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Factorizing formal contexts from closures of necessity operators
Roberto G. Aragón, Jesús Medina, Eloísa Ramírez-Poussa
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 90%
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Summary
This paper analyzes the factorization of formal contexts in Formal Concept Analysis (FCA) using necessity operators. It extends a method for obtaining independent subcontexts from Boolean data to the fuzzy framework using the multi-adjoint framework. The study investigates properties of pairs of sets forming closures under necessity operators, characterizing minimal independent contexts via supremum irreducible elements in a complemented complete lattice.
Entities (10)
Relation Signals (7)
Roberto G. Aragón → affiliatedwith → University of Cádiz
confidence 99% · Roberto G. Aragón ... Department of Mathematics, University of Cádiz
Formal Concept Analysis → uses → Concept Lattice
confidence 95% · The tools provided by FCA can properly manipulate data and extract relevant information from it, representing the information by means of the algebraic structure of a complete lattice
Multi-adjoint framework → generalizes → Formal Concept Analysis
confidence 92% · One of the most versatile fuzzy extensions of this theory is the one given by the multi-adjoint framework
Necessity operator → usedfor → Factorization
confidence 90% · Factorizing formal contexts from closures of necessity operators
Necessity operator → forms → Complemented complete lattice
confidence 88% · analyzing the pairs of closed subsets under the necessity operator which form a complemented complete lattice
Independent subcontext → derivedfrom → Necessity operator
confidence 85% · by using modal operators a context is split into two or more independent subcontexts
Adjoint triple → includes → Gödel t-norm
confidence 85% · Some examples of adjoint triples are the Gödel, product and Łukasiewicz t-norms
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Abstract
Abstract:Factorizing datasets is an interesting process in a multitude of approaches, but many times it is not possible or efficient the computation of a factorization of the dataset. A method to obtain independent subcontexts of a formal context with Boolean data was proposed in~\cite{dubois:2012}, based on the operators used in possibility theory. In this paper, we will analyze this method and study different properties related to the pairs of sets from which a factorization of a formal context arises. We also inspect how the properties given in the classical case can be extended to the fuzzy framework, which is essential to obtain a mechanism that allows the computation of independent subcontexts of a fuzzy context.
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- Source: https://arxiv.org/abs/2604.09582v1
- Canonical: https://arxiv.org/abs/2604.09582v1
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Factorizing formal contexts from closures of necessity operators111Partially supported by the 2014–2020 ERDF Operational Programme in collaboration with the State Research Agency (AEI) in projects PID2019-108991GB-I00 and PID2022-137620NB-I00, with the Ecological and Digital Transition Projects 2021 of the Ministry of Science and Innovation in project TED2021-129748B-I00, with the Department of Economy, Knowledge, Business and University of the Regional Government of Andalusia in project FEDER-UCA18-108612, and by the European Cooperation in Science & Technology (COST) Action CA17124. Roberto G. Aragón222Corresponding author. Jesús Medina Eloísa Ramírez-Poussa Department of Mathematics, University of Cádiz. Spain Email: roberto.aragon,jesus.medina,eloisa.ramirez@uca.es (November 20, 2023) Abstract Factorizing datasets is an interesting process in a multitude of approaches, but many times it is not possible or efficient the computation of a factorization of the dataset. A method to obtain independent subcontexts of a formal context with Boolean data was proposed in [18], based on the operators used in possibility theory. In this paper, we will analyze this method and study different properties related to the pairs of sets from which a factorization of a formal context arises. We also inspect how the properties given in the classical case can be extended to the fuzzy framework, which is essential to obtain a mechanism that allows the computation of independent subcontexts of a fuzzy context. keywords: Formal concept analysis, factorization, multi-adjoint framework, independent subcontext †journal: Computational and Applied Mathematics 1 Introduction The factorization of relational data represents a relevant research line since the seventies [3, 20, 24, 25, 26, 27, 28, 35]. Being able to factorize a (real) dataset makes possible to reduce the complexity of information processing and, therefore, to obtain the solution of the problem to be solved in a more efficient way [4, 9]. Moreover, there exist another two fundamental aspects: the extracted factors reveal important information about the whole data and the factors can be considered as new variables, originally hidden in the data and revealed by factorization. A mathematical theory for extracting knowledge from relational databases is Formal Concept Analysis (FCA, for short) introduced at the beginning of the eighties [21, 39]. In particular, the databases considered in this theory are called contexts and are composed of a set of objects, a set of attributes and a relationship between these sets. The tools provided by FCA can properly manipulate data and extract relevant information from it, representing the information by means of the algebraic structure of a complete lattice [34, 36, 37, 41] . One of the most versatile fuzzy extensions of this theory is the one given by the multi-adjoint framework [1, 11, 14, 30, 32]. The characteristics of this framework provide it with a greater capacity to model real problems. One of the most important research lines within FCA is the factorization of formal contexts. In this regard, different procedures have been already developed, such as [5, 6, 7, 18, 29, 38]. In [18], the authors make use of tools from possibility theory to obtain independent subcontexts of a formal context. Specifically, by using modal operators a context is split into two or more independent subcontexts of smaller size which can be studied separately more easily and from which the original context can be recovered. Recently, in [2] an initial study was presented about the necessity operators applied to the factorization of formal contexts in FCA. This study continues in this paper analyzing the pairs of closed subsets under the necessity operator which form a complemented complete lattice [23]. The minimal independent contexts will be characterized by the supremum irreducible elements of this lattice and these pairs will also satisfy interesting relationships with property-oriented concepts and formal concepts. In this paper, we continue with the study initiated in [2], providing additional properties that are satisfied by blocks of concepts corresponding to the independent subcontexts in the classical FCA. Moreover, we will consider the necessity operators in the fuzzy case and analyze the pairs of closures obtained from them. We will prove that the set of these pairs also forms a complete lattice, but the relationship between these pairs and the independent subcontexts is naturally unclear, in general. Hence, we will study a fuzzy frame in which the translation of the properties of the classical case will be satisfied. This paper is organized as follows: Section 2 recalls different preliminaries notions and results required in this work. In Section 3, properties related to the blocks of concepts associated with independent subcontexts in classical FCA are introduced. The extensions of some of these properties to the fuzzy framework are given in Section 4 together with additional properties to determine intervals of concepts in the fuzzy environment. Finally, we expose some conclusions and prospects of future works in Section 5. 2 Preliminaries In this section, we recall several notions concerning FCA and its fuzzy generalization to the multi-adjoint framework, which have been taken into consideration to develop this paper. We will split this section into two parts, in order to expose preliminary notions as clearly as possible. The first part will be focused on classical modal operators and basic notions in FCA and the second part will concern the generalizations of these notions to the multi-adjoint framework. 2.1 Formal concept analysis First of all, a context in FCA is a triple (A,B,R)(A,B,R) where A is a set of attributes, B is a set of objects and R⊆A×BR A× B is a relationship, such that (a,b)∈R(a,b)∈ R if the object b∈Bb∈ B possesses the attribute a∈Aa∈ A. Moreover, the mappings :↑2B→2A 2^B→ 2^A and :↓2A→2B 2^A→ 2^B defined as follow: X↑ X =a∈A∣(a,b)∈R, for all b∈X =\a∈ A (a,b)∈ R, for all b∈ X\ Y↓ Y =b∈B∣(a,b)∈R, for all a∈Y =\b∈ B (a,b)∈ R, for all a∈ Y\ with X⊆BX B and Y⊆AY A, are called derivation operators. The pair (↑,↓)( , ) forms an antitone Galois connection [21]. Furthermore, a pair of sets X⊆BX B and Y⊆AY A, satisfying that X↑=YX =Y and Y↓=XY =X, is called a concept and denoted as ⟨X,Y⟩ X,Y . In addition, all the concepts together with the inclusion ordering on the left argument (or the opposite of the inclusion order on the right argument) has the algebraic structure of a complete lattice, which is called concept lattice and it is denoted by (A,B,R)C(A,B,R), that is, for each ⟨X1,Y1⟩,⟨X2,Y2⟩∈(A,B,R) X_1,Y_1 , X_2,Y_2 (A,B,R), we have that ⟨X1,Y1⟩⪯⟨X2,Y2⟩ X_1,Y_1 X_2,Y_2 if X1⊆X2X_1 X_2 (or, equivalently, Y2⊆Y1Y_2 Y_1). Besides, the derivation operators previously presented are not the only operators that can be defined on a formal context, there are other modal operators that can be defined for X⊆BX B and Y⊆AY A [16, 22, 40]. The mappings :↑π2B→2A _π 2^B→ 2^A, :↓N2A→2B ^N 2^A→ 2^B, defined as follow: X↑π X _π =a∈A∣ there exists b∈X, such that (a,b)∈R =\a∈ A there exists b∈ X, such that (a,b)∈ R\ Y↓N Y ^N =b∈B∣ if (a,b)∈R, then a∈Y, for all a∈A, =\b∈ B if (a,b)∈ R, then a∈ Y, for all a∈ A\, are called possibility and necessity operator, respectively. Analogously, the mappings :↑N2B→2A _N 2^B→ 2^A, :↓π2A→2B ^π 2^A→ 2^B are defined as: X↑N X _N =a∈A∣ if (a,b)∈R, then b∈X, for all b∈B =\a∈ A if (a,b)∈ R, then b∈ X, for all b∈ B\ Y↓π Y ^π =b∈B∣ there exists a∈Y, such that (a,b)∈R =\b∈ B there exists a∈ Y, such that (a,b)∈ R\ The pairs (↑π,↓N)( _π, ^N) and (↑N,↓π)( _N, ^π) form isotone Galois connections and, as a consequence, the necessity operators preserve the intersection of subsets and the possibility operators preserve unions, that is, (X1∩X2)↑N=X1↑N∩X2↑N(X_1∩ X_2) _N=X_1 _N∩ X_2 _N and (X1∪X2)↑π=X1↑π∪X2↑π(X_1∪ X_2) _π=X_1 _π∪ X_2 _π, for all X1,X2⊆BX_1,X_2 B, and analogously for attributes [17]. From these pairs, we obtain the concept lattices called property-oriented concept lattice and object-oriented concept lattice [8, 22, 31]. In addition, we need to recall the notion of join-irreducible element of a lattice. Definition 1 ([15]) Given a lattice (L,⪯)(L, ), such that ∨ is the join operator, and an element x∈Lx∈ L verifying: 1. If L has a bottom element ⊥ , then x≠⊥x≠ . 2. If x=y∨zx=y z, then x=yx=y or x=zx=z, for all y,z∈Ly,z∈ L. The element x is called join-irreducible (∨ -irreducible) element of L. Another important elements we can find in a lattice are the atoms. Definition 2 Let (L,⪯)(L, ) be a lattice with bottom element ⊥ . An element x∈Lx∈ L verifying that ⊥≺x x and there is no y∈Ly∈ L such that ⊥≺y≺x y x is called an atom of L. A coatom of L is defined dually. 2.2 Multi-adjoint framework As we previously mentioned, in this paper we will consider the fuzzy generalization of FCA given by the multi-adjoint framework [32]. We recall the generalizations of the previous notions to the multi-adjoint frame. First of all, the operators called adjoint triples [10] need to be recalled. Definition 3 Let (P1,≤1)(P_1, _1), (P2,≤2)(P_2, _2), (P3,≤3)(P_3, _3) be posets and &:P1×P2→P3 \& P_1× P_2→ P_3, ↙:P3×P2→P1 P_3× P_2→ P_1, ↖:P3×P1→P2 P_3× P_1→ P_2 be mappings, then (&,↙,↖)( \& , , ) is an adjoint triple with respect to P1,P2,P3P_1,P_2,P_3 if: x≤1z↙yiffx&y≤3ziffy≤2z↖x _1z y \!\! iff\!\! x \& y _3z \!\! iff\!\! y _2z x (1) where x∈P1x∈ P_1, y∈P2y∈ P_2 and z∈P3z∈ P_3. Condition (1) is called adjoint property. Notice that, if an operator &:P1×P2→P3 \& P_1× P_2→ P_3 is part of an adjoint triple, then the operators ↙:P3×P2→P1 P_3× P_2→ P_1, ↖:P3×P1→P2 P_3× P_1→ P_2 are uniquely determined [13]. Moreover, the following proposition can be useful when we have to handle with these operators. Proposition 4 ([13]) Let (&,↙,↖)(\&, , ) be an adjoint triple with respect to the posets (P1,≤1)(P_1, _1), (P2,≤2)(P_2, _2) and (P3,≤3)(P_3, _3), then the following properties are satisfied: 1. ⊥1&y=⊥3 _1\&y= _3, ⊤3↙y=⊤1 _3 y= _1, for all y∈P2y∈ P_2, when (P1,≤1,⊥1,⊤1)(P_1, _1, _1, _1) and (P3,≤3,⊥3,⊤3)(P_3, _3, _3, _3) are bounded posets. 2. x&⊥2=⊥3x\& _2= _3, ⊤3↖x=⊤2 _3 x= _2, for all x∈P1x∈ P_1, when (P2,≤2,⊥2,⊤2)(P_2, _2, _2, _2) and (P3,≤3,⊥3,⊤3)(P_3, _3, _3, _3) are bounded posets. 3. When the infimum exists: (⋀z′∈Zz′)↙y=⋀z′∈Z(z′↙y) ( _z ∈ Zz ) y= _z ∈ Z(z y) for any Z⊆P3Z P_3 and y∈P2y∈ P_2. Some examples of adjoint triples are the Gödel, product and Łukasiewicz t-norms together with their residuated implications, which will be used in different examples of this paper. Notice that, the Gödel, product and Łukasiewicz t-norms are commutative, therefore the residuated implications satisfy that ↙G=↖G ^G= _G, ↙P=↖P ^P= _P and ↙L=↖L ^L= _L. Example 5 Given m∈ℕm , the set [0,1]m[0,1]_m is a regular partition of [0,1][0,1] in m pieces, for example [0,1]4=0,0.25,0.5,0.75,1[0,1]_4=\0,0.25,0.5,0.75,1\ divide the unit interval in four pieces. A discretization of the product t-norm is the operator &P∗:[0,1]4×[0,1]8→[0,1]10 \& ^*_P [0,1]_4×[0,1]_8→[0,1]_10 defined, for each x∈[0,1]4x∈[0,1]_4 and y∈[0,1]8y∈[0,1]_8 as: x&P∗y=⌈10⋅x⋅y⌉10x \& ^*_Py= 10· x· y 10 where ⌈_⌉ \,\_\, is the ceiling function. In this case the residuated implications ↙P∗:[0,1]10×[0,1]8→[0,1]4 ^*_P [0,1]_10×[0,1]_8→[0,1]_4, ↖P∗:[0,1]10×[0,1]4→[0,1]8 ^*_P [0,1]_10×[0,1]_4→[0,1]_8 are defined as: z↙P∗x z ^*_Px =⌊4⋅z←x⌋4z↖P∗y = 4· z← x 4 z ^*_Py =⌊8⋅z←y⌋8 = 8· z← y 8 where z←x=1if x≤yz/xotherwisez← x= cases1& if x≤ y\\ z/x& otherwise cases for all z,x∈[0,1]z,x∈[0,1], and ⌊_⌋ \,\_\, is the floor function. Similar adjoint triples can be obtained from the Gödel and Łukasiewicz t-norms [12]. Furthermore, in order to define the notion of context in the multi-adjoint framework, we have to fix an algebraic structure called multi-adjoint frame. Definition 6 A multi-adjoint frame is a tuple (L1,L2,P,&1,…,&n)(L_1,L_2,P, \& _1,…, \& _n), where (L1,⪯1)(L_1, _1) and (L2,⪯2)(L_2, _2) are complete lattices, (P,≤)(P,≤) is a poset and (&i,↙i,↖i)( \& _i, ^i, _i) is an adjoint triple with respect to L1,L2,PL_1,L_2,P, for all i∈1,…,ni∈\1,…,n\. Once a multi-adjoint frame is fixed, the notion of context in this frame is defined as follows. Definition 7 Let (L1,L2,P,&1,…,&n)(L_1,L_2,P, \& _1,…, \& _n) be a multi-adjoint frame, a context is a tuple (A,B,R,σ)(A,B,R,σ) such that A and B are non-empty sets (usually interpreted as attributes and objects, respectively), R is a P-fuzzy relation R:A×B→PR A× B→ P and σ:A×B→1,…,nσ A× B→\1,…,n\ is a mapping which associates any element in A×BA× B to some particular adjoint triple of the frame. In addition, the generalization of derivation operators :↑L2B→L1A L_2^B→ L_1^A and :↓L1A→L2B L_1^A→ L_2^B are given as follows: g↑(a) g (a) = = infR(a,b)↙σ(a,b)g(b)∣b∈B \R(a,b) ^σ(a,b)g(b) b∈ B\ f↓(b) f (b) = = infR(a,b)↖σ(a,b)f(a)∣a∈A \R(a,b) _σ(a,b)f(a) a∈ A\ for all g∈L2Bg∈ L_2^B, f∈L1Af∈ L_1^A and a∈Aa∈ A, b∈Bb∈ B, where L2BL_2^B and L1AL_1^A denote the set of mappings g:B→L2g B→ L_2 and f:A→L1f A→ L_1, respectively. Equivalently, a multi-adjoint concept is a pair ⟨g,f⟩ g,f , where g∈L2Bg∈ L_2^B is a fuzzy subset of objects and f∈L1Af∈ L_1^A is a fuzzy subset of attributes, satisfying that g↑=fg =f and g↓=g =g. Furthermore, the set of multi-adjoint concepts together with the usual ordering form a complete lattice. Definition 8 The multi-adjoint concept lattice associated with a multi-adjoint frame (L1,L2,P,&1,…,&n)(L_1,L_2,P, \& _1,…, \& _n) and a context (A,B,R,σ)(A,B,R,σ) given, is the set ℳ=⟨g,f⟩∣g∈L2B,f∈L1A and g↑=f,f↓=gM=\ g,f g∈ L_2^B,f∈ L_1^A and g =f,f =g\ where the ordering is defined by ⟨g1,f1⟩⪯⟨g2,f2⟩ if and only if g1⪯2g2 g_1,f_1 g_2,f_2 if and only if g_1 _2g_2 (equivalently f2⪯1f1f_2 _1f_1), for all ⟨g1,f1⟩,⟨g2,f2⟩∈ℳ g_1,f_1 , g_2,f_2 . Similarly to what we have shown for the classical environment, the operators ↑ and ↓ are not the only operators that can be defined on a context. The classical definitions of the necessity and possibility operators were also generalized in a fuzzy framework [30], where two complete lattices (L1,⪯1),(L2,⪯2)(L_1, _1),(L_2, _2) and a poset (P,≤)(P,≤) are fixed. A multi-adjoint property-oriented frame is given by (L1,L2,P,&1,…,&n)(L_1,L_2,P, \& _1,…, \& _n), where (&i,↙i,↖i)( \& _i, ^i, _i) is an adjoint triple with respect to P, L2L_2, L1L_1 for all i∈1,…,ni∈\1,…,n\. In this multi-adjoint algebra, the necessity and possibility operators are the mappings :↓NL1A→L2B ^N L_1^A→ L_2^B, :↑πL2B→L1A _π L_2^B→ L_1^A, defined as: g↑π(a) g _π(a) = = supR(a,b)&σ(a,b)g(b)∣b∈B \R(a,b) \& _σ(a,b)g(b) b∈ B\ f↓N(b) f ^N(b) = = inff(a)↖σ(a,b)R(a,b)∣a∈A \f(a) _σ(a,b)R(a,b) a∈ A\ for all a∈A,b∈B,g∈L2Ba∈ A,b∈ B,g∈ L_2^B and f∈L1Af∈ L_1^A. A multi-adjoint object-oriented frame is the tuple (L1,L2,P,&1,…,&n)(L_1,L_2,P, \& _1,…, \& _n), where (&i,↙i,↖i)( \& _i, ^i, _i) is an adjoint triple with respect to L1L_1, P, L2L_2 for all i∈1,…,ni∈\1,…,n\. In this algebra these operators are given by the mappings :↑NL2B→L1A _N L_2^B→ L_1^A, :↓πL1A→L2B ^π L_1^A→ L_2^B defined as: g↑N(a) g _N(a) = = infg(b)↙σ(a,b)R(a,b)∣b∈B \g(b) ^σ(a,b)R(a,b) b∈ B\ f↓π(b) f ^π(b) = = supf(a)&σ(a,b)R(a,b)∣a∈A \f(a) \& _σ(a,b)R(a,b) a∈ A\ for all a∈A,b∈B,g∈L2Ba∈ A,b∈ B,g∈ L_2^B and f∈L1Af∈ L_1^A. Notice that the pairs of operators (↑π,↓N)( _π, ^N) and (↑N,↓π)( _N, ^π) are isotone Galois connections. Moreover, a pair ⟨g,f⟩ g,f , where g∈L2Bg∈ L_2^B is a fuzzy subset of objects and f∈L1Af∈ L_1^A is a fuzzy subset of attributes, satisfying that g↑π=fg _π=f and f↓N=gf ^N=g is called property-oriented concept. Additionally, a multi-adjoint property-oriented concept lattice is denoted by ℳπNM_π N. Lastly, the fuzzy sets g∈L2Bg∈ L_2^B and f∈L1Af∈ L_1^A such that g(b)=⊤2g(b)= _2, for all b∈Bb∈ B, and f(a)=⊤1f(a)= _1, for all a∈Aa∈ A, are denoted as g⊤g_ and f⊤f_ , respectively. Similarly, when g(b)=⊥2g(b)= _2, for all b∈Bb∈ B, and f(a)=⊥1f(a)= _1, for all a∈Aa∈ A, they are denoted as g⊥g_ and f⊥f_ , respectively. 3 Properties of the factorization in FCA In [18], one of the four set-functions of possibility theory is used to characterize independent subcontexts (i.e. contexts that have no common objects and no common properties from which the original context can be recovered) of a given finite context. In particular, the pair of operators of actual necessity, (↑N,↓N)( _N, ^N) is considered to decompose the relation R of a context (A,B,R)(A,B,R) with finite sets A and B, by computing the following intersection: R∗=⋂(X×Y)∪(Xc×Yc)∣X⊆B,Y⊆A,X↑N=Y,Y↓N=XR^*= \(X× Y)∪(X^c×Y^c) X B,Y A,X _N=Y,~Y ^N=X\ where XcX^c and YcY^c are the complements of X and Y, respectively. From the study presented in [18], we can bring to light some interesting properties from a pair of subsets X⊆BX B, Y⊆AY A, that satisfies the following equalities: X↑N=Y and Y↓N=X,X _N=Y and Y ^N=X, (2) The set of all pairs satisfying Expression (2) is denoted by NC_N, that is, N=(X,Y)∣X⊆B,Y⊆A,X↑N=Y,Y↓N=XC_N=\(X,Y) X B,Y A,X _N=Y,~Y ^N=X\ Different properties of this decomposition will be analyzed in this section, which will be paramount in the extension of this factorization procedure to more general environments, such as the fuzzy case provided by the multi-adjoint framework. In the following, we assume that the data table has neither empty rows nor attributes that are possessed by all objects in the context, this kind of contexts are called normalized. Notice that, this assumption is not a real restriction, but if some of the objects/attributes have no attribute/object or have all the attributes/objects, they can be removed at the beginning of the procedure and taken into consideration at the end. In addition, since the context is normalized, the pairs ⟨B,∅⟩ B, and ⟨∅,A⟩ ,A are concepts, indeed the are the top and bottom elements of the associated concept lattice, respectively. As a starting point, we recall that each pair (X,Y)(X,Y) belonging to NC_N determines an independent subcontext of the original context [18]. As a consequence, its complement (Xc,Yc)(X^c,Y^c), where XcX^c and YcY^c are the complements of X and Y respectively, also belongs to NC_N and thus, determines another independent subcontext. Lemma 9 Given a context (A,B,R)(A,B,R) and a pair (X,Y)∈N(X,Y) _N, the complement of the pair (X,Y)(X,Y) also belongs to NC_N, that is, (X,Y)c=(Xc,Yc)∈N(X,Y)^c=(X^c,Y^c) _N. Proof 1 We will proceed by reductio ad absurdum. We will suppose that (X,Y)∈N(X,Y) _N and (Xc,Yc)∉N(X^c,Y^c) _N. Since (Xc)↑N≠Yc(X^c) _N≠ Y^c, we have to distinguish two different cases: 1. There exists a∈(Xc)↑Na∈(X^c) _N such that a∉Yca∉ Y^c. Therefore, we have that a∈Ya∈ Y. But we know that (X,Y)∈N(X,Y) _N, as a consequence, the equality Y=X↑NY=X _N holds and we have that a∈X↑Na∈ X _N. Since a∈(Xc)↑Na∈(X^c) _N and a∈X↑Na∈ X _N, we obtain the following chain a∈X↑N∩(Xc)↑N=(X∩Xc)↑N=∅↑N=∅a∈ X _N∩(X^c) _N=(X∩ X^c) _N= _N= which leads us to a contradiction. 2. There exists a∈Yca∈ Y^c such that a∉(Xc)↑Na∉(X^c) _N. Since a∉(Xc)↑Na∉(X^c) _N, applying the definition of the operator ↑N _N, there exists b∈Bb∈ B such that (a,b)∈R(a,b)∈ R and b∉Xcb∉ X^c; that is, b∈X=Y↓Nb∈ X=Y ^N from which it can be concluded that a∈Ya∈ Y which is a contradiction, since a∉Ya∉ Y. Therefore, there may be different ways of factorizing the original context into independent subcontexts, depending on the cardinality of the set NC_N. Since we are interested in reducing the complexity in the data processing as much as possible, we should consider the minimal independent subcontexts. However, pairs in NC_N do not entail any minimality. Therefore, we have to find those pairs that generate minimal subcontexts. In this paper, we propose to obtain these minimal subcontexts from a different approach to the one given in [18]. The elements of NC_N equipped with the operations 1. (X1,Y1)⊔(X2,Y2)=(X1∪X2,Y1∪Y2)(X_1,Y_1) (X_2,Y_2)=(X_1∪ X_2,Y_1∪ Y_2) 2. (X1,Y1)⊓(X2,Y2)=(X1∩X2,Y1∩Y2)(X_1,Y_1) (X_2,Y_2)=(X_1∩ X_2,Y_1∩ Y_2) 3. (X,Y)c=(Xc,Yc)(X,Y)^c=(X^c,Y^c) where ⊔ , ⊓ and c represent the supremum, infimum and complement operators, have the structure of a complemented complete lattice (this was proved in a more general framework in [23]), with the inclusion order on the left argument or on the right argument, that is, (X1,Y1)≤(X2,Y2)(X_1,Y_1)≤(X_2,Y_2) if X1⊆X2X_1 X_2 or, equivalently, Y1⊆Y2Y_1 Y_2. Hence, (N,≤)(C_N,≤) is a complete lattice [15], where (B,A)(B,A) and (∅,∅)( , ) are the top and bottom elements, respectively. From this point of view, we analyze new properties of the decomposition of a formal context, using the necessity operators. Next, several of these properties will be highlighted and some examples to illustrate the decomposition of a context as well as the introduced properties are given. The first result we present in this paper relates the elements in NC_N that determine minimal subcontexts to the ∨ -irreducible elements of the complete lattice associated with the elements in NC_N. Proposition 10 Let (A,B,R)(A,B,R) be a formal context. An ∨ -irreducible element of NC_N is an atom of NC_N. Proof 2 Let us consider a ∨ -irreducible element (X∗,Y∗)(X^*,Y^*) of NC_N. We proceed by reduction ad absurdum. We suppose that there exists an element (X,Y)(X,Y) of NC_N such that (X,Y)≠(∅,∅)(X,Y)≠( , ) and (X,Y)<(X∗,Y∗)(X,Y)<(X^*,Y^*). Therefore, the pair (X∗ ,Y∗ )(X^* X,Y^* Y) is also lesser than (X∗,Y∗)(X^*,Y^*). If we prove that this pair belongs to NC_N, then we could obtain the ∨ -irreducible element as union of the pairs (X,Y)(X,Y) and (X∗ ,Y∗ )(X^* X,Y^* Y) which is a contradiction. Hence, we have the following equalities: (X∗ )↑N=(X∗∩Xc)↑N=(X∗)↑N∩(Xc)↑N=Y∗∩Yc=Y∗ (X^* X) _N=(X^*∩ X^c) _N=(X^*) _N∩(X^c) _N=Y^*∩ Y^c=Y^* Y Recall that the second equality holds since the necessity operator satisfies that (X1∩X2)↑N=(X1)↑N∩(X2)↑N(X_1∩ X_2) _N=(X_1) _N∩(X_2) _N. Moreover, by Lemma 9, we obtain the third equality. The other equality (Y∗ )↓N=X∗ (Y^* Y) ^N=X^* X holds in a similar way. Therefore, (X∗ ,Y∗ )∈N(X^* X,Y^* Y) _N and consequently, we have that (X∗,Y∗)=(X,Y)⊔(X∗ ,Y∗ )(X^*,Y^*)=(X,Y) (X^* X,Y^* Y) which is a contradiction. From now on, we will denote the ∨ -irreducible elements of NC_N as (X∗,Y∗)(X^*,Y^*). In addition, we can go further and determine the smallest independent subcontexts by means of the ∨ -irreducible elements of NC_N. Proposition 11 Given a context (A,B,R)(A,B,R), the set N∗C_N^* of all ∨ -irreducible elements of (N,≤)(C_N,≤) determines a partition on the set of attributes A and on the set of objects B. Proof 3 First of all, given the set N∗=(Xi∗,Yi∗)∣i∈IC_N^*=\(X^*_i,Y^*_i) i∈ I\ of all ∨ -irreducible elements of NC_N, where I is an index set, we will prove that N∗C_N^* determines a partition of the set of objects. From the descendent chain condition of the concept lattice [15], which arise because the context is finite, we have that any element different from the bottom element can be expressed as supremum of ∨ -irreducible elements of NC_N. As we previously commented, the supremum in (N,≤)(C_N,≤) is defined by the union, and therefore, we have that the top element of NC_N, (B,A)(B,A), satisfies that B=⋃i∈IXi∗B= _i∈ IX^*_i. On the other hand, by Proposition 10, we have that every pair (Xi∗,Yi∗),(Xj∗,Yj∗)∈N∗(X^*_i,Y^*_i),(X^*_j,Y^*_j) _N^* with i≠ji≠ j satisfies that Xi∗∩Xj∗=∅X^*_i∩ X^*_j= . Therefore, the set Xi∗∣i∈I\X_i^* i∈ I\ is a partition of the set of objects B. Similarly, it is obtained that the set Yi∗∣i∈I\Y_i^* i∈ I\ is a partition of the set of attributes A. The following example illustrates the previous results. Example 12 Let us consider the formal context (A,B,R)(A,B,R) associated with the data given in Table 1. In addition, the list of concepts and the Hasse diagram of the concept lattice is shown in Figure 1. R b1b_1 b2b_2 b3b_3 b4b_4 b5b_5 b6b_6 a1a_1 0 1 1 1 0 0 a2a_2 0 0 0 1 0 0 a3a_3 1 0 0 0 0 0 a4a_4 0 0 0 0 1 1 a5a_5 0 0 1 0 0 0 a6a_6 0 0 0 0 1 0 Table 1: Relation of the formal context of Example 12. C0=⟨∅,A⟩C_0= ,A C1=⟨b5,a4,a6⟩C_1= \b_5\,\a_4,a_6\ C2=⟨b5,b6,a4⟩C_2= \b_5,b_6\,\a_4\ C3=⟨b1,a3⟩C_3= \b_1\,\a_3\ C4=⟨b4,a1,a2⟩C_4= \b_4\,\a_1,a_2\ C5=⟨b3,a1,a5⟩C_5= \b_3\,\a_1,a_5\ C6=⟨b2,b3,b4,a1,a2,a5⟩C_6= \b_2,b_3,b_4\,\a_1,a_2,a_5\ C7=⟨B,∅⟩C_7= B, C0C_0 C1C_1 C5C_5 C2C_2 C3C_3 C4C_4 C7C_7 C6C_6 Figure 1: List of concepts and concept lattice of Example 12. This context is normalized. Therefore, we can apply Propositions 10 and 11, and compute the elements of the set NC_N. The list of elements of NC_N and the complete lattice333Note that the numbers that appear in the nodes of the complete lattice refer to the subscripts of the corresponding pairs in NC_N. associated with (N,≤)(C_N,≤) are given in Figure 2. (X0,Y0)=(∅,∅)(X_0,Y_0)=( , ) (X1,Y1)=(b5,b6,a4,a6)(X_1,Y_1)=(\b_5,b_6\,\a_4,a_6\) (X2,Y2)=(b2,b3,b4,a1,a2,a5)(X_2,Y_2)=(\b_2,b_3,b_4\,\a_1,a_2,a_5\) (X3,Y3)=(b2,b3,b4,b5,b6,a1,a2,a4,a5,a6)(X_3,Y_3)=(\b_2,b_3,b_4,b_5,b_6\,\a_1,a_2,a_4,a_5,a_6\) (X4,Y4)=(b1,a3)(X_4,Y_4)=(\b_1\,\a_3\) (X5,Y5)=(b1,b5,b6,a3,a4,a6)(X_5,Y_5)=(\b_1,b_5,b_6\,\a_3,a_4,a_6\) (X6,Y6)=(b1,b2,b3,b4,a1,a2,a3,a5)(X_6,Y_6)=(\b_1,b_2,b_3,b_4\,\a_1,a_2,a_3,a_5\) (X7,Y7)=(B,A)(X_7,Y_7)=(B,A) 011223344556677 Figure 2: List of elements and lattice of NC_N of Example 12. We can see that the ∨ -irreducible elements are (X1,Y1)(X_1,Y_1), (X2,Y2)(X_2,Y_2) and (X4,Y4)(X_4,Y_4), which have no concept less than themselves, except the bottom of the lattice, as Proposition 10 states. Moreover, we have that X1∪X2∪X4=BX_1∪ X_2∪ X_4=B, X1∩X2=∅X_1∩ X_2= , X1∩X4=∅X_1∩ X_4= and X2∩X4=∅X_2∩ X_4= , as Proposition 11 asserts. On the other hand, by their construction, one may think that there is no relation between the pairs of NC_N and the concepts associated with the formal context, but there exists a relevant relationship as it is shown in the following result. Specifically, this result relates the pairs of NC_N to the property-oriented concepts. Proposition 13 Given formal context (A,B,R)(A,B,R) and a pair (X,Y)∈N(X,Y) _N, satisfying that X≠∅X≠ and X≠BX≠ B, we have that the pair ⟨X,X↑π⟩ X,X _π is a concept in the property-oriented framework. Proof 4 The proof straightforwardly follows by the properties of the Galois connection (↑π,↓N)( _π, ^N). Specifically, if (X,Y)∈N(X,Y) _N, we obtain that X↑π↓N=Y↓N↑π↓N=Y↓N=X _π ^N=Y ^N _π ^N=Y ^N=X The following property relates the pairs of NC_N to the formal concepts associated with the context. Proposition 14 Let (A,B,R)(A,B,R) be a formal context and (X,Y)∈N(X,Y) _N. If X↑≠∅X ≠ , then the pair ⟨X,X↑⟩ X,X is a concept of (A,B,R)C(A,B,R), that is, X↑↓=X =X. Proof 5 If X=∅X= , the result trivially holds since X↑=∅↑=AX = =A and X↓↑=∅↓↑=∅X = = because of (A,B,R)(A,B,R) is normalized and therefore, ⟨∅,A⟩ ,A is a concept. Otherwise, let us consider X≠∅X≠ . Since the pair (↑,↓)( , ) is a Galois connection, we have that X⊆X↑↓X X . As a consequence, due to X≠∅X≠ , then X↑↓≠∅X ≠ . Now, we consider b0∈X↑↓=b∈B∣(a,b)∈R, for all a∈X↑b_0∈ X =\b∈ B (a,b)∈ R, for all a∈ X \, and we will prove that b0∈Xb_0∈ X. In particular, taking any a0∈X↑=a∈A∣(a,b)∈R, for all b∈Xa_0∈ X =\a∈ A (a,b)∈ R, for all b∈ X\, by b0∈X↑↓b_0∈ X , we have that (a0,b0)∈R(a_0,b_0)∈ R and, in addition, we have that (a0,b)∈R(a_0,b)∈ R, for all b∈Xb∈ X, but ∅≠X=X↑N↓N=b∈B∣ if (a,b)∈R then a∈X↑N ≠ X=X _N ^N=\b∈ B if (a,b)∈ R then a∈ X _N\. Thus, a0∈X↑N=a∈A∣ if (a,b)∈R then b∈Xa_0∈ X _N=\a∈ A if (a,b)∈ R then b∈ X\ and therefore, since we have that (a0,b0)∈R(a_0,b_0)∈ R and a0∈X↑Na_0∈ X _N, we can conclude b0∈Xb_0∈ X. Dually, we can obtain that the pair ⟨Y↓,Y⟩ Y ,Y is a concept of (A,B,R)C(A,B,R), when the condition Y↓≠∅Y ≠ holds. The following result shows when the top element of the independent subcontext differentiates from the top of the original concept lattice. Proposition 15 Let (A,B,R)(A,B,R) be a formal context and (X,Y)∈N(X,Y) _N, with X≠∅X≠ and X≠BX≠ B. If X↑≠∅X ≠ , then ⟨X,X↑⟩ X,X is a coatom of (A,B,R)C(A,B,R). Proof 6 We will proceed by reductio ad absurdum. We will suppose that there exists a concept ⟨X0,Y0⟩ X_0,Y_0 such that ⟨X,X↑⟩≺⟨X0,Y0⟩≺⟨B,∅⟩ X,X X_0,Y_0 B, , that is, X⊂X0⊂BX⊂ X_0⊂ B and ∅⊂Y0⊂X↑ ⊂ Y_0⊂ X . This means that there exists b0∈X0∖Xb_0∈ X_0 X. Due to X0=Y0↓X_0=Y_0 , we have that (a,b0)∈R(a,b_0)∈ R, for all a∈Y0a∈ Y_0. Moreover, due to Y0≠∅Y_0≠ (because ∅⊂Y0 ⊂ Y_0), we can consider a0∈Y0a_0∈ Y_0. Since Y0⊂X↑=a∈A∣(a,b)∈R, for all b∈XY_0⊂ X =\a∈ A (a,b)∈ R, for all b∈ X\, we have that (a0,b)∈R(a_0,b)∈ R holds, for all b∈Xb∈ X. Now, given b′∈Xb ∈ X, which exists due to by hypothesis X≠∅X≠ , since b′∈X=X↑N↓N=b∈B∣ if (a,b)∈R then a∈X↑N,b ∈ X=X _N ^N=\b∈ B if (a,b)∈ R then a∈ X _N\, we have that a0∈X↑N=a∈A∣ if (a,b)∈R then b∈Xa_0∈ X _N=\a∈ A if (a,b)∈ R then b∈ X\. Therefore, from a0∈X↑Na_0∈ X _N and (a0,b0)∈R(a_0,b_0)∈ R, we can conclude that b0∈Xb_0∈ X, which contradicts the hypothesis. Consequently, in the previous result it is shown that the top element of the concept lattice (A,B,R)C(A,B,R) is the concept directly greater than the concept ⟨X,X↑⟩ X,X associated with the pair (X,Y)∈N(X,Y) _N. A similar result can be stated by duality for the pair ⟨Y↓,Y⟩ Y ,Y . Now, the following result analyzes the relationship between ⟨X,X↑⟩ X,X and ⟨Y↓,Y⟩ Y ,Y . Proposition 16 Given a formal context (A,B,R)(A,B,R) and a pair (X,Y)∈N(X,Y) _N with X≠BX≠ B, where X↑≠∅X ≠ and Y↓≠∅Y ≠ , then we have that the concept generated by ⟨X,X↑⟩ X,X is greater than ⟨Y↓,Y⟩ Y ,Y , that is, Y↓⊆XY X. Proof 7 Given b0∈Y↓=b∈B∣(a,b)∈R, for all a∈Yb_0∈ Y =\b∈ B (a,b)∈ R, for all a∈ Y\. Hence, for a0∈Ya_0∈ Y we have that (a0,b0)∈R(a_0,b_0)∈ R and since Y=Y↓N↑N=a∈A∣ if (a,b′)∈R then b′∈Y↓NY=Y ^N _N=\a∈ A if (a,b )∈ R then b ∈ Y ^N\, we have straightforwardly that b0∈Y↓N=Xb_0∈ Y ^N=X. Note that the pairs (X,X↑)(X,X ) and (Y↓,Y)(Y ,Y) do not necessarily have to be concepts from the original concept lattice. This happens when X↑=∅X = (or Y↓=∅Y = ) hold. In this case, we can find more than one concept that satisfies Proposition 15 and Proposition 16, that is, there exist maximal elements satisfying these properties (equivalently with the minimal elements, when Y↓=∅Y = holds). Therefore, when we find several independent subcontexts in this situation, the top elements of the concept lattices associated with these independent subcontexts are identified by the top element of the original concept lattice (similarly, when the condition Y↓=∅Y = holds, the bottom elements of the concept lattices are identified by the bottom element of the original concept lattice). In the following example, the previously introduced properties are illustrated. Example 17 Let us consider a context (A,B,R)(A,B,R) where the set of attributes is given by A=a1,a2,a3,a4,a5,a6,a7,a8A=\a_1,a_2,a_3,a_4,a_5,a_6,a_7,a_8\, the set of objects is B=b1,b2,b3,b4,b5,b6,b7B=\b_1,b_2,b_3,b_4,b_5,b_6,b_7\ and the relationship R:A×B→0,1R A× B→\0,1\ is given in Table 2. R b1b_1 b2b_2 b3b_3 b4b_4 b5b_5 b6b_6 b7b_7 a1a_1 0 1 1 1 0 0 0 a2a_2 0 0 0 1 0 0 0 a3a_3 1 0 0 0 0 0 0 a4a_4 0 0 0 0 0 1 1 a5a_5 0 1 1 0 0 0 0 a6a_6 0 0 0 0 1 1 0 a7a_7 0 0 1 0 0 0 0 a8a_8 0 0 0 0 1 1 1 Table 2: Relation of the formal context of Example 17. First of all, we have to compute all pairs of NC_N. These pairs, except the trivial ones, are the following: (X1,Y1)= (X_1,Y_1)= (b5,b6,b7,a4,a6,a8) (\b_5,b_6,b_7\,\a_4,a_6,a_8\) (X2,Y2)= (X_2,Y_2)= (b1,a3) (\b_1\,\a_3\) (X3,Y3)= (X_3,Y_3)= (b2,b3,b4,a1,a2,a5,a7) (\b_2,b_3,b_4\,\a_1,a_2,a_5,a_7\) (X4,Y4)= (X_4,Y_4)= (b1,b5,b6,b7,a3,a4,a6,a8) (\b_1,b_5,b_6,b_7\,\a_3,a_4,a_6,a_8\) (X5,Y5)= (X_5,Y_5)= (b1,b2,b3,b4,a1,a2,a3,a5,a7) (\b_1,b_2,b_3,b_4\,\a_1,a_2,a_3,a_5,a_7\) (X6,Y6)= (X_6,Y_6)= (b2,b3,b4,b5,b6,b7,a1,a2,a4,a5,a7) (\b_2,b_3,b_4,b_5,b_6,b_7\,\a_1,a_2,a_4,a_5,a_7\) If we analyze the first pair, (X1,Y1)(X_1,Y_1), we can see that X1↑=a8≠∅X_1 =\a_8\≠ and, moreover, a8↓=b5,b6,b7=X\a_8\ =\b_5,b_6,b_7\=X. This means that X1↑↓=X1X_1 =X_1, that is, ⟨X1,X1↑⟩ X_1,X_1 is a formal concept as Proposition 14 states. In particular, ⟨X1,X1↑⟩=C8 X_1,X_1 =C_8 as it can be seen in the list of concepts shown in Figure 3, which also shows the concept lattice associated with the context. In this concept lattice can be checked that C8C_8 is less than the concept C10C_10, which is the maximum element of the lattice, and there is no intermediate concept between them, that is, the pair ⟨X1,X1↑⟩ X_1,X_1 also satisfies Proposition 15. Now, let us check that the pair (X1,Y1)(X_1,Y_1) also satisfies Proposition 16. For that, we need to see that Y1↓≠∅Y_1 ≠ . Indeed, Y1↓=b6≠∅Y_1 =\b_6\≠ , that is, ⟨Y1↓,Y1⟩=C1 Y_1 ,Y_1 =C_1 and moreover Y1↓=b6⊆b5,b6,b7=XY_1 =\b_6\ \b_5,b_6,b_7\=X as Proposition 16 states. As a conclusion, the pair (X1,Y1)(X_1,Y_1) satisfies the three properties we have presented, that is, the pair (X1,Y1)(X_1,Y_1) determines one block (interval) of concepts bounded by the concepts C8C_8 and C1C_1, as can be seen in Figure 3. C0=⟨∅,A⟩C_0= ,A C1=⟨b6,a4,a6,a8⟩C_1= \b_6\,\a_4,a_6,a_8\ C2=⟨b3,a1,a5,a7⟩C_2= \b_3\,\a_1,a_5,a_7\ C3=⟨b5,b6,a6,a8⟩C_3= \b_5,b_6\,\a_6,a_8\ C4=⟨b6,b7,a4,a8⟩C_4= \b_6,b_7\,\a_4,a_8\ C5=⟨b1,a3⟩C_5= \b_1\,\a_3\ C6=⟨b4,a1,a2⟩C_6= \b_4\,\a_1,a_2\ C7=⟨b2,b3,a1,a5⟩C_7= \b_2,b_3\,\a_1,a_5\ C8=⟨b5,b6,b7,a8⟩C_8= \b_5,b_6,b_7\,\a_8\ C9=⟨b2,b3,b4,a1⟩C_9= \b_2,b_3,b_4\,\a_1\ C10=⟨B,∅⟩C_10= B, C0C_0 C1C_1 C2C_2 C3C_3 C4C_4 C5C_5 C6C_6 C7C_7 C8C_8 C10C_10 C9C_9 Figure 3: List of concepts and concept lattice of Example 17. On the other hand, if we focus on the pair (X3,Y3)(X_3,Y_3) and we follow a similar procedure to the one given for the pair (X1,Y1)(X_1,Y_1), we can check that ⟨X3,X3↑⟩=⟨b2,b3,b4,a1⟩=C9 X_3,X_3 = \b_2,b_3,b_4\,\a_1\ =C_9 and C9C_9 is less than C10C_10 and there is no intermediate concept between C9C_9 and C10C_10, that is, the pair satisfies Proposition 14 and Proposition 15. However, if we compute the extent of Y3Y_3 we obtain that Y3↓=∅Y_3 = and, therefore, (Y3↓,Y3)(Y_3 ,Y_3) is not a concept and does not satisfy any of the presented properties. However, the pair (X3,Y3)(X_3,Y_3) still defines a block of concepts comprised by C0C_0, C2C_2, C6C_6, C7C_7 and C9C_9. In other words, the fact that the pair (Y3↓,Y3)(Y_3 ,Y_3) is not a concept means that the bottom element of the concept lattice associated with the subcontext generated by (X3,Y3)(X_3,Y_3) is identified by the bottom element of the original concept lattice ⟨∅,A⟩ ,A . Notice that, in all the previous results, we are requiring that X↑≠∅X ≠ , that is, there exists at least one attribute in the subcontext which is shared by all the objects of the considered subcontext (X,Y,RX×Y)(X,Y,R_X× Y) where RX×YR_X× Y denotes the restriction of the relation R to the subsets X and Y. Equivalently, Y↓≠∅Y ≠ indicates the existence of at least one object that possesses all the attributes of the subcontext. 4 Toward the factorization of contexts in multi-adjoint frameworks We are interested in generalizing the factorization of formal contexts into independent subcontexts in the fuzzy environment provided by the multi-adjoint framework. With that goal, in this section we study how the properties and results presented in the previous section can be translated into the multi-adjoint framework. In order to simplify the results presented in this section, we will consider a multi-adjoint frame with only one adjoint triple. Therefore, we fix the frame (L1,L2,P,&,↙,↖)(L_1,L_2,P,\&, , ) and the context (A,B,R)(A,B,R), where the mapping σ does not appear since it is not necessary when we consider a single adjoint triple. As we commented in the previous section, to find independent subcontexts in the classical case, one of the restrictions imposed on the relation of the context is that it should be normalized. The fuzzy relations of the contexts in this fuzzy environment should also keep the direct extension of this condition to the fuzzy case, i.e., there are no rows or columns with all values equal to bottom or with all values different from bottom. In this environment, we consider the pair (g,f)(g,f), with g∈L2Bg∈ L_2^B and f∈L1Af∈ L_1^A, satisfying the equalities g↑N=f and f↓N=g,g _N=f and f ^N=g, but considering the generalized versions of the necessity operators, given in Section 2.2. Specifically, we will consider the set: ℱN=(g,f)∣g∈L2B,f∈L1A,f↓N=g,g↑N=fF_N=\(g,f) g∈ L_2^B,f∈ L_1^A,f ^N=g,g _N=f\ Note that ℱN≠∅F_N≠ since it has at least one element as the following lemma states. Lemma 18 Given a frame (L1,L2,P,&,↙,↖)(L_1,L_2,P,\&, , ) and a context (A,B,R)(A,B,R), then the pair (g⊤,f⊤)(g_ ,f_ ) is an element of the set ℱNF_N. Proof 8 We have to prove that the pair (g⊤,f⊤)(g_ ,f_ ) satisfies that f⊤↓N=g⊤f_ ^N=g_ and g⊤↑N=f⊤g_ _N=f_ . We will prove that g⊤↑N=f⊤g_ _N=f_ , the other equality is obtained analogously. For every a∈Aa∈ A, we have that: g⊤↑N(a) g_ _N(a) =infg⊤(b)↙R(a,b)∣b∈B = \g_ (b) R(a,b) b∈ B\ =inf⊤2↙R(a,b)∣b∈B = \ _2 R(a,b) b∈ B\ =(∗)inf⊤1∣b∈B (*)= \ _1 b∈ B\ =⊤1 = _1 =f⊤(a) =f_ (a) Note that (∗)(*) holds by Proposition 4. Therefore, g⊤↑N=f⊤g_ _N=f_ . However, one might think that the pair (g⊥,f⊥)(g_ ,f_ ) is also an element of ℱNF_N, but this is not true in general as the following example illustrates. Example 19 Let us consider the frame ([0,1],[0,1],[0,1],≤,≤,≤,&Ł)([0,1],[0,1],[0,1],≤,≤,≤,\&_ ), where &Ł\&_ is the Łukasiewicz conjunctor defined for every x,y∈[0,1]x,y∈[0,1] as: x&Ły=max0,x+y−1x\&_ y= \0,x+y-1\ The corresponding residuated implications ↙Ł,↖Ł:[0,1]×[0,1]→[0,1] , _ :[0,1]×[0,1]→[0,1] are defined for every y,z∈[0,1]y,z∈[0,1] as: z↙Ły=z↖Ły=min1,1−y+zz y=z _ y= \1,1-y+z\ R b1b_1 b2b_2 a1a_1 0.5 0 a2a_2 0 0.75 Table 3: Fuzzy relation R of Example 19. In addition, we consider the context (A,B,R)(A,B,R) where A=a1,a2A=\a_1,a_2\, B=b1,b2B=\b_1,b_2\ and the fuzzy relation R is given in Table 3. Now, in order to show that (g⊥,f⊥)(g_ ,f_ ) does not belong to ℱNF_N, it is sufficient to verify that g⊥↑N≠f⊥g_ _N≠ f_ . Thus, we apply the operator ↑N _N to g⊥g_ , for every a∈Aa∈ A, that is: g⊥↑N(a1) g_ _N(a_1) =infg⊥(b)↙ŁR(a1,b)∣b∈B = \g_ (b) R(a_1,b) b∈ B\ =inf0↙Ł0.5,0↙Ł0 = \0 0.5,0 0\ =inf0.5,1 = \0.5,1\ =0.5 =0.5 g⊥↑N(a2) g_ _N(a_2) =infg⊥(b)↙ŁR(a2,b)∣b∈B = \g_ (b) R(a_2,b) b∈ B\ =inf0↙Ł0,0↙Ł0.75 = \0 0,0 0.75\ =inf1,0.25 = \1,0.25\ =0.25 =0.25 Clearly, g⊥↑N≠f⊥g_ _N≠ f_ . Therefore, we have that (g⊥,f⊥)∉ℱN(g_ ,f_ ) _N. On the other hand, it is known that the set NC_N has the structure of a complete lattice, as we commented in the previous section. The following result proves that the fuzzy extension of NC_N, that is, the set ℱNF_N also has the structure of a complete lattice. Proposition 20 Let (L1,L2,P,&,↙,↖)(L_1,L_2,P,\&, , ) be a frame and let (A,B,R)(A,B,R) be a context. Then, the elements in ℱNF_N form a complete lattice with the ordering (g1,f1)≤(g2,f2)(g_1,f_1)≤(g_2,f_2) if and only if g1⪯2g2g_1 _2g_2, or equivalently, if and only if f1⪯1f2f_1 _1f_2. Proof 9 Let us consider a family of pairs (gi,fi)i∈I⊆ℱN\(g_i,f_i)\_i∈ I _N, with I an index set. Now, we will prove (⋀i∈Igi,⋀i∈Ifi)∈ℱN( _i∈ Ig_i, _i∈ If_i) _N. (⋀i∈Igi)↑N(a) ( _i∈ Ig_i ) _N(a) =inf(⋀i∈Igi(b))↙R(a,b)∣b∈B = \ ( _i∈ Ig_i(b) ) R(a,b) b∈ B\ =(∗)inf⋀i∈I(gi(b)↙R(a,b))∣b∈B (*)= \ _i∈ I(g_i(b) R(a,b)) b∈ B\ =⋀i∈I(infgi(b)↙R(a,b)∣b∈B) = _i∈ I ( \g_i(b) R(a,b) b∈ B\ ) =⋀i∈Igi↑N(a) = _i∈ Ig_i _N(a) =⋀i∈Ifi(a) = _i∈ If_i(a) Note that (∗)(*) holds by Proposition 4. Analogously, we obtain that (⋀i∈Ifi)↓N=⋀i∈Igi( _i∈ If_i) ^N= _i∈ Ig_i. Moreover, (⋀i∈Igi,⋀i∈Ifi)( _i∈ Ig_i, _i∈ If_i) clearly is the greatest lower bound of (gi,fi)i∈I\(g_i,f_i)\_i∈ I. Therefore, ⋀i∈I(gi,fi)=(⋀i∈Igi,⋀i∈Ifi)∈ℱN _i∈ I(g_i,f_i)= ( _i∈ Ig_i, _i∈ If_i ) _N Furthermore, by Lemma 18, ℱNF_N has a top element, this is (g⊤,f⊤)∈ℱN(g_ ,f_ ) _N Thus, ℱNF_N forms a complete lattice. Although we have seen that the elements of NC_N and ℱNF_N have the same algebraic structure, the properties satisfied by the elements of NC_N and ℱNF_N have remarkable differences. For example, in the classical case, the non-existence of independent blocks entails that the set NC_N only contains the trivial pairs. However, in the fuzzy case, even when there are no independent subcontexts the set ℱNF_N may have more elements. This fact is illustrated in the following example. Example 21 Consider the multi-adjoint frame ([0,1]4,[0,1]4,[0,1]4,≤,≤,≤,&P∗)([0,1]_4,[0,1]_4,[0,1]_4,≤,≤,≤,\&^*_P) where &P∗\&^*_P is the discretization of the product conjunctor (Example 5). The context is given by the set of attributes A=a1,a2,a3A=\a_1,a_2,a_3\, the set of objects B=b1,b2,b3B=\b_1,b_2,b_3\ and the relation R1:A×B→[0,1]4R_1 A× B→[0,1]_4 shown on the left side of Figure 4. R1R_1 b1b_1 b2b_2 b3b_3 a1a_1 0.5 0.5 1 a2a_2 0.25 1 0 a3a_3 0 0.75 0.25 Figure 4: Fuzzy relation R1R_1 and concept lattice of Example 21. We can observe that non-independent blocks of concepts exist, as it can be checked in the concept lattice shown in Figure 4. In this case, the set ℱNF_N contains several elements, despite not containing independent subcontexts. For instance, the pair (g,f)=(b1/1,b2/0.5,b3/0.5,a1/0.5,a2/0.5,a3/0.5)(g,f)=(\b_1/1,b_2/0.5,b_3/0.5\,\a_1/0.5,a_2/0.5,a_3/0.5\) satisfies that g↑N=fg _N=f and f↓N=gf ^N=g, that is, it is an element of ℱNF_N. Consequently, the existence of pairs in ℱNF_N does not determine independent blocks of concepts, that is, independent subcontexts. Notice that the fuzzy relation R1R_1 is not normalized because the first row has all the values different from zero. Nevertheless, if the fuzzy relation of the context were normalized, it would be possible to find these independent blocks of concepts, as it happens in the context (A,B,R2)(A,B,R_2) and it is shown in Figure 5. R2R_2 b1b_1 b2b_2 b3b_3 a1a_1 0.5 0 1 a2a_2 0 0.5 0 a3a_3 0.75 0 0.25 Figure 5: Fuzzy relation R2R_2 and concept lattice of Example 21. (g⊥,f⊥)=(b1/0,b2/0,b3/0,a1/0,a2/0,a3/0)(g_ ,f_ )=(\b_1/0,b_2/0,b_3/0\,\a_1/0,a_2/0,a_3/0\) (g1,f1)=(b1/0,b2/1b3/0,a1/0,a2/1,a3/0)(g_1,f_1)=(\b_1/0,b_2/1b_3/0\,\a_1/0,a_2/1,a_3/0\) (g2,f2)=(b1/0.25,b2/0,b3/0.25,a1/0.25,a2/0,a3/0.25)(g_2,f_2)=(\b_1/0.25,b_2/0,b_3/0.25\,\a_1/0.25,a_2/0,a_3/0.25\) (g3,f3)=(b1/0.25,b2/0,b3/0.5,a1/0.5,a2/0,a3/0.25)(g_3,f_3)=(\b_1/0.25,b_2/0,b_3/0.5\,\a_1/0.5,a_2/0,a_3/0.25\) (g4,f4)=(b1/0.5,b2/0,b3/0.25,a1/0.25,a2/0,a3/0.5)(g_4,f_4)=(\b_1/0.5,b_2/0,b_3/0.25\,\a_1/0.25,a_2/0,a_3/0.5\) (g5,f5)=(b1/0.5,b2/0,b3/1,a1/1,a2/0,a3/0.5)(g_5,f_5)=(\b_1/0.5,b_2/0,b_3/1\,\a_1/1,a_2/0,a_3/0.5\) (g6,f6)=(b1/0.5,b2/1,b3/0.25,a1/0.25,a2/1,a3/0.5)(g_6,f_6)=(\b_1/0.5,b_2/1,b_3/0.25\,\a_1/0.25,a_2/1,a_3/0.5\) (g7,f7)=(b1/0.5,b2/1,b3/0.5,a1/0.5,a2/1,a3/0.5)(g_7,f_7)=(\b_1/0.5,b_2/1,b_3/0.5\,\a_1/0.5,a_2/1,a_3/0.5\) (g8,f7)=(b1/0.5,b2/1,b3/0.75,a1/0.75,a2/1,a3/0.5)(g_8,f_7)=(\b_1/0.5,b_2/1,b_3/0.75\,\a_1/0.75,a_2/1,a_3/0.5\) (g9,f9)=(b1/0.5,b2/1,b3/1,a1/1,a2/1,a3/0.5)(g_9,f_9)=(\b_1/0.5,b_2/1,b_3/1\,\a_1/1,a_2/1,a_3/0.5\) (g10,f10)=(b1/1,b2/0,b3/1,a1/1,a2/0,a3/1)(g_10,f_10)=(\b_1/1,b_2/0,b_3/1\,\a_1/1,a_2/0,a_3/1\) (g⊤,f⊤)=(b1/1,b2/1,b3/1,a1/1,a2/1,a3/1)(g_ ,f_ )=(\b_1/1,b_2/1,b_3/1\,\a_1/1,a_2/1,a_3/1\) Table 4: Some pairs of ℱNF_N in the context (A,B,R2)(A,B,R_2) of Example 21. The set ℱNF_N in the context (A,B,R2)(A,B,R_2) consists of 20 pairs, some of them are listed in Table 4. For instance, the pair (g10,f10)(g_10,f_10) characterizes the block delimited by the concepts C2C_2 and C9C_9 shown in the concept lattice in Figure 5. However, not all of them provide independent blocks, which raises the need to carry out the study of more properties of the set ℱNF_N. The following result shows the relation between the possibility and necessity operators applied to the same fuzzy set of objects g of a considered pair (g,f)∈ℱN(g,f) _N. Proposition 22 Given a frame (L1,L2,P,&,↙,↖)(L_1,L_2,P,\&, , ), a context (A,B,R)(A,B,R) and a pair (g,f)∈ℱN(g,f) _N, we have that g↑π(a)⪯1g↑N(a)g _π(a) _1g _N(a), for all a∈Aa∈ A. Proof 10 Given (g,f)∈ℱN(g,f) _N, and b∈Bb∈ B, we have that the expression: g(b)=g↑N↓N(b)=infinfg(b′)↙R(a′,b′)∣b′∈B↖R(a′,b)∣a′∈Ag(b)=g _N ^N(b)= \ \g(b ) R(a ,b ) b ∈ B\ R(a ,b) a ∈ A\ Therefore, by Proposition 4, we have that g(b)=inf(g(b′)↙R(a′,b′))↖R(a′,b)∣a′∈A,b′∈Bg(b)= \(g(b ) R(a ,b )) R(a ,b) a ∈ A,b ∈ B\. Thus, given a′∈Aa ∈ A and b′∈Bb ∈ B g(b)⪯1(g(b′)↙R(a′,b′))↖R(a′,b)g(b) _1(g(b ) R(a ,b )) R(a ,b) and applying the adjoint property (Equivalence (1)) we obtain that R(a′,b)&g(b)⪯1g(b′)↙R(a′,b′)R(a ,b)\&g(b) _1g(b ) R(a ,b ) Now, we can apply the supremum on the left argument of the previous inequality and infimum in the right argument, obtaining that: supR(a′,b)&g(b)∣b∈B⪯1infg(b′)↙R(a′,b′)∣b′∈B \R(a ,b)\&g(b) b∈ B\ _1 \g(b ) R(a ,b ) b ∈ B\ Therefore, g↑π(a′)⪯1g↑N(a′),g _π(a ) _1g _N(a ), for all a′∈Aa ∈ A. The following property relates the closure operator that arises from the composition of the mappings of the pair (↑π,↓N)( _π, ^N) to the composition of the necessity operators ↑N _N and ↓N ^N. Proposition 23 Let (L1,L2,P,&,↙,↖)(L_1,L_2,P,\&, , ) be a frame, (A,B,R)(A,B,R) be a context and (g,f)∈ℱN(g,f) _N. Then, g↑π↓Ng _π ^N=g=g, and therefore ⟨g,g↑π⟩∈ℳπN g,g _π _π N. Proof 11 By Proposition 22, we have that g↑π⪯1g↑Ng _π _1g _N. If we apply the operator ↓N ^N, we have that g↑π↓N⪯2g↑N↓Ng _π ^N _2g _N ^N, since ↓N ^N is order preserving. In addition, we have that g⪯2g↑π↓Ng _2g _π ^N, since the composition of the mappings in the pair (↑π,↓N)( _π, ^N) is a closure operator and g↑N↓N=gg _N ^N=g, by hypothesis. Therefore, we obtain that g⪯2g↑π↓N⪯2g↑N↓N=g _2g _π ^N _2g _N ^N=g Thus, we can conclude that the pair ⟨g,g↑π⟩ g,g _π is a concept of the property-oriented concept lattice. Hence, the previous result asserts that the pair ⟨g,g↑π⟩ g,g _π is always a property-oriented concept, whenever the pair (g,f)(g,f) is in ℱNF_N, as in the Boolean case. Nevertheless, the extension of Proposition 14 to the fuzzy framework does not hold, that is, the pair (g,g↑)(g,g ) is not a concept in general, as the following example shows. Example 24 Considering again the context (A,B,R2)(A,B,R_2) of Example 21, we focus on the pair of ℱNF_N: (g2,f2)=(b1/0.25,b2/0,b3/0.25,a1/0.25,a2/0,a3/0.25)(g_2,f_2)=(\b_1/0.25,b_2/0,b_3/0.25\,\a_1/0.25,a_2/0,a_3/0.25\) Applying the operator ↑ to the fuzzy subset g2g_2, we obtain that: (g2,g2↑)=(b1/0.25,b2/0,b3/0.25,a1/1,a2/0,a3/1)(g_2,g_2 )=(\b_1/0.25,b_2/0,b_3/0.25\,\a_1/1,a_2/0,a_3/1\) However, although g2↑g_2 is different from g⊥g_ , if we compute g2↑↓g_2 we have that g2↑↓=b1/0.5,b2/0,b3/0.5g_2 =\b_1/0.5,b_2/0,b_3/0.5\. Therefore, we can conclude that g2↑↓≠g2g_2 ≠ g_2 and, as a consequence, the pair (g2,g2↑)(g_2,g_2 ) is not a concept. As a consequence, the pairs (g,f)(g,f) in ℱNF_N do not provide in general a concept in ℳM either with extent g or intent f. Therefore, Proposition 15 and Proposition 16 cannot be extended to the multi-adjoint frame either. Due to these properties given in the Boolean case are not satisfied in the fuzzy framework, it is necessary to set some assumptions which allow us to obtain good properties related to the factorization of fuzzy contexts. 4.1 ⊤ -normalized contexts From now on, we will consider a framework for which interesting results about the factorization of contexts in the fuzzy case can be obtained. Specifically, we will consider the following kind of formal context. Definition 25 A normalized context (A,B,R)(A,B,R), satisfying that for every a∈Aa∈ A, there exist ba∈Bb_a∈ B such that R(a,ba)=⊤PR(a,b_a)= _P is called ⊤ -normalized context. Notice that, the assumption on R means that the fuzzy subsets associated with every row of R be a normal fuzzy subset, in this case, we will say that R is normal by rows. This fact is not restrictive because, for example, if we are handling real numbers, it is usual to normalize (divide by the greatest element of the row) in order to obtain values in the unit interval, if they are not in this interval, or to extend them, if they are mainly concentrated in a part of the unit interval. Moreover, we will consider the frame ([0,1],≤,&G)([0,1],≤,\&_G) where &G\&_G is the Gödel conjunctor, that is, the operator defined for every x,y∈[0,1]x,y∈[0,1] as: x&Gy=minx,yx\&_Gy= \x,y\ The corresponding residuated implications ↙G,↖G:[0,1]×[0,1]→[0,1] ^G, _G:[0,1]×[0,1]→[0,1] are defined for every y,z∈[0,1]y,z∈[0,1] as: z↙Gy=z↖Gy=1 if y≤z otherwise z ^Gy=z _Gy= \ array[]l1& if y≤ z\\ z& otherwise array . Under these new hypotheses, the opposite inequality to the one given in Proposition 22 also holds, as the following result shows. Proposition 26 Given the frame ([0,1],≤,&G)([0,1],≤,\&_G), a ⊤ -normalized context (A,B,R)(A,B,R) and a pair (g,f)∈ℱN(g,f) _N, it is satisfied that g↑N≤g↑πg _N≤g _π. Proof 12 Let us consider an arbitrary attribute a∈Aa∈ A and a pair (g,f)∈ℱN(g,f) _N. By definition of the necessity operator, we have that: g↑N(a)=infg(b)↙GR(a,b)∣b∈Bg _N(a)= \g(b) ^GR(a,b) b∈ B\ Furthermore, taking into account the definition of the Gödel residuated implication, the previous equality can be written as follows: g↑N(a)=infg(b)∣b∈B and R(a,b)≰g(b)g _N(a)= \g(b) b∈ B and R(a,b) g(b)\ Notice that, if the set g(b)∣b∈B and R(a,b)≰g(b)\g(b) b∈ B and R(a,b) g(b)\ is the empty set, then its infimum is the maximum of the frame, that is, 1 in this case. In addition, since the context is a ⊤ -normalized context, there exists at least one object b′b such that R(a,b′)=1R(a,b )=1. Now, we have to distinguish two cases: 1. If R(a,b′)≤g(b′)R(a,b )≤ g(b ), then R(a,b′)&Gg(b′)=minR(a,b′),g(b′)=1R(a,b )\&_Gg(b )= \R(a,b ),g(b )\=1 and, as a consequence, 1=supR(a,b)&Gg(b)∣b∈B=g↑π(a)1= \R(a,b)\&_Gg(b) b∈ B\=g _π(a). Therefore, the inequality g↑N(a)≤g↑π(a)g _N(a)≤ g _π(a) holds. 2. Otherwise, if R(a,b′)≰g(b′)R(a,b ) g(b ), then g(b′)∈g(b)∣b∈B and R(a,b)≰g(b)g(b )∈\g(b) b∈ B and R(a,b) g(b)\. Therefore, we have that: g↑N(a)=infg(b)∣b∈B and R(a,b)≰g(b)≤g(b′)g _N(a)= \g(b) b∈ B and R(a,b) g(b)\≤ g(b ) On the other hand, since R(a,b′)≰g(b′)R(a,b ) g(b ) the following chain holds: g(b′)=minR(a,b′),g(b′)≤supR(a,b)&Gg(b)∣b∈B=g↑π(a)g(b )= \R(a,b ),g(b )\≤ \R(a,b)\&_Gg(b) b∈ B\=g _π(a) Therefore, from both inequalities we have that g↑N(a)≤g↑π(a)g _N(a)≤ g _π(a), for all a∈Aa∈ A. Consequently, from the previous items we conclude that g↑N≤g↑πg _N≤ g _π. Furthermore, we can also establish a relationship between the derivation operators and the possibility operators, as the following result states. Proposition 27 Given the frame ([0,1],≤,&G)([0,1],≤,\&_G), a ⊤ -normalized context (A,B,R)(A,B,R) and a pair (g,f)∈ℱN(g,f) _N satisfying that for every a∈Aa∈ A, there exists b∈Bb∈ B such that g(b)≰R(a,b)g(b) R(a,b), then the inequality g↑≤g↑πg ≤g _π holds. Proof 13 Let us consider an arbitrary attribute a∈Aa∈ A and a pair (g,f)∈ℱN(g,f) _N satisfying g(b′)≰R(a,b′)g(b ) R(a,b ), with b′∈Bb ∈ B. By definition, we have that g↑(a)=infR(a,b)↙Gg(b)∣b∈Bg (a)= \R(a,b) _Gg(b) b∈ B\. Taking into account the definition of the Gödel residuated implication, the previous equality can be written as: g↑(a)=infR(a,b)∣b∈B and g(b)≰R(a,b)g (a)= \R(a,b) b∈ B and g(b) R(a,b)\ Notice that the set R(a,b)∣b∈B and g(b)≰R(a,b)\R(a,b) b∈ B and g(b) R(a,b)\ is not empty, since R(a,b′)R(a,b ) belongs to it. Therefore, we have that: g↑(a)=infR(a,b)∣b∈B and g(b)≰R(a,b)≤R(a,b′)g (a)= \R(a,b) b∈ B and g(b) R(a,b)\≤ R(a,b ) In addition, we know that b′∈Bb ∈ B satisfies that g(b′)≰R(a,b′)g(b ) R(a,b ), from which the following chain is deduced: R(a,b′)=minR(a,b′),g(b′)≤supR(a,b)&Gg(b)∣b∈B=g↑π(a)R(a,b )= \R(a,b ),g(b )\≤ \R(a,b)\&_Gg(b) b∈ B\=g _π(a) Consequently, from both inequalities we obtain that g↑≤g↑πg ≤ g _π. The following result shows another particular case in which the inequality given in the previous result is also satisfied. Proposition 28 Given the frame ([0,1],≤,&G)([0,1],≤,\&_G), a ⊤ -normalized context (A,B,R)(A,B,R) and a pair (g,f)∈ℱN(g,f) _N, if for any a∈Aa∈ A there exists b∈Bb∈ B such that R(a,b)=g(b)=1R(a,b)=g(b)=1, then g↑π=g⊤g _π=g_ and, in particular g↑≤g↑πg ≤g _π is satisfied. Proof 14 We consider a∈Aa∈ A such that there exists b′∈Bb ∈ B such that R(a,b′)=g(b′)=1R(a,b )=g(b )=1, then g↑π(a)=supR(a,b)&Gg(b)∣b∈B=1g _π(a)= \R(a,b)\&_Gg(b) b∈ B\=1 and the inequality g↑(a)≤g↑π(a)g (a)≤ g _π(a) trivially holds. As we have already seen in Example 21, the existence of pairs of ℱNF_N does not imply that these pairs determine independent blocks of concepts. In what follows, we will show under what conditions these pairs provide intervals or blocks of concepts (that could not be independent). Taking into account Proposition 26 and Proposition 27, we can find a relationship between the two concepts ⟨g↑↓,g↑⟩ g ,g and ⟨f↓,f↓↑⟩ f ,f obtained from pairs in ℱNF_N. Proposition 29 Let (A,B,R)(A,B,R) be ⊤ -normalized context, the frame ([0,1],≤,&G)([0,1],≤,\&_G) and a pair (g,f)∈ℱN(g,f) _N satisfying that for every a∈Aa∈ A there exists b∈Bb∈ B such that g(b)≰R(a,b)g(b) R(a,b). Then, the inequality f↓≤g↑↓f ≤g holds and, therefore, ⟨f↓,f↓↑⟩⪯⟨g↑↓,g↑⟩ f ,f g ,g . Proof 15 Applying the operator ↓ to the inequality given by Proposition 27, we obtain that g↑π↓≤g↑↓g _π ≤ g , since the operator ↓ is order-reversing. Moreover, taking into consideration Proposition 22 and Proposition 26, the equality g↑π=g↑Ng _π=g _N holds. Therefore, replacing g↑πg _π by g↑Ng _N in the first inequality, we have that f↓=g↑N↓=g↑π↓≤g↑↓f =g _N =g _π ≤ g . Furthermore, the following result shows another situation in which the inequality shown in the previous proposition is also satisfied. Proposition 30 Let (A,B,R)(A,B,R) be a ⊤ -normalized context, the frame ([0,1],≤,&G)([0,1],≤,\&_G) and a pair (g,f)∈ℱN(g,f) _N. Given a∈Aa∈ A if there exists b∈Bb∈ B satisfying that R(a,b)=g(b)=1R(a,b)=g(b)=1, the inequality f↓≤g↑↓f ≤g is satisfied, and so ⟨f↓,f↓↑⟩⪯⟨g↑↓,g↑⟩ f ,f g ,g . Proof 16 Similarly to the proof given in Proposition 29, but taking into account Proposition 28 instead of Proposition 27. Notice that, if we consider in Definition 25 that the fuzzy subsets associated with the columns of R be normal fuzzy subsets, that is, R is normal by columns instead of by rows, then we obtain Propositions 26, 27, and 28 related to the mapping f of the pair (g,f)∈ℱN(g,f) _N. As a consequence, Propositions 29 and 30 also hold, if R is normal by columns. In the following example, we illustrate all the previous results. Example 31 Given the frame ([0,1],≤,&G)([0,1],≤,\&_G), and the context (A,B,R1)(A,B,R_1) composed of the set of attributes A=a1,a2,a3A=\a_1,a_2,a_3\, the set of objects B=b1,b2,b3B=\b_1,b_2,b_3\ and the relation R1:A×B→[0,1]R_1 A× B→[0,1] defined in Table 5. R1R_1 b1b_1 b2b_2 b3b_3 a1a_1 0.5 0.25 0 a2a_2 0.5 1.0 0 a3a_3 0 0 0.75 Table 5: Fuzzy relation R1R_1 of Example 31. As we can observe, the fuzzy relation R1R_1 is a normalized context, but it is not a normal fuzzy relation, therefore it is not a ⊤ -normalized context. This fact gives rise to not satisfying the hypotheses of Proposition 26. For instance, the pair (g,f)=(b1/1,b2/0.5,b3/0,a1/1,a2/0.5,a3/0)(g,f)=(\b_1/1,b_2/0.5,b_3/0\,\a_1/1,a_2/0.5,a_3/0\) belongs to ℱNF_N and, if we apply the possibility operator to the fuzzy subset g, we obtain that: g↑N=f=a1/1,a2/0.5,a3/0≰a1/0.5,a2/0.5,a3/0=g↑πg _N=f=\a_1/1,a_2/0.5,a_3/0\ \a_1/0.5,a_2/0.5,a_3/0\=g _π R2R_2 b1b_1 b2b_2 b3b_3 a1a_1 1 0.25 0 a2a_2 0.5 1 0 a3a_3 0 0 1 Table 6: Fuzzy relation R2R_2 of Example 31. Now, we consider a ⊤ -normalized context (A,B,R2)(A,B,R_2) in the same frame and where the fuzzy relation R2R_2 is given in Table 6. We can compute all pairs from ℱNF_N, some of them are listed in Table 7. The associated concept lattice of the context along with the list of concepts are shown in Figure 6. (g⊥,f⊥)=(b1/0,b2/0,b3/0,a1/0,a2/0,a3/0)(g_ ,f_ )=(\b_1/0,b_2/0,b_3/0\,\a_1/0,a_2/0,a_3/0\) (g1,f1)=(b1/0,b2/0,b3/0.5,a1/0,a2/0,a3/0.5)(g_1,f_1)=(\b_1/0,b_2/0,b_3/0.5\,\a_1/0,a_2/0,a_3/0.5\) (g2,f2)=(b1/0.75,b2/0.5,b3/0,a1/0.75,a2/0.5,a3/0)(g_2,f_2)=(\b_1/0.75,b_2/0.5,b_3/0\,\a_1/0.75,a_2/0.5,a_3/0\) (g3,f3)=(b1/1,b2/0.75,b3/0,a1/1,a2/0.75,a3/0)(g_3,f_3)=(\b_1/1,b_2/0.75,b_3/0\,\a_1/1,a_2/0.75,a_3/0\) (g4,f4)=(b1/0.75,b2/0.5,b3/0.5,a1/0.75,a2/0.5,a3/0.5)(g_4,f_4)=(\b_1/0.75,b_2/0.5,b_3/0.5\,\a_1/0.75,a_2/0.5,a_3/0.5\) (g5,f5)=(b1/0,b2/0,b3/0.5,a1/0,a2/0,a3/0.5)(g_5,f_5)=(\b_1/0,b_2/0,b_3/0.5\,\a_1/0,a_2/0,a_3/0.5\) (g⊤,f⊤)=(b1/1,b2/1,b3/1,a1/1,a2/1,a3/1)(g_ ,f_ )=(\b_1/1,b_2/1,b_3/1\,\a_1/1,a_2/1,a_3/1\) Table 7: Some pairs of ℱNF_N in the context (A,B,R2)(A,B,R_2) of Example 31. Taking into account (g1,f1)(g_1,f_1), it easy to check that g1(b)≤R2(a3,b)g_1(b)≤ R_2(a_3,b) for every b∈Bb∈ B. Therefore, we are not under the hypothesis of Proposition 27. Thus, we can observe that Proposition 27 does not hold for a3∈Aa_3∈ A, that is, g1↑(a3)=1≰0.75=g1↑π(a3)g_1 (a_3)=1 0.75=g_1 _π(a_3) However, considering the pair (g2,f2)(g_2,f_2) which satisfies the hypothesis of Proposition 27, we obtain that Proposition 29 also holds for the pair (g2,f2)(g_2,f_2), that is, ⟨f2↓,f2↓↑⟩⪯⟨g2↑↓,g2↑⟩ f_2 ,f_2 g_2 ,g_2 as we show below: f2↓=b1/1,b2/0.25,b3/0≤b1/1,b2/1,b3/0=g2↑↓f_2 =\b_1/1,b_2/0.25,b_3/0\≤\b_1/1,b_2/1,b_3/0\=g_2 Notice that these concepts correspond to C3C_3 and C5C_5, respectively, providing an interval of concepts in the concept lattice. C0=⟨b1/0,b2/0,b3/0,a1/1.0,a2/1.0,a3/1.0⟩C_0= \b_1/0,b_2/0,b_3/0\,\a_1/1.0,a_2/1.0,a_3/1.0\ C1=⟨b1/0.5,b2/0.25,b3/0,a1/1.0,a2/1.0,a3/0⟩C_1= \b_1/0.5,b_2/0.25,b_3/0\,\a_1/1.0,a_2/1.0,a_3/0\ C2=⟨b1/0,b2/0,b3/1,a1/0,a2/0,a3/1⟩C_2= \b_1/0,b_2/0,b_3/1\,\a_1/0,a_2/0,a_3/1\ C3=⟨b1/1,b2/0.25,b3/0,a1/1.0,a2/0.5,a3/0⟩C_3= \b_1/1,b_2/0.25,b_3/0\,\a_1/1.0,a_2/0.5,a_3/0\ C4=⟨b1/0.5,b2/1,b3/0,a1/0.25,a2/1,a3/0⟩C_4= \b_1/0.5,b_2/1,b_3/0\,\a_1/0.25,a_2/1,a_3/0\ C5=⟨b1/1,b2/1,b3/0,a1/0.25,a2/0.5,a3/0⟩C_5= \b_1/1,b_2/1,b_3/0\,\a_1/0.25,a_2/0.5,a_3/0\ C6=⟨b1/1,b2/1,b3/1,a1/0,a2/0,a3/0⟩C_6= \b_1/1,b_2/1,b_3/1\,\a_1/0,a_2/0,a_3/0\ Figure 6: List of concepts and concept lattice associated with the context (A,B,R2)(A,B,R_2) of Example 31. On the other hand, according to Proposition 30, we can verify that (g3,f3)(g_3,f_3) satisfies that g3(b1)=R(a1,b1)=1g_3(b_1)=R(a_1,b_1)=1 and, in this occasion, we obtain a block of concepts delimited by C1C_1 and C5C_5, that is, f3↓=b1/1,b2/1,b3/0≤b1/0.5,b2/0.25,b3/0=g3↑↓f_3 =\b_1/1,b_2/1,b_3/0\≤\b_1/0.5,b_2/0.25,b_3/0\=g_3 Similarly, the block which is only composed of the concept C2C_2 is determined by the pair (g5,f5)(g_5,f_5), since ⟨f5↓,f5↓↑⟩=⟨g5↑↓,g5↑⟩=C2 f_5 ,f_5 = g_5 ,g_5 =C_2. Notice that there are some pairs of ℱNF_N that provide an interval which represents the whole concept lattice. For instance, Proposition 29 holds for the pair (g4,f4)(g_4,f_4) and this pair determines the interval C0=⟨f4↓,f4↓↑⟩⪯⟨g4↑↓,g4↑⟩=C6C_0= f_4 ,f_4 g_4 ,g_4 =C_6. Note that, the corresponding concept lattice of the latter context in this example contains two independent blocks of concepts. Thanks to the results obtained in this paper, we have been able to establish intervals of concepts within the concept lattice, from the pairs in ℱNF_N. One of these intervals delimits one independent block. Specifically, this block has been determined from the pair (g3,f3)(g_3,f_3) in ℱNF_N. Therefore, the following step will be to discover what elements in ℱNF_N can characterize the independent blocks of concepts in a concept lattice. Consequently, we can find different intervals or blocks of concepts by means of pairs of ℱNF_N under certain assumptions, as we have proven in the previous results and illustrated in Example 31. Notice that the introduced consequences can be easily proved considering in the frame the Product conjunctor instead of the Gödel conjunctor. 5 Conclusions and future work In this paper, we have continued with the study initiated in [2], on the necessity operator when it is used to factorize a formal context into independent subcontexts, which can provide particular consequences/information from the whole dataset and from which the original context can be recovered. We have presented several properties that are satisfied by blocks of concepts associated with independent subcontexts in classical FCA. For example, we have identified the bounds associated with each block of concepts, that is, we have identified the interval of the concept lattice associated with each element in NC_N. Furthermore, we have shown how some of the properties in the classical case are extended to the fuzzy framework. We have also provided a specific fuzzy frame in which we have obtained additional properties, such as, the determination of intervals of concepts by means of the elements ℱNF_N. In the near future, we are interested in providing a mechanism to factorize formal contexts in the multi-adjoint framework. Moreover, we will continue with the study of more properties related to independent subcontexts in both, the classical and the fuzzy framework. In particular, we will study under what conditions the intervals of concepts, determined by means of elements in ℱNF_N, characterize independent subcontexts. We will also consider alternative ways to address the decomposition of formal contexts into smaller subcontexts. Furthermore, factorization techniques can be very useful to analyze the information contained in real databases, as it has been shown in [4, 9]. 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