = r /  ( r ) Vr ; Gp. f1 / 0.95 , Gp. f 2 / 0.95 , Gp. f3 /1 , and Gp. f 4 /1 are fuzzy methods, which return values of corresponding properties of the geographic place gp /  ( gp ) , i.e. gp.get _ latitude() → round ( gp.latitude, 1) , gp.get _ longitude() → round ( gp.longitude, 1) , gp.get _ name() → gp.name, gp.get _ region() → gp.region. Let us use the fuzzy homogeneous class of objects Gp / 0.9 to define another fuzzy homogeneous class of objects Tr / 0.95 , which determines the concept of transfer from one geographic place to another and has the following structure: ( Tr p1 = ( place1 , ( gp1 , Gp ) ) /1, p2 = ( place2 , ( gp2 , Gp ) ) /1, p3 = ( distance, ( dst , km ) ) / 0.94, p4 = ( transport , ( t  Vt , str ) ) / 0.81, ( ) p5 = ( duration, ( dr , h ) ) / 0.87, p6 = price, ( p  V p , UAH ) / 0.91, f1 = get _ place1 ( tr , GP ) /1, f 2 = get _ place2 ( tr , GP ) /1, f3 = get _ distance ( tr , km ) / 0.98, f 4 = get _ transport ( tr , str ) /1, f5 = get _ duration ( tr , h ) / 0.94, f 6 = get _ price ( tr , UAH ) / 0.96 ) / 0.95 where Tr. p1 /1 and Tr. p2 /1 are fuzzy quantitative properties, which define two different geographic places for the transfer tr /  ( tr ) between them, i.e. tr. place1 = gp1 /  ( gp1 ) , tr. place2 = gp2 /  ( gp2 ) ; Tr. p3 / 0.94 is a fuzzy quantitative property, which means a distance between two geographic places Tr. p1 /1 and Tr. p2 /1 , and is defined in the following way tr.distance = d1 + d 2 , d1 = ( tr. place1.get _ latitude() − tr. place2 .get _ latitude() ) , 2 d1 = ( tr. place1.get _ longitude() − tr. place2 .get _ longitude() ) ; 2 Tr. p4 / 0.81 is a fuzzy quantitative property, which means a kind of transport for a transfer between two geographic places and is defined as the following fuzzy set Vt = bus / 0.82, train / 0.67, plane / 0.93 , where elements of the fuzzy set Vt are possible kinds of transport for a transfer, i.e. tr.transport = t /  ( t ) Vt ; Tr. p5 / 0.87 is a fuzzy quantitative property, which means duration of the transfer between two geographic places and is defined in the following way  Db , if tr.transport = bus / 0.82,  tr.duration =  Dt , if tr.transport = train / 0.67,  D , if tr.transport = plane / 0.93,  p −  − ( ) where Db = xi /  xi , durationb /1, xi /  xi + + ( ) , i = 1,... is a fuzzy set, such that tr.distance tr.distance  durationb  , 120 80 tr.distance xi− = durationb − 4  i,  durationb − 4  i  durationb , 120 xi− 120 − tr.distance  ( xi− ) = −  i− ,  i− = 1 −  ( xi− ) − ( xi− ) ,  ( xi− ) = 1 −  ( xi− ) , durationb 120 − tr.distance tr.distance xi+ = durationb + 4  i, durationb  durationb + 4  i  , 80 tr.distance − xi+  80  (x ) = + −  i+ ,  i+ = 1 −  ( xi+ ) − ( xi+ ) ,  ( xi+ ) = 1 −  ( xi+ ) , tr.distance − durationb  80 i   Dt = xi− /  ( xi− ) , durationt /1, xi+ /  ( xi+ ) , i = 1,... is a fuzzy set, such that tr.distance tr.distance  durationt  , 80 50 tr.distance xi− = durationt − 3  i,  durationt − 3  i  durationt , 80 xi−  80 − tr.distance  ( xi− ) = −  i− ,  i− = 1 −  ( xi− ) − ( xi− ) ,  ( xi− ) = 1 −  ( xi− ) , durationt  80 − tr.distance tr.distance xi+ = durationt + 3  i, durationt  durationt + 3  i  , 50 tr.distance − xi+  50  ( xi+ ) = −  i+ ,  i+ = 1 −  ( xi+ ) − ( xi+ ) ,  ( xi+ ) = 1 −  ( xi+ ) , tr.distance − durationt  50 −  − ( ) and Dp = xi /  xi , duration p /1, xi /  xi + + ( ) , i = 1,... is a fuzzy set, such that tr.distance tr.distance  duration p  , 500 450 tr.distance xi− = duration p − 5  i,  duration p − 5  i  duration p , 500 xi−  500 − tr.distance  ( xi− ) = −  i− ,  i− = 1 −  ( xi− ) − ( xi− ) ,  ( xi− ) = 1 −  ( xi− ) , duration p  500 − tr.distance tr.distance xi+ = duration p + 5  i, duration p  duration p + 5  i  , 450 tr.distance − xi+  450  ( xi+ ) = −  i+ ,  i+ = 1 −  ( xi+ ) − ( xi+ ) ,  ( xi+ ) = 1 −  ( xi+ ) ; tr.distance − duration p  450 Tr. p6 / 0.91 is a fuzzy quantitative property, which means a price of a transfer between two geographic places and is defined as follows  Pb , if tr.transport = bus / 0.82,  tr. price =  Pt , if tr.transport = train / 0.67,  P , if tr.transport = plane / 0.93;  p − − ( ) where Pb = xi /  xi , priceb /1, xi /  xi + + ( ) , i = 1,... is a fuzzy set, such that tr.distance  20  priceb  tr.distance  60, xi− = priceb − 4  i, tr.distance  20  priceb − 4  i  priceb , xi− − tr.distance  20  (x ) = − −  i− ,  i− = 1 −  ( xi− ) − ( xi− ) ,  ( xi− ) = 1 −  ( xi− ) , priceb − tr.distance  20 i xi+ = priceb + 4  i, priceb  priceb + 4  i  tr.distance  60, tr.distance  60 − xi+  ( xi+ ) = −  i+ ,  i+ = 1 −  ( xi+ ) − ( xi+ ) ,  ( xi+ ) = 1 −  ( xi+ ) , tr.distance  60 − priceb   Pt = xi− /  ( xi− ) , pricet /1, xi+ /  ( xi+ ) , i = 1,... is a fuzzy set, such that tr.distance 15  pricet  tr.distance  35, xi− = pricet − 2  i, tr.distance 15  pricet − 2  i  pricet , xi− − tr.distance 15  ( xi− ) = −  i− ,  i− = 1 −  ( xi− ) − ( xi− ) ,  ( xi− ) = 1 −  ( xi− ) , pricet − tr.distance 15 xi+ = pricet + 2  i, pricet  pricet + 2  i  tr.distance  35, tr.distance  35 − xi+  ( xi+ ) = −  i+ ,  i+ = 1 −  ( xi+ ) − ( xi+ ) ,  ( xi+ ) = 1 −  ( xi+ ) , tr.distance  35 − pricet −  − ( ) and Pp = xi /  xi , price p /1, xi /  xi + + ( ) , i = 1,... is a fuzzy set, such that tr.distance  55  price p  tr.distance  85, xi− = price p − 3  i, tr.distance  55  price p − 3  i  price p , xi− − tr.distance  55  ( xi− ) = −  i− ,  i− = 1 −  ( xi− ) − ( xi− ) ,  ( xi− ) = 1 −  ( xi− ) , price p − tr.distance  55 xi+ = price p + 3  i, price p  price p + 3  i  tr.distance  85, tr.distance  85 − xi+  ( xi+ ) = −  i+ ,  i+ = 1 −  ( xi+ ) − ( xi+ ) ,  ( xi+ ) = 1 −  ( xi+ ) ; tr.distance  85 − price p Tr. f1 /1 , Tr. f 2 /1 , Tr. f3 / 0.98 , Tr. f 4 /1 , Tr. f5 / 0.94 , and Tr. f6 / 0.96 are fuzzy methods, which return values of corresponding properties of the transfer tr /  ( tr ) , i.e. tr.get _ place1 () → tr. place1 , tr.get _ place2 () → tr. place2 , tr.get _ distance() → round ( tr.distance, 0 ) , tr.get _ transport () → tr.transport , tr.get _ duration() → tr.duration, tr.get _ price() → tr. price. To enrich the example, let us define a fuzzy homogeneous class of objects Jrn / 0.87 , which determines the concept of a journey through the sequence of geographic places and has the following structure: ( ( ) Jrn p1 = transfers, ( ( tr1 , Tr ) ,..., ( trn , Tr ) ) / 0.93, p2 = ( distannce, ( dst , km ) ) / 0.88 p3 = ( duration, ( dr , h ) ) / 0.79, p4 = ( price, ( p, UAH ) ) / 0.84, f1 = get _ transfer ( jrn, i, Tr ) /1, f 2 = get_distance ( jrn, km ) / 0.91, f3 = get _ duration ( jrn, h ) / 0.89, f 4 = get _ price ( jrn, UAH ) / 0.82, f5 = compute _ discount ( jrn, UAH ) / 0.78 ) / 0.87 where Jrn. p1 / 0.93 is a fuzzy quantitative property, defining a sequence of transfers between different geographic places in the scope of the journey jrn /  ( jrn ) , i.e. jrn.transfers = ( tr1 /  ( tr1 ) ,..., trn /  ( trn ) ) , such that tri . place2 = tri +1. place1 , i = 1, n − 1 ; Jrn. p2 / 0.88 is a fuzzy quantitative property, meaning the total distance of the transfer, during the journey jrn /  ( jrn ) , and is defined in the following way n jrn.distance =  jrn.transfers[i].get _ distance(); i =1 Jrn. p3 / 0.79 is a fuzzy quantitative property, meaning the total duration of the transfer, during the journey jrn /  ( jrn ) , and is defined in the following way n a jrn.duration =  , i =1 b m a =   ( jrn.transfers[i ].get _ duration()[ j ])  jrn.transfers[i].get _ duration()[ j ], j =1 m b =   ( jrn.transfers[i ].get _ durtion()[ j ]) , j =1 where n = jrn.transfers , m = jrn.transfers[i].duration ; Jrn. p4 / 0.84 is a fuzzy quantitative property, meaning the total price of the transfer, during the journey jrn /  ( jrn ) , and is defined in the following way n a jrn. price =    − jrn.compute _ diccount (), i =1  b  m a =   ( jrn.transfers[i].get _ price()[ j ])  jrn.transfers[i].get _ price()[ j], j =1 m b =   ( jrn.transfers[i ].get _ price()[ j ]) , j =1 where n = jrn.transfers , m = jrn.transfers[i]. price ; Jrn. f1 /1 , Jrn. f 2 / 0.91 , Jrn. f3 / 0.89 , and Jrn. f 4 / 0.82 are fuzzy methods, returning values for corresponding properties of the journey object jrn /  ( jrn ) , i.e. jrn.get _ transfer (i ) → jrn.transfers[i ], jrn.get _ distance() → jrn.distance, jrn.get _ duration() → round ( jrn.duration, 0 ) , jrn.get _ price() → round ( jrn. price, 1) ; Jrn. f5 / 0.87 is a fuzzy method, calculating a discount on the price of a journey, depending on its distance, and that is defined as follows n jrn.compute _ discount () →  jrn.transfers[i].price  d , i =1 where d is defined by the following set of rules 0.15, if jrn.transfer[i ].distance  1000,  0.12, if jrn.transfer[i ].distance  800,   0.09, if jrn.transfer[i ].distance  600, d =  0.05, if jrn.transfer[i ].distance  500,  0.02, if jrn.transfer[i ].distance  300,   0.00, if jrn.transfer[i ].distance  300. Let us analyze the properties and methods of the fuzzy homogeneous class of objects Jrn / 0.87 , to detect its internal semantic dependencies. As we see, fuzzy property Jrn. p1 / 0.93 meaning transfer between two geographic places, is defined without using any other class members, therefore, it determines a corresponding fuzzy atom of the fuzzy class Jrn / 0.87 , i.e. A1 ( Jrn / 0.87 ) = Jrn. p1 / 0.93. Fuzzy property Jrn. p2 / 0.88 , meaning the total distance of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M1 ( Jrn / 0.87 ) = ( Jrn. p2 / 0.88, Jrn. p1 / 0.93) . Fuzzy property Jrn. p3 / 0.79 , meaning the total duration of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M 2 ( Jrn / 0.87 ) = ( Jrn. p3 / 0.79, Jrn. p1 / 0.93) . Fuzzy method Jrn. f1 /1 , returning transfers list during the journey, is defined using the property Jrn. p1 / 0.93 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M 3 ( Jrn / 0.87 ) = ( Jrn. f1 /1, Jrn. p1 / 0.93) . Fuzzy method Jrn. f5 / 0.78 , returning the total price of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M 4 ( Jrn / 0.87 ) = ( Jrn. f5 / 0.78, Jrn. p1 / 0.93) . Fuzzy property Jrn. p4 / 0.84 , meaning the total price of the transfer during the journey, is defined using the property Jrn. p1 / 0.93 and fuzzy method Jrn. f5 / 0.78 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M 5 ( Jrn / 0.87 ) = ( Jrn. p4 / 0.84, Jrn. f5 / 0.78, Jrn. p1 / 0.93) . Fuzzy method Jrn. f 2 / 0.91 , returning the total distance of the transfer during the journey, is defined using the property Jrn. p2 / 0.88 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M 6 ( Jrn / 0.87 ) = ( Jrn. f 2 / 0.91, Jrn. p2 / 0.88, Jrn. p1 / 0.93) . Fuzzy method Jrn. f3 / 0.89 , returning the discount on the price of a journey, is defined using the property Jrn. p3 / 0.79 , therefore, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M 7 ( Jrn / 0.87 ) = ( Jrn. f3 / 0.89, Jrn. p3 / 0.79, Jrn. p1 / 0.93) . Fuzzy method Jrn. f 4 / 0.82 , returning the total price of the transfer during the journey, is defined using the properties Jrn. p4 / 0.84 , Jrn. p1 / 0.93 and fuzzy method Jrn. f5 / 0.78 , consequently, it determines a fuzzy molecule of the fuzzy class Jrn / 0.87 , i.e. M 8 ( Jrn / 0.87 ) = ( Jrn. f 4 / 0.8, Jrn. p4 / 0.84, Jrn. f5 / 0.78, Jrn. p1 / 0.93) . All atoms and molecules of the class Jrn / 0.87 , define its internal dependencies, i.e. ISD ( Jrn / 0.87 ) =  A1 ( Jrn / 0.87 ) , M1 ( Jrn / 0.87 ) ,..., M 8 ( Jrn / 0.87 ). Analyzing detected internal semantic dependencies of the fuzzy class Jrn / 0.87 , we can find some similarities and intersections among them. To observe the connections among different dependencies we visualized them in Fig. 1. The orange nodes depict corresponding internal semantic dependencies, while the violet ones mean the attributes of the fuzzy class. As we can see, all molecules of the fuzzy class Jrn / 0.87 contain the atom A1 ( Jrn / 0.87 ) , i.e. A1 ( Jrn / 0.87 )  M i ( Jrn / 0.87 ) , where i = 1,8 , while the bigger molecules contain some of the smaller ones, i.e. M 1 ( Jrn / 0.87 )  M 6 ( Jrn / 0.87 ) , M 2 ( Jrn / 0.87 )  M 7 ( Jrn / 0.87 ) , M 4 ( Jrn / 0.87 )  M 5 ( Jrn / 0.87 ) , M 4 ( Jrn / 0.87 )  M 8 ( Jrn / 0.87 ) , M 5 ( Jrn / 0.87 )  M 8 ( Jrn / 0.87 ) . The directed arrows mean the dependencies between a pair of attributes. Figure 1. Internal semantic dependencies of the fuzzy homogeneous class of objects Jrn / 0.87 . Analyzing the structure of each internal semantic dependency, illustrated in Fig. 1, we can construct a dependency graph combining all of them, i.e. G ( Jrn / 0.87 ) = ( A ( Jrn / 0.87 ) , DL ( Jrn / 0.87 ) ) , where A ( Jrn / 0.87 ) is a set of attributes of the fuzzy class Jrn / 0.87 , i.e. A ( Jrn / 0.87 ) =  Jrn. p1 / 0.93, Jrn. p2 / 0.88, Jrn. p3 / 0.79, Jrn. p4 / 0.84, Jrn. f1 /1, Jrn. f 2 / 0.91, Jrn. f3 / 0.89, Jrn. f 4 / 0.82, Jrn. f5 / 078 , and DL ( Jrn / 0.87 ) is a set of dependency links among the attributes of the fuzzy class Jrn / 0.87 , i.e. DL ( Jrn / 0.87 ) =  Jrn. p1 / 0.93 ⊥, Jrn. p2 / 0.88  Jrn. p1 / 0.93, Jrn. p3 / 0.79  Jrn. p1 / 0.93, Jrn. f1 /1  Jrn. p1 / 0.93, Jrn. f 5 / 0.78  Jrn. p1 / 0.93, Jrn. p4 / 0.84  Jrn. f5 / 0.78, Jrn. f 2 / 0.91  Jrn. p2 / 0.88, Jrn. f 3 / 0.89  Jrn. p3 / 0.79, Jrn. f 4 / 0.82  Jrn. p4 / 0.84 . The graph of internal semantic dependencies G ( Jrn / 0.87 ) is represented in Fig. 2. Violet nodes represent attributes of a fuzzy class Jrn / 0.87 , edges depict dependency relations among the attributes, and edge titles mean the numbers of the molecules, which contain corresponding dependencies. To simplify the graph, we denote its nodes using only attribute identifiers. Figure 2. Graph of internal semantic dependencies of the fuzzy homogeneous class of objects Jrn / 0.87 . Such representation of the internal semantic dependencies of the class Jrn / 0.87 allows us to use graph-based interpretation of its subclasses. Therefore, each subgraph of graph G ( Jrn / 0.87 ) defines an appropriate subclass of the fuzzy class Jrn / 0.87 . However, as we mentioned above, not all subclasses are semantically consistent, i.e. do not contradict the internal semantic dependencies of a fuzzy class. Therefore, let us define the satisfiability of internal semantic dependencies for the fuzzy homogeneous class of objects. Definition 4. Any subclass SC (T ) / M ( SC ) of a fuzzy homogeneous class of objects T / M (T ) is semantically consistent if only if its graph of internal semantic dependencies ( ( G ( SC (T ) / M ( SC ) ) = A ( SC (T ) / M ( SC ) ) , DL A ( SC (T ) / M ( SC ) ) , )) ( ) ( ) ( ) where A SC (T ) / M ( SC )  A T / M (T ) , DL SC (T ) / M ( SC )  DL T / M (T ) , ( ) satisfies the following conditions:  ( u, v )  DL (T / M (T ) ) | ( u , v )  DL ( SC (T ) / M ( SC ) ) → → u  A ( SC (T ) / M ( SC ) )  v  A ( SC (T ) / M ( SC ) ) , v  A (T / M (T ) ) | v  A ( SC (T ) / M ( SC ) )  u  A (T / M (T ) )   ( u, v )  DL (T / M (T ) ) → ( u , v )  DL ( SC (T ) / M ( SC ) ) . Those subclasses whose internal semantic dependency graphs do not satisfy these conditions are semantically inconsistent. 4. Conceptual Identification Space Reducing Let us compute the complete decomposition D ( Jrn / 0.87 ) of the fuzzy class Jrn / 0.87 , using an algorithm for decomposing fuzzy homogeneous classes of objects via constraint- based filtering, proposed in [12], with the following configuration (T / M (T ) = Jrn / 0.87, C = ISD ( Jrn / 0.87) , N = 1,...,8 , M = [0,1],  = 2) , where T / M (T ) is a fuzzy homogeneous class of objects, C is a set of constraints (internal semantic dependencies) defined by molecules of the class T / M (T ) , N is a sequence of required subclasses cardinalities, M is an interval that defines admitted fuzziness of each subclass, and  is an accuracy to calculate a measure of fuzziness for subclasses. As a result, the algorithm constructed 71 semantically consistent proper non-empty subclasses of the fuzzy class Jrn / 0.87 , among 510 possible ones, namely 1 subclass of cardinality 1, i.e. SC11 ( Jrn ) / 0.93 = ( p1 / 0.93) , 4 subclasses of cardinality 2, i.e. SC12 ( Jrn ) / 0.91 = ( p1 / 0.93, p2 / 0.88 ) , SC22 ( Jrn ) / 0.86 = ( p1 / 0.93, p3 / 0.79 ) , SC72 ( Jrn ) / 0.97 = ( p1 / 0.93, f1 /1) , SC292 ( Jrn ) / 0.86 = ( p1 / 0.93, f 5 / 0.78 ) , 9 subclasses of cardinality 3, i.e. SC13 ( Jrn ) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79 ) , SC53 ( Jrn ) / 0.94 = ( p1 / 0.93, p2 / 0.88, f1 /1) , SC63 ( Jrn ) / 0.91 = ( p1 / 0.93, p3 / 0.79, f1 /1) , SC113 ( Jrn ) / 0.91 = ( p1 / 0.93, p2 / 0.88, f 2 / 0.91) , 3 SC22 ( Jrn ) / 0.87 = ( p1 / 0.93, p3 / 0.79, f3 / 0.89 ) , SC573 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, f5 / 0.78 ) , SC583 ( Jrn ) / 0.83 = ( p1 / 0.93, p3 / 0.79, f5 / 0.78 ) , 3 SC60 ( Jrn ) / 0.85 = ( p1 / 0.93, p4 / 0.84, f5 / 0.78 ) , 3 SC63 ( Jrn ) / 0.9 = ( p1 / 0.93, f1 /1, f5 / 0.78 ) , 14 subclasses of cardinality 4. i.e. SC24 ( Jrn ) / 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1) , SC64 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f 2 / 0.91) , SC104 ( Jrn ) / 0.93 = ( p1 / 0.93, p2 / 0.88, f1 /1, f 2 / 0.91) , SC164 ( Jrn ) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f 3 / 0.89 ) , SC214 ( Jrn ) / 0.9 = ( p1 / 0.93, p3 / 0.79, f1 /1, f 3 / 0.89 ) , SC714 ( Jrn ) / 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f 5 / 0.78 ) , SC724 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f 5 / 0.78 ) , SC734 ( Jrn ) / 0.84 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f 5 / 0.78 ) , SC754 ( Jrn ) / 0.9 = ( p1 / 0.93, p2 / 0.88, f1 /1, f 5 / 0.78 ) , SC746 ( Jrn ) / 0.88 = ( p1 / 0.93, p3 / 0.79, f1 /1, f 5 / 0.78 ) , SC784 ( Jrn ) / 0.89 = ( p1 / 0.93, p4 / 0.84, f1 /1, f 5 / 0.78 ) , SC814 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, f 2 / 0.91, f 5 / 0.78 ) , SC924 ( Jrn ) / 0.85 = ( p1 / 0.93, p3 / 0.79, f 3 / 0.89, f 5 / 0.78 ) , 4 SC109 ( Jrn ) / 0.84 = ( p1 / 0.93, p4 / 0.84, f 4 / 0.82, f5 / 0.78 ) , 16 subclasses of cardinality 5. i.e. SC35 ( Jrn ) / 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f 2 / 0.91) , SC85 ( Jrn ) / 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f 3 / 0.89 ) , SC125 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f 2 / 0.91, f 3 / 0.89 ) , SC575 ( Jrn ) / 0.84 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f5 / 0.78 ) , SC585 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f5 / 0.78 ) , SC595 ( Jrn ) / 0.89 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f5 / 0.78 ) , 5 SC60 ( Jrn ) / 0.87 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f5 / 0.78) , 5 SC62 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f 2 / 0.91, f5 / 0.78) , 5 SC63 ( Jrn ) / 0.87 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f 2 / 0.91, f5 / 0.78) , 5 SC66 ( Jrn ) / 0.9 = ( p1 / 0.93, p2 / 0.88, f1 /1, f 2 / 0.91, f5 / 0.78) , 5 SC72 ( Jrn ) / 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f3 / 0.89, f5 / 0.78) , 5 SC74 ( Jrn ) / 0.85 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f3 / 0.89, f5 / 0.78) , 5 SC77 ( Jrn ) / 0.88 = ( p1 / 0.93, p3 / 0.79, f1 /1, f3 / 0.89, f5 / 0.78) , SC935 ( Jrn ) / 0.85 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f 4 / 0.82, f 5 / 0.78 ) , SC945 ( Jrn ) / 0.83 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f 4 / 0.82, f5 / 0.78 ) , 5 SC99 ( Jrn ) / 0.87 = ( p1 / 0.93, p4 / 0.84, f1 /1 f4 / 0.82, f5 / 0.78) , 14 subclasses of cardinality 6. i.e. SC46 ( Jrn ) / 0.9 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f 2 / 0.91, f3 / 0.89 ) , 6 SC29 ( Jrn ) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f5 / 0.78 ) , SC306 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f 2 / 0.91, f 5 / 0.78 ) , SC316 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f 2 / 0.91, f 5 / 0.78 ) , SC326 ( Jrn ) / 0.9 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f 2 / 0.91, f 5 / 0.78 ) , SC356 ( Jrn ) / 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f3 / 0.89, f5 / 0.78 ) , SC366 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f3 / 0.89, f5 / 0.87 ) , SC386 ( Jrn ) / 0.87 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f 3 / 0.89, f 5 / 0.78 ) , 6 SC40 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f 2 / 0.91, f3 / 0.89, f5 / 0.78) , SC506 ( Jrn ) / 0.84 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f 4 / 0.82, f 5 / 0.78 ) , SC526 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f 4 / 0.82, f 5 / 0.78 ) , SC536 ( Jrn ) / 0.86 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f 4 / 0.82, f5 / 0.78 ) , SC566 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f 2 / 0.91, f 4 / 0.82, f 5 / 0.78 ) , SC676 ( Jrn ) / 0.84 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f 3 / 0.89, f 4 / 0.82, f 5 / 0.78 ) , 9 subclasses of cardinality 7. i.e. SC97 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f 2 / 0.91, f 5 / 0.78 ) , SC107 ( Jrn ) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f 3 / 0.89, f 5 / 0.78 ) , SC117 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f 2 / 0.91, f 3 / 0.89, f 5 / 0.78 ) , SC127 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, f1 /1, f 2 / 0.91, f 3 / 0.89, f 5 / 0.78 ) , SC167 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f 4 / 0.82, f 5 / 0.78 ) , SC177 ( Jrn ) / 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f 2 / 0.91, f 4 / 0.82, f 5 / 0.78 ) , SC197 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p4 / 0.84, f1 /1, f 2 / 0.91, f 4 / 0.82, f 5 / 0.78 ) , 7 SC22 ( Jrn ) / 0.85 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f3 / 0.89, f 4 / 0.82, f5 / 0.78) , 7 SC25 ( Jrn ) / 0.86 = ( p1 / 0.93, p3 / 0.79, p4 / 0.84, f1 /1, f3 / 0.89, f 4 / 0.82, f5 / 0.78) , and 4 subclasses of cardinality 8. i.e. SC28 ( Jrn ) / 0.88 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f 2 / 0.91, f3 / 0.89, f5 / 0.78) , SC38 ( Jrn ) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f 2 / 0.91, f 4 / 0.82, f5 / 0.78 ) , SC48 ( Jrn ) / 0.87 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f1 /1, f3 / 0.89, f 4 / 0.82, f5 / 0.78 ) , SC58 ( Jrn ) / 0.86 = ( p1 / 0.93, p2 / 0.88, p3 / 0.79, p4 / 0.84, f 2 / 0.91, f3 / 0.89, f 4 / 0.82 f5 / 0.78 ) . All obtained subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 are elements of its semantically consistent decomposition D ( Jrn / 0.87 ) . Analyzing graphs of internal semantic dependencies for each constructed subclass of fuzzy class Jrn / 0.87 , we can see, that all of them are semantically consistent, according to Def. 4. It is known that the power set of a certain set is a partially ordered set, which defines a complete bounded lattice [21]. Therefore, the power set of the set of properties and methods of the fuzzy homogeneous class of objects Jrn / 0.87 is a poset ( PS ( Jrn / 0.87 ) , ) , which defines a complete bounded lattice of subclasses of the fuzzy class Jrn / 0.87 . i.e. L ( Jrn / 0.87 ) = ( PS ( Jrn / 0.87 ) , , , , 0, 1) , where  and  are the least upper bound (join) and the greatest lower bound (meet) operations, defined on the set PS ( Jrn / 0.87 ) i.e. SC1 / M ( SC1 )  PS ( Jrn / 0.87 ) , SC2 / M ( SC2 )  PS ( Jrn / 0.87 ) → → SC1 / M ( SC1 )  SC2 / M ( SC2 )  PS ( Jrn / 0.87 ) , SC1 / M ( SC1 )  SC2 / M ( SC2 )  PS ( Jrn / 0.87 ) , and where 0 is the least element SC1 ( Jrn ) / 0.0 , and 1 is the greatest elements 0 SC19 ( Jrn ) / 0.87 of the lattice, i.e. SC / M ( SC )  PS ( Jrn / 0.87 ) → SC10 ( Jrn ) / 0.0  SC / M ( SC ) = SC10 ( Jrn ) / 0.0, SC19 ( Jrn ) / 0.87  SC / M ( SC ) = SC19 ( Jrn ) / 0.87. As we know, the cardinality of PS ( Jrn / 0.87 ) is equal to 2n = 29 = 512 , where n = Jrn / 0.87 , therefore, illustrating the Hasse diagram of such a lattice is a non-trivial task, because of its size. Therefore, let us construct and illustrate the tower of subclass lattice L ( Jrn / 0.87 ) , using the corresponding approach proposed in [20] (see Fig. 3). As we can see, Fig. 3 represents three objects similar to tower buildings, which consist of sections and floors of a certain capacity. The tower sections are vertical columns of floors Figure 3. Tower of the complete bounded subclass lattice of the fuzzy homogeneous class of objects Jrn / 0.87 . represented by grey and lime circles. The capacity of all floors, within the particular section, is represented by yellow circles with a corresponding number. The tower of the subclass lattice L ( Jrn / 0.87 ) is depicted on the left side of Fig. 3. The circles colored in gray can be interpreted as the unlighted tower floors because they mean semantically inconsistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 . The circles colored in lime have an opposite interpretation since they mean semantically consistent subclasses, detected by the decomposition algorithm. The second and third towers depicted in the middle and on the right in Fig. 3 are towers of sublattices of the subclass lattice L ( Jrn / 0.87 ) , which contain only semantically inconsistent and only semantically consistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 , respectively. Comparing the number of subclasses of both kinds, we can see, that there are only 71 semantically consistent proper non-empty subclasses among the 510 formally possible. In more detail, this comparison can be represented by Tab. 1. The first row of the table means the cardinality of subclasses, while the second and the third rows contain the number of all formally possible and all semantically consistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 of certain cardinality. The fourth row represents the ratio third row to the second row in percent. According to [20], the decomposition consistency of the fuzzy homogeneous class of objects Jrn / 0.87 is approximately equal to 13.9% , i.e. Table 1 Quantitative analysis of subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 . Cardinality 1 2 3 4 5 6 7 8 Total Possible Subclasses 9 36 84 126 126 84 36 9 510 Consistent Subclasses 1 4 9 14 16 14 9 4 71 Decomposition Consistency 11% 11% 11% 11% 13% 17% 25% 44% 14% D ( Jrn / 0.87 ) 71 DC ( Jrn / 0.87 ) = 100% =  13.9%. PS ( Jrn / 0.87 ) − 2 510 It means that only 13.9% of all possible proper non-empty subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 are semantically consistent ones. Consequently, we can reduce the search space for the conceptual identification within the semantically consistent decomposition of a fuzzy homogeneous class of objects. To do this, let us construct the sublattice of the subclass lattice L ( Jrn / 0.87 ) , which contains only semantically consistent subclasses of the fuzzy class Jrn / 0.87 . According to the definition provided in [21], the sublattice of the lattice L is a subset X of L , such that a  X , b  X → a  b  X , a  b  X . It is obvious that the empty subclass SC10 ( Jrn ) / 0.0  Jrn / 0.87 , as well as the subclass SC19 ( Jrn ) / 0.87  Jrn / 0.87 , are semantically consistent subclasses of the fuzzy homogeneous class of objects Jrn / 0.87 , therefore the set of all its semantically consistent subclasses is defined as follows CS ( Jrn / 0.87 ) = D ( Jrn / 0.87 )  SC10 ( Jrn ) / 0.0, SC19 ( Jrn ) / 0.87. Figure 4. Hasse diagram of complete bounded semantically consistent subclass lattice of the fuzzy homogeneous class of objects Jrn / 0.87 . Therefore, a set of all semantically consistent subclasses CS ( Jrn / 0.87 )  PS ( Jrn / 0.87 ) of the fuzzy class of objects Jrn / 0.87 defines a carrier (CS ( Jrn / 0.87 ) , ) of the consistent subclass lattice CSL ( Jrn / 0.87 ) , which is a sublattice of the subclass lattice L ( Jrn / 0.87 ) , since SC1 / M ( SC1 )  CS ( Jrn / 0.87 ) , SC2 / M ( SC2 )  CS ( Jrn / 0.87 ) → → SC1 / M ( SC1 )  SC2 / M ( SC2 )  CS ( Jrn / 0.87 ) , SC1 / M ( SC1 )  SC2 / M ( SC2 )  CS ( Jrn / 0.87 ) , where  and  are the least upper bound (join) and the greatest lower bound (meet) operations, defined on the set CS ( Jrn / 0.87 ) . In addition, the carrier of the sublattice CSL ( Jrn / 0.87 ) contains the least element SC10 ( Jrn ) / 0.0 , and the greatest element SC19 ( Jrn ) / 0.87 , which makes it the a bounded one, since SC / M ( SC )  CS ( Jrn / 0.87 ) → → SC01 ( Jrn ) / 0.0  SC / M ( SC ) = SC01 ( Jrn ) / 0.0, SC19 ( Jrn ) / 0.87  SC / M ( SC ) = SC19 ( Jrn ) / 0.87. Consequently, the semantically consistent sublattice of subclasses lattice of the fuzzy homogeneous class of objects Jrn / 0.87 is defined as follows CSL ( Jrn / 0.87 ) = ( CS ( Jrn / 0.87 ) , , , , 0, 1) , where 0 means the least element, while 1 means the greatest element of the lattice. The Hasse diagram of the lattice CSL ( Jrn / 0.87 ) is depicted in the Fig. 4. The superscript of each lattice node means the cardinality of the corresponding subclass of the fuzzy class Jrn / 0.87 , and the subscript indicates the number of the subclass among all possible subclasses of such cardinality. Constructing the sublattice CSL ( Jrn / 0.87 ) allows us to reduce the subclass identification space by 512 / 73  7 times and to perform the subclass identification, analyzing only semantically consistent subclasses of the fuzzy class Jrn / 0.87 . 5. Identification of Consistent Fuzzy Knowledge To develop the identification of consistent fuzzy knowledge within the decomposition of fuzzy homogeneous classes of objects via constraint-based filtering, we modified the corresponding decomposition algorithm, proposed in [19], adding subclasses and superclasses detection procedures (see Algorithm 1). The main idea of the algorithm is to ( ) detect sets of all semantically consistent subclasses S  SC / M ( SC ) and superclasses S  ( SC / M ( SC ) ) for the selected semantically consistent subclass SC / M ( SC ) of the fuzzy homogeneous class of objects T / M (T ) . As the input parameters, the algorithm uses the following: a fuzzy homogeneous class of objects T / M (T ) as a space for the fuzzy knowledge identification; a semantically consistent subclass SC / M ( SC ) of the fuzzy class T / M (T ) as an identification target; a set of constraints C = ISD (T / M (T ) ) defined by molecules of the fuzzy class T / M (T ) , to detect its semantically consistent subclasses; a required accuracy  for computation of the measure of fuzziness of subclasses and superclasses of the class SC / M ( SC ) . In general, the procedure of the identification of consistent fuzzy knowledge can be split into a few successive stages. In the first stage, the algorithm constructs all formally possible subclasses of the fuzzy class T / M (T ) , which have a cardinality greater or less than the cardinality of the subclass SC / M ( SC ) . This reduces the identification space again, avoiding subclasses of the same cardinality as the SC / M ( SC ) subclass, since such subclasses definitely are not superclass or subclass of the subclass SC / M ( SC ) . Algorithm 1. Identification of Consistent Fuzzy Knowledge Require: T / M (T ) , SC / M ( SC ) , C ,  Ensure: S  , S  1: t := {}; 2: S  := {}; 3: S  := {}; T / M (T ) 4: for i = 1,..., 2 − 1 do 5: if binary(i).count(1)  SC / M ( SC ) then 6: ( ) for a j /  a j  T / M (T ) , j = 1,..., T / M (T ) do 7: if ( i & (1 j ) )  0 then 8: ( t.add a j /  ( a j ) ; ) 9: satisfy := true; 10: for all c  C do 11: if is_satisfy ( t , c ) = false then 12: staisfy := false; 13: break; 14: if satisfy then 15: M ( t ) = compute_fuzziness ( t ,  ) ; 16: ( ) if is_subclass SC / M ( SC ) , t / M ( t ) then 17: S  .add ( t / M ( t ) ) ; 18: ( ) if is_superclass SC / M ( SC ) , t / M ( t ) then 19: S .add ( t / M ( t ) ) ;  20: t := {}; 21: return S  , S  . In the second stage, the algorithm detects all semantically consistent subclasses of the fuzzy homogeneous class of objects T / M (T ) , among previously generated, performing the constraint-based filtering, using Procedure 1. It verifies the satisfiability of each constraint c  C , defined by molecules of the fuzzy class T / M (T ) , for the subclass SC / M ( SC ) . In general, the subclass SC / M ( SC ) can satisfy or not satisfy the constraint c  C as well as the constraint can be inapplicable to the subclass, therefore, the procedure can return as a value true, false, or none, respectively. In the third stage, the algorithm computes the fuzziness for each semantically consistent potential subclass or superclass of the class SC / M ( SC ) , using Procedure 2. After that, the algorithm verifies the subclass and (or) superclass relation between the detected semantically consistent subclasses of the fuzzy class T / M (T ) and subclass SC / M ( SC ) , using Procedure 3 and Procedure 4, respectively. Procedure 1. is_satisfy ( t , c ) Input: t , c Output: satisfy → { true, false, none } 1: satisfy := none; 2: if c  0  t then 3: satisfy := false; 4: for i 1,..., c do 5: for all a /  ( a )  c i  do 6: if a /  ( a )  t then 7: satisfy := true; 8: else 9: satisfy := false; 10: break; 11: if satisfy then 12: return satisfy; 13: return satisfy. Procedure 2. compute_fuzziness ( t ,  ) Input: t ,  Output: M ( t ) →  0, 1 1: sum := 0; 2: for all a /  ( a )  t do 3: sum := sum +  ( a ) ; ( ( 4: M ( t ) := round sum / max t , 1 ,  ; ) ) 5: return M ( t ) . ( Procedure 3. is_subclass SC / M ( SC ) , t / M ( t ) ) Input: SC / M ( SC ) , t / M ( t ) Output: is_subclass → { true, false } 1: if SC / M ( SC )  t / M ( t ) then 2: return false; 3: for all a /  ( a )  SC / M ( SC ) do 4: if a /  ( a )  t / M ( t ) then 5: return false; 6: return true. ( Procedure 4. is_superclass SC / M ( SC ) , t / M ( t ) ) Input: SC / M ( SC ) , t / M ( t ) Output: is_superclass → { true, false } 1: if SC / M ( SC )  t / M ( t ) then 2: return false; 3: for all a /  ( a )  t / M ( t ) do 4: if a /  ( a )  SC / M ( SC ) then 5: return false; 6: return true. As a result, the algorithm computes a set of all semantically consistent proper non-empty ( subclasses S  SC / M ( SC ) ) and superclasses S ( SC / M ( SC )) for the subclass  SC / M ( SC ) . To demonstrate the conceptual identification of fuzzy knowledge using Algorithm 1, let us apply the proposed approach to the subclass SC29 ( Jrn ) / 0.87 of the fuzzy homogeneous 6 class of objects Jrn / 0.87 , described in the previous section. Consequently, we obtain the following sets of proper non-empty subclasses S  ( SC29 6 ( Jrn ) / 0.87 ) = SC575 ( Jrn ) / 0.84, SC585 ( Jrn ) / 0.88, SC595 ( Jrn ) / 0.89, 5 SC60 ( Jrn ) / 0.87, SC24 ( Jrn ) / 0.9, SC714 ( Jrn ) / 0.85, SC724 ( Jrn ) / 0.86, SC734 ( Jrn ) / 0.84, SC754 ( Jrn ) / 0.9, SC764 ( Jrn ) / 0.88, SC784 ( Jrn ) / 0.89, SC13 ( Jrn ) / 0.87, SC53 ( Jrn ) / 0.94, SC63 ( Jrn ) / 0.91, SC573 ( Jrn ) / 0.86, SC583 ( Jrn ) / 0.83, SC60 3 ( Jrn ) / 0.85, SC633 ( Jrn ) / 0.9, SC12 ( Jrn ) / 0.91, SC22 ( Jrn ) / 0.86, SC72 ( Jrn ) / 0.97, SC292 ( Jrn ) / 0.86, SC11 ( Jrn ) / 0.93 , and superclasses S  ( SC29 6 ( Jrn ) / 0.87 ) = SC97 ( Jrn ) / 0.88, SC107 ( Jrn ) / 0.87, SC167 ( Jrn ) / 0.86, SC28 ( Jrn ) / 0.88, SC38 ( Jrn ) / 0.87, SC48 ( Jrn ) / 0.87 . As was noted above, subclasses SC1 ( Jrn ) / 0.0 , and SC1 ( Jrn ) / 0.87 are semantically 0 9 consistent subclasses of the fuzzy class Jrn / 0.87 , therefore the identification space for the subclass SC29 ( Jrn ) / 0.87 is defined as follows 6 6 IS29 ( Jrn / 0.87 ) = S  ( SC296 ( Jrn ) / 0.87 )  S  ( SC296 ( Jrn ) / 0.87 )   SC10 ( Jrn ) / 0.0, SC19 ( Jrn ) / 0.87 . Thus, the identification space IS29 6 ( Jrn / 0.87 ) of the subclass SC29 ( Jrn ) / 0.87 defines the 6 ( carrier IS29 6 ) ( Jrn / 0.87 ) ,  of the identification space lattice ISL29 ( Jrn / 0.87 ) , which is 6 a sublattice of the consistent subclass lattice CSL ( Jrn / 0.87 ) and subclass lattice L ( Jrn / 0.87 ) , since IS296 ( Jrn / 0.87 )  CS ( Jrn / 0.87 )  PS ( Jrn / 0.87 ) and SC1 / M ( SC1 )  IS29 6 ( Jrn / 0.87 ) , SC2 / M ( SC2 )  IS296 ( Jrn / 0.87 ) → → SC1 / M ( SC1 )  SC2 / M ( SC2 )  IS296 ( Jrn / 0.87 ) , SC1 / M ( SC1 )  SC2 / M ( SC2 )  IS296 ( Jrn / 0.87 ) , where  and  are the least upper bound (join) and the greatest lower bound (meet) operations, defined on the set IS SC29 6 ( ) ( Jrn ) / 0.87 . Since the carrier of the lattice ISL629 ( Jrn / 0.87 ) contains the least element SC10 ( Jrn ) / 0.0 , and the greatest element SC19 ( Jrn ) / 0.87 , it makes the identification sublattice ISL629 ( Jrn / 0.87 ) a bounded one, since SC / M ( SC )  IS296 ( Jrn / 0.87 ) → → SC01 ( Jrn ) / 0.0  SC / M ( SC ) = SC01 ( Jrn ) / 0.0, SC19 ( Jrn ) / 0.87  SC / M ( SC ) = SC19 ( Jrn ) / 0.87. Thus, the identification lattice for the subclass SC29 ( Jrn ) / 0.87 of the fuzzy homogeneous 6 class of objects Jrn / 0.87 is defined as follows ISL629 ( Jrn / 0.87 ) = ( IS29 6 ( Jrn / 0.87 ) , , , , 0, 1) , where 0 means the least element, while 1 means the greatest element of the lattice. The Hasse diagram of the lattice ISL29 ( Jrn / 0.87 ) is depicted in the Fig. 5. The yellow node 6 means the subclass SC29 6 ( Jrn ) / 0.87 , while the lime nodes represent its subclasses and superclasses. Figure 5. Hasse diagram of complete bounded identification space lattice of the subclass 6 SC29 ( Jrn ) / 0.87  Jrn / 0.87. Let us define a concept of a subclass neighborhood within the identification lattice, which will provide an opportunity to consider the so-called subclass locus instead of the subclass itself. Definition 5. The neighborhood of the subclass SC / M ( SC ) of a fuzzy homogeneous class ( ) ( of objects T / M (T ) is a pair N SC / M ( SC ) = Sn , Sn , where   )  ( ) Sn = SC1 / M SC1 ,..., SCk / M SCk , ( ) Sn  IS ( SC / M ( SC ) ) ,  ( ) Sn = SC1 / M SC1 ,..., SCw / M SCw , ( ) Sn  IS ( SC / M ( SC ) ) , are sets of subclasses and superclasses of the subclass SC / M ( SC ) such that ( ) ( ) SCi / M SCi  Sn , i = 1, k → SCi / M SCi  SC / M ( SC ) , SC / M ( SC )  S , j = 1, w → SC / M ( SC )  SC / M ( SC ) ,  j  j  n  j  j and where IS ( SC / M ( SC ) ) is a carrier of the subclass identification lattice ISL ( SC / M ( SC ) ) . Using concept of subclass neighborhood, let us define a notion of a subclass neighborhood measure. Definition 6. The neighborhood measure of the subclass SC / M ( SC ) of a fuzzy ( homogeneous class of objects T / M (T ) is a pair N  SC / M ( SC ) = n , n , where n   ) ( ) and n are subclass and superclass neighborhood, respectively, and defined in the following way  S n ( SC / M ( SC ) ) n = , S  ( SC / M ( SC ) )  S n ( SC / M ( SC ) ) n = . S  ( SC / M ( SC ) ) Now let us consider an example of the subclass neighborhood and its measure for the subclass SC29 ( Jrn ) / 0.87 . Suppose that neighborhood of the subclass SC29 ( Jrn ) / 0.87 is 6 6 defined in the following way N ( SC29 6 ( Jrn ) / 0.87 ) = Sn , Sn , ( ) Sn = SC595 ( Jrn ) / 0.89, SC60 5 ( Jrn ) / 0.87, SC784 ( Jrn ) / 0.89 , Sn = SC97 ( Jrn ) / 0.88, SC167 ( Jrn ) / 0.86, SC38 ( Jrn ) / 0.87 , ( then its neighborhood measure is N  SC29 ( Jrn ) / 0.87 = 0.13 , 0.43 , where 6   ) ( )  Sn ( SC29 6 ( Jrn ) / 0.87 ) 3 n = =  0.13, S ( SC  6 29 ( Jrn ) / 0.87 ) 24  Sn ( SC29 6 ( Jrn ) / 0.87 ) 3 n = =  0.43. S  ( SC29 6 ( Jrn ) / 0.87 ) 7 It is clear that n   0,1 and n   0,1 , therefore if their values are closer to 0 , this means that the subclass and superclass neighborhood is low, but if it is closer to 1 then their neighborhood is high. Using a neighborhood N SC29 6 ( Jrn ) / 0.87( ) of the subclass 6 SC29 ( Jrn ) / 0.87 we can consider a subclass locus defined by its subclasses and superclasses instead of the subclass itself. 6. Conclusions In this paper, we analyzed known approaches to the conceptual identification of fuzzy knowledge using a formal concept analysis and its fuzzy extension. Since the FCA/FFCA- based computation of formal concepts can construct semantically inconsistent concepts, which are impossible or unreal within a modeled domain, we proposed another new lattice- based approach to the conceptual identification of fuzzy knowledge. We study the conceptual identification of subclasses within the decomposition of fuzzy homogeneous classes of objects. The proposed approach allows us to identify semantically consistent subclasses within the decomposition of a fuzzy homogeneous class of objects constructing corresponding sub-class lattice. Such lattice is considered as a space for conceptual identification of any of its elements via detection of its subclasses and superclasses. Identification of a specific subclass involves the construction of a corresponding identification lattice, which is a sub-lattice of the subclass lattice. To implement the approach, we developed the corresponding identification algorithm, extending the algorithm for the decomposition of fuzzy homogeneous classes of objects via constraint- based filtering, proposed in [18]. As a result, the algorithm constructs all semantically consistent subclasses of a fuzzy homogeneous class of objects and then, verifies the subclass-superclass relation between each of them and a subclass, which needs to be identified. To demonstrate the conceptual identification of fuzzy knowledge using the developed algorithm, we provided an example of conceptual identification of a semantically consistent subclass of a fuzzy homogeneous class of objects, which defines a fuzzy concept of a journey through the sequence of geographic places. To visualize the identification process, the corresponding identification lattice was constructed. Using this lattice, we introduced the notions of subclass neighborhood and its measure. The proposed approach can be extended for the conceptual identification of fuzzy knowledge within the fuzzy conceptual hierarchies and scaling of big concept lattices. Acknowledgements This research has been supported by the National Academy of Science of Ukraine (project 0123U103273 Development of Algorithms and Software Tools for the Analysis of Object- Oriented Dynamic Networks). References [1] G. Birkhoff, Lattice Theory, 3rd ed., volume 25 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, Rhode Island, USA, 1973. [2] P. Cordero, M. Enciso, D. Lopez, A. Mora, A conversational recommender system for diagnosis using fuzzy rules, Expert Syst. With Appl. 154 (2020). doi: 10.1016/j.eswa.2020.113449. [3] C. De Maio, G. Fenza, V. Loia, S. Senatore, Hierarchical web resources retrieval by exploiting Fuzzy Formal Concept Analysis. Inform. Process. Manage. 48 (2012) 399– 418. doi: 10.1016/j.ipm.2011.04.003. [4] C. De Maio, M. Gallo, F. Hao, V. Loia, E. Yang, Fine-Grained Context-aware Ad Targeting on Social Media Platforms, in: Proc. 2020 IEEE Int. Conf. Systems, Man, and Cybernetics (SMC), Toronto, ON, Canada, 2020, pp. 3059–3065. doi: 10.1109/SMC42975.2020.9282827. [5] B. Ganter, R. Wille, Formal Concept Analysis: Mathematical Foundations. Springer, Berlin, Heidelberg, 1999. doi: 10.1007/978-3-642-59830-2. [6] J. Gao, F. Hao, Z Pei, G. Min, Learning Concept Interestingness for Identifying Key Structures From Social Networks. IEEE Trans. Netw. Sci. Eng. 8 (2021) 3220–3232. doi: 10.1109/TNSE.2021.3107529. [7] F. Hao, J. Gao, J. Chen, A. Nasridinov, G. Min, Skyline (  , k ) -Cliques Identification From Fuzzy Attributed Social Networks. IEEE Trans. Comput. Social Syst. 9 (2022) 1075–1086. doi: 10.1109/TCSS.2021.3101152. [8] F. Hao, Z. Pei, L. T. Yang, Diversified top- k maximal clique detection in Social Internet of Things. Future Gen. Comput. Syst. 107 (2020) 408–417. doi: 10.1016/j.future.2020.02.023. [9] M. H. Ibrahim, R. Missaoui, J. Vaillancourt, Cross-Face Centrality: A New Measure for Identifying Key Nodes in Networks Based on Formal Concept Analysis. IEEE Access. 8 (2020) 206901–206913. doi: 10.1109/ACCESS.2020.3038306. [10] M. H. Ibrahim, R. Missaoui, J. Vaillancourt, Identifying Influential Nodes in Two-Mode Data Networks Using Formal Concept Analysis. IEEE Access. 9 (2021) 159549– 159565. doi: 10.1109/ACCESS.2021.3131987. [11] L. Kwuida, R. Missaoui, Formal Concept Analysis and Extensions for Complex Data Analytics, in: R. Missaoui, L. Kwuida, T. Abdessalem (Eds.), Complex Data Analytics with Formal Concept Analysis, Springer, Cham., 2022, pp. 1–15. doi: 10.1007/978-3- 030-93278-7_1. [12] H. Meznia, T. Abdeljaoued, A cloud services recommendation system based on Fuzzy Formal Concept Analysis. Data & Knowl. Eng. 116 (2018) 100–123. doi: 10.1016/j.datak.2018.05.008. [13] S. M. Neto, S. Dias, R. Missaoui, L. Zarate, M. Song, Identification of Substructures in Complex Networks using Formal Concept Analysis. Int. J. Web Inf. Syst. 14 (2018) 281–298. doi: 10.1108/IJWIS-10-2017-0067. [14] K. Ravi, V. Ravi, P. Sree Rama Krishna Prasad, Fuzzy formal concept analysis based opinion mining for CRM in financial services. Appl. Soft Comput. 60 (2017) 786–807. doi: 10.1016/j.asoc.2017.05.028. [15] P. R. Silva, S. M. Dias, W. C. Brandao, M. A. Song, L. E. Zarate, Professional Competence Identification Through Formal Concept Analysis, in: S. Hammoudi, M. Smialek, O. Camp, J. Filipe (Eds.) Enterprise Information Systems. ICEIS 2017, volume 321 of LNBIP, Springer, 2018, pp. 34–56. doi: 10.1007/978-3-319-93375-7_3. [16] P. Sokol, L. Antoni, O. Kridlo, E. Markova, K. Kovacova, S. Krajci, The analysis of digital evidence by Formal concept analysis, in: Proc. 16th Int. Conf. Concept Lattices and Their Applications (CLA 2022), Tallinn, Estonia, 2022, pp. 147–158. [17] P. Sokol, L. Antoni, O. Kridlo, E. Markova, K. Kovacova, S. Krajci, Formal concept analysis approach to understand digital evidence relationships. Int. J. Approx. Reason. 159 (2023). doi: 10.1016/j.ijar.2023.108940. [18] D. O. Terletskyi, S. V. Yershov, Decomposition of Fuzzy Homogeneous Classes of Objects, in: A. Lopata, D. Gudonienė, R. Butkienė (Eds.), Information and Software Technologies. ICIST 2022, volume 1665 of CCIS, Springer, Cham., 2022, pp. 43–63. doi: 10.1007/978-3-031-16302-9_4. [19] D. O. Terletskyi, S. V. Yershov, Fuzzy Conceptual Knowledge Extraction and Retrieval Within Fuzzy Classes Decomposition, in: Proc. 9th Int. Sci. Pract. Conf. Inf. Technol. Implement. (IT&I), CEUR Workshop Proceedings, volume 3347, Kyiv, Ukraine, 2022, pp. 195–211. [20] R. Wille, Conceptual Knowledge Processing: Theory and Practice, in: K. E. Wolff, D. E. Palchunov, N. G. Zagoruiko, U. Andelfinger, (Eds.), Knowledge Processing and Data Analysis. KPP KONT 2007, volume 6581 of LNCS, Springer, Berlin, Heidelberg, 2007, pp. 1–25. doi: 10.1007/978-3-642-22140-8_1. [21] Y. Yang, F. Hao, B. Pang, G. Min, Y. Wu, Dynamic Maximal Cliques Detection and Evolution Management in Social Internet of Things: A Formal Concept Analysis Approach. IEEE Trans. Netw. Sci. Eng. 9 (2021) 1020–1032. doi: 10.1109/TNSE.2021.3067939. [22] Y, Yang, S. Peng, D.-S. Park, F. Hao, H. Lee, A Novel Community Detection Method of Social Networks for the Well-Being of Urban Public Spaces. Land. 11 (2022) 716. doi: 10.3390/land11050716. [23] F. Zheng, R. Abeysinghe, L. Cui, Identification of missing concepts in biomedical terminologies using sequence-based formal concept analysis. BMC Med. Inform. Decis. Mak. 21 (Suppl 7) (2021). doi: 10.1186/s12911-021-01592-w. [24] F. Zheng, L. Cui, A Lexical-based Formal Concept Analysis Method to Identify Missing Concepts in the NCI Thesaurus, in: Proc. 2020 IEEE Int. Conf. Bioinformatics and Biomedicine (BIBM), Seoul, Korea (South), 2020, pp. 1757–1760. doi: 10.1109/BIBM49941.2020.9313186.