Formal Concept Analysis of higher order Ondrej Krı́dlo, Patrik Mihalčin, Stanislav Krajči and Lubomı́r Antoni University of Pavol Jozef Šafárik, Košice, Slovakia? Abstract. The second order formal context is a formal context such that its object and attribute sets are disjoint unions of object and at- tribute sets of external formal contexts. Every subset of object or at- tribute set will be evaluated from concept lattice of corresponding ex- ternal formal context. The paper provides a method how to compute such second order formal concepts by using of bonds between external formal contexts or by using of heterogeneous formal contexts methods. Last part of the paper shows how this structure generalizes homogenic fuzzy formal context and its derivation operators. 1 Motivation example Imagine the following situation as a motivation. Lets have a group of people that everybody knows each other (schoolmates, co-workers, etc.). All of them are going to travel somewhere together and you (as an organizer of the trip) would like to know their requirements for accommodation. Consider the following formal context. Set of objects represents a group of co-workers (Anna, Bob, Cyril, David, Erik). Set of attributes expresses their requirements (TV, Wellness, Closeness to a city center, Restaurant in a hotel). An example of such context is in the following table. Lets denote the following table as P as preferences. P TV W Ce R Anna • • • Bob ◦ ◦ • • Cyril ◦ • • ◦ David • • ◦ Erik • • ◦ • A particular formal concept of the given context describes a set of co-workers such that these people together have a common requirements for accommoda- tion. In addition, there are another two formal contexts. The first one describes a friendship relation inside the group of such people (denoted as F). The second one describes a situation about hotels and services they offer (denoted as H). ? Partially supported by grant VEGA 1/0832/12 and APVV-0035-10. c paper author(s), 2013. Published in Manuel Ojeda-Aciego, Jan Outrata (Eds.): CLA 2013, pp. 117–128, ISBN 978–2–7466–6566–8, Laboratory L3i, University of La Rochelle, 2013. Copying permitted only for private and academic purposes. 118 Ondrej Krı́dlo, Patrik Mihalčin, Stanislav Krajči and Lubomı́r Antoni F Anna Bob Cyril David Erik H TV W C R Anna • ◦ ◦ H1 • • • • Bob ◦ • • ◦ ◦ H2 • • • Cyril ◦ • • • • H3 • David ◦ • • • H4 • • • Erik ◦ • • • All contexts are filled by truth degrees from the following set {• = true, ◦ = middle, ” ” = false}. Computing of L-concepts is based on Lukasiewicz logic. Now, we aim at connecting such table data with known intercontextual mech- anisms in order to obtain closed sets of friends from F that are able to stay in any of a closed set of hotels from H. The hotels of this closed set offer as much requirements from P as it gets. 2 Preliminaries 2.1 Basics Formal Concept Analysis (FCA) as an applied Lattice Theory [7] has become a very useful tool for discovering of hidden knowledge inside a data of object- attribute table, so called formal contexts. Fundamental construction of FCA is a Galois connection between complete lattices of all subsets of objects and attributes. A Galois connection consists of two mappings such that a compo- sition of these mappings form a closure operators on each subsets of complete lattice. Pair of closed subset of objects and subset of attributes connected to each other by the Galois connection is called formal concept. The set of formal concepts forms a complete lattice. The mentioned notions were generalized over a fuzzy logic based on a complete residuated lattice. The notions of order, Galois connection and complete lattice were also generalized by Bělohlávek in [3–6]. Definition 1. Complete residuated lattice is an algebra hL, ∧, ∨, 0, 1, ⊗, →i, where – hL, ∧, ∨, 0, 1i is a complete lattice with top 1 and bottom 0, – hL, ⊗, 1i is a commutative monoid, – h⊗, →i is an adjoint pair, i.e. a ⊗ b ≤ c is equivalent to a ≤ b → c for any a, b, c ∈ L. Definition 2. L-fuzzy formal context C is a triple hB, A, ri, where r : B × A → L is an L-fuzzy binary relation and L is the complete residuated lattice. Definition 3. Let hB, A, ri be an L-fuzzy formal context. Lets define a pair of derivation operators h↑, ↓i of the form ↑: LB −→ LA and ↓: LA −→ LB , where ^ ↑ (f )(a) = (f (b) → r(b, a)) for any f ∈ LB and a ∈ A, b∈B ^ ↓ (g)(b) = (g(a) → r(b, a)) for any g ∈ LA and b ∈ B. a∈A Formal Concept Analysis of higher order 119 Lemma 1. Let h↑, ↓i be a pair of derivation operators defined on an L-fuzzy for- mal context hB, A, ri. A pair h↑, ↓i forms a Galois connection between complete lattices of all L-sets of objects LB and attributes LA . Definition 4. Let C = hB, A, ri be an L-fuzzy formal context. Formal concept is a pair of L-sets hf, gi ∈ LB × LA such that ↑ (f ) = g and ↓ (g) = f . The set of all L-concepts of C will be denoted by FCL(C). Object or attribute part of any concept is called extent or intent. Sets of all extents or intents of C will be denoted as Ext(C) or Int(C), respectively. 2.2 Bonds and Chu correspondences FCA provides the useful methods how to connect two formal contexts. A struc- ture of the so called Chu correspondence was introduced by Mori [13, 14] that is very close to the notion of bond [7]. The notions of Chu correspondence and bond were extended into L-fuzzy Chu correspondence and L-bond in [8]. The corresponding notions are introduced now. Definition 5. Let Ci = hBi , Ai , ri i for i ∈ {1, 2} be two L-fuzzy formal contexts. Pair of L-multimappings ϕ = hϕL , ϕR i such that – ϕL : B1 −→ Ext(C2 ), – ϕR : A2 −→ Int(C1 ), where ↑2 (ϕL (o1 ))(a2 ) =↓1 (ϕR (a2 ))(o1 ) for any (o1 , a2 ) ∈ B1 × A2 , is said to be an L-Chu correspondence between C1 and C2 . Set of all L-Chu correspondences between L-contexts C1 and C2 will be denoted by L-ChuCors(C1 , C2 ). Definition 6. Let Ci = hBi , Ai , ri i for i ∈ {1, 2} be two L-fuzzy formal contexts. L-multimapping β : B1 −→ Int(C2 ), such that β t : A2 −→ Ext(C1 ), where β t (a2 )(o1 ) = β(o1 )(a2 ) for any (o1 , a2 ) ∈ B1 × A2 , is said to be an L-bond. Set of all L-bonds beyween L-contexts C1 and C2 will be denoted by L-Bonds(C1 , C2 ). Lemma 2. Let Ci = hBi , Ai , ri i for i ∈ {1, 2} be two L-fuzzy formal contexts. Each set L-Bonds(C1 , C2 ) and L-ChuCors(C1 , C2 ) forms a complete lattice and, moreover, there exists a dual isomorphism between them. The dual isomorphism between bonds and Chu correspondences is based on the following construction. Consider two L-fuzzy formal contexts Ci = hBi , Ai , ri i for i ∈ {1, 2} and let β ∈ L-Bonds(C1 , C2 ), then hϕβL , ϕβR i such that for any (o1 , a2 ) ∈ B1 × A2 ϕβL (o1 ) =↓2 (β(o1 )) and ϕβR (a2 ) =↑1 (β t (a2 )) is an L-Chu correspondence from L-ChuCors(C1 , C2 ). On the other hand, let ϕ ∈ L-ChuCors(C1 , C2 ). Then βϕ defined as βϕ (o1 )(a2 ) =↓1 (ϕR (a2 ))(o1 ) =↑2 (ϕL (o1 ))(a2 ) for any (o1 , a2 ) ∈ B1 × A2 is an L-bond from L-Bonds(C1 , C2 ). 120 Ondrej Krı́dlo, Patrik Mihalčin, Stanislav Krajči and Lubomı́r Antoni 2.3 Categorical relationship to fuzzy Galois connection Categories ChuCors and L-ChuCors of classical or fuzzy formal contexts and classical or fuzzy Chu correspondences are described in [9, 13]. Important cate- gorical property of ∗-autonomism is also proved in mentioned papers. The con- tinuation of categorical research in [10] resulted in a categorical equivalence of L-ChuCors and a category L-CLLOS of so called completely lattice L-ordered sets and monotone fuzzy Galois connections. Equivalence is proved by construct- ing of equivalence functor between these categories. Definition 7. Lets define a functor Γ : L-ChuCors −→ L-CLLOS in the fol- lowing way: 1. Γ (C) = hhL-FCL(C), ≈i, i for any L-context C 2. Γ (ϕ) = hλϕ ϕ ϕ L , λR i for any ϕ ∈ L-ChuCors(C1 , C2 ) such that λL : FCL(C1 ) −→ ϕ FCL(C2 ) and λR : FCL(C2 ) −→ FCL(C1 ) λϕ L (hf, ↑1 (f )i) = h↓2 ↑2 (ϕL+ (f )), ↑2 (ϕL+ (f ))i λϕ R (h↓2 (g), gi) = h↓1 (ϕR+ (g)), ↑1 ↓1 (ϕR+ (g))i for any two L-concepts hf, ↑1 (f )i ∈ FCL(C1 ) and h↓2 (g), gi ∈ FCL(C2 ), Y X Y W multifunction ω : X −→ L is ωX+ : L −→ L defined as where for any ω+ (f )(y) = x∈X f (x) ⊗ ω(x)(y) for any f ∈ L and y ∈ Y . In [10] is proved that Γ is the equivalence functor. Hence, it holds for the particular two L-concepts that     hf, ↑1 (f )i 1 λϕ ϕ R (h↓2 (g), gi) = λL (hf, ↑1 (f )i) 2 h↓2 (g), gi . Lemma 3. Consider two L-contexts Ci = hBi , Ai , ri i for i ∈ {1, 2} and let ϕ ∈ L-ChuCors(C1 , C2 ). A functor Γ (ϕ) is a fuzzy Galois connection between hhFCL(C1 ), ≈1 i, 1 i and hhFCL(C2∗ ), ≈2 i, 2 i where C2∗ = hA2 , B2 , r2t i. Proof.  Due to order reversing of dual L-context  C2∗ for any two L-concepts  we ϕ ϕ obtain hf, ↑1 (f )i 1 λR (h↓2 (g), gi) = h↓2 (g), gi 2 λL (hf, ↑1 (f )i) . t u 3 Formal concept analysis of second order Once we have introduced preliminaries, the formal context of second order and the corresponding results are presented now in details. Definition 8.S Consider S two non-empty index sets I and J and an L-fuzzy for- mal context h i∈I Bi , j∈J Aj , ri, whereby – Bi1 ∩ Bi2 = ∅ for any i1 , i2 ∈ I, – Aj1S∩ Aj2 = ∅Sfor any j1 , j2 ∈ J, – r : i∈I Bi × j∈J Aj −→ L. Formal Concept Analysis of higher order 121 Moreover, consider two non-empty sets of L-contexts notated – {Ci = hBi , Ti , pi i : i ∈ I} – {Dj = hOj , Aj , qj i : j ∈ J}. Formal context of second order is a tuple D[ [ [ E Bi , {Ci ; i ∈ I}, Aj , {Dj ; j ∈ J}, ri,j , i∈I j∈J (i,j)∈I×J where ri,j : Bi × Aj −→ L defined as ri,j (o, a) = r(o, a) for any o ∈ Bi and a ∈ Aj . Q In what follows, consider the below used notation. S Lets have an L-set f : i∈I Xi −→ L for a non-empty universe set X = i∈I Xi , where Xi1 ∩ Xi2 = ∅ for any i1 , i2 ∈ I. Then f i : Xi −→ L is defined as f i (x) = f (x) for an arbitrary x ∈ Xi and i ∈ I. With the help of functor Γ , we define the mappings between products of fuzzy concept lattices of objects and attributes formal contexts of the following form: Definition 9. Lets define the mappings h⇑, ⇓i as follows Y Y Y Y ⇑: FCL(Ci ) −→ FCL(Dj ) and ⇓: FCL(Dj ) −→ FCL(Ci ) i∈I j∈J j∈J i∈I ^ Y j i ⇑ (Φ) = λijL (Φ ), for any Φ ∈ FCL(Ci ) i∈I i∈I ^ Y ⇓ (Ψ )i = λijR (Ψ j ), for any Ψ ∈ FCL(Dj ) j∈J j∈J such that λij = hλijL , λijR i = Γ (ϕρij ), where _ ρij = {β ∈ L-Bonds(Ci , Dj ) : (∀(oi , aj ) ∈ Bi × Aj )β(oi )(aj ) ≤ rij (oi , aj )}. Lemma 4. Let {hf, gi} ∪ {hfk , gk i : k ∈ K} be a non-empty set of L-concepts of any L-context and K be a non-empty index set. Then ^ ^ hf, gi  hfk , gk i = (hf, gi  hfk , gk i). k∈K k∈K Proof. Let the L-context be of the form hB, A, ri. Hence ^ ^ ^   ^ ^  hf, gi  hfk , gk i = f (o) → fk (o) = f (o) → fk (o) k∈K o∈B k∈K o∈B k∈K ^ ^  ^  = f (o) → fk (o) = hf, gi  hfk , gk i . k∈K o∈B k∈K t u 122 Ondrej Krı́dlo, Patrik Mihalčin, Stanislav Krajči and Lubomı́r Antoni Lemma Q 5. Pair of mappings h⇑, Q⇓i forms a Galois connection between complete lattices h i∈I FCL(Ci ), vI i and h j∈J FCL(Dj ), vJ i. Proof. Proof is provided in fuzzy ordering as a generalization of classical one. ^ ^ ^  Ψ vJ ⇑ (Φ) = (Ψ j j ⇑ (Φ)j ) = Ψ j j λijL (Φi ) j∈J j∈J i∈I ^^ ^^ = (Ψ j j λijL (Φi )) = (Φi i λijR (Ψ j )) i∈I j∈J i∈I j∈J ^ ^  ^ i j = Φ i λijR (Ψ ) = (Φi i ⇓ (Ψ )i ) i∈I j∈J i∈I = Φ vI ⇓ (Ψ ). t u 3.1 Simplification In this subsection will be presented a method that simplifies the previous con- sideration. Definition 10. Let Ci = hBi , Ai , ri i for i ∈ {1, 2} be two L-fuzzy contexts and let β be an arbitrary L-bond between C1 and C2 . Consider the following pair of mappings ↑β : LB1 −→ LA2 and ↓β : LA2 −→ LB1 such that ^ ^ ↑β (f )(a) = (f (o) → β(o)(a)), ↓β (g)(o) = (g(a) → β(o)(a)) o∈B1 a∈A2 for any f ∈ LB1 and g ∈ LA2 . Lemma 6. Let Ci = hBi , Ai , ri i for i ∈ {1, 2} be two L-fuzzy contexts and let β be an arbitrary L-bond between C1 and C2 . A pair h↑β , ↓β i forms a Galois connection between complete lattices hExt(C1 ), ≤i and hInt(C2 ), ≤i, where ≤ is ordering based on fuzzy sets inclusion. Proof. Proof of the fact that h↑β , ↓β i forms a Galois connection between hLB1 , ≤i and hLA2 , ≤i is simple, h↑β , ↓β i is a pair of derivation operators for L-context hB1 , A2 , β r i, where binary L-relation β r is defined as β r (o1 , a2 ) = β(o1 )(a2 ). Now, we will show that h↑β , ↓β i is a pair of mappings between complete lattices of extents and intents of C1 and C2 , respectively. First, let f be an extent of C1 . ^ ↑β (f )(a) = (f (o) → β(o)(a)) o∈B1 ^ = (f (o) →↑2 (ϕβL (o))(a)) o∈B1 ^  ^  = f (o) → (ϕβL (o)(b) → r2 (b, a)) o∈B1 b∈B2 Formal Concept Analysis of higher order 123 ^ ^ = (f (o) → (ϕβL (o)(b) → r2 (b, a))) b∈B2 o∈B1 ^ ^ = ((ϕβL (o)(b) ⊗ f (o)) → r2 (b, a))) b∈B2 o∈B1 ^ _  = (ϕβL (o)(b) ⊗ f (o)) → r2 (b, a)) b∈B2 o∈B1 ^ = (ϕβL+ (f )(b) → r2 (b, a))) b∈B2 =↑2 (ϕβL+ (f ))(a). So ↑β (f ) is an intent of C2 . Proof of ↓β (g) is an extent of C1 is easy to obtain similarly with equality β(o)(a) =↓1 (ϕβR (o))(a). t u We define an L-context such that its L-concept lattice is isomorphic to a complete lattice of all second order formal concepts. Definition 11. Let K be a second order formal context of the form D[ [ [ E K= Bi , {Ci : i ∈ I}, Aj , {Dj : j ∈ J}, rij . i∈I j∈J (i,j)∈I×J b Lets define an L-context K D[ [ [ E b= K Bi , Aj , ρij , i∈I j∈J (i,j)∈I×J where _ ρij = {β ∈ L-Bonds(Ci , Dj ) : (∀(oi , aj ) ∈ Bi × Aj )β(oi )(aj ) ≤ rij (oi , aj )}. b are isomorphic. Lemma 7. Concept lattices of K and K b and o ∈ Bi . Proof. Let hΦ, Ψ i be an L-concept of K ^ ^ Φi (o) = (↓Kb (Ψ ))i (o) = (Ψ j (a) → ρij (o)(a)) j∈J a∈Aj ^ ^ = (Ψ j (a) →↓i (ϕρij R (a))(o)) j∈J a∈Aj ^ = ↓i (ϕρij R+ (Ψ j ))(o). j∈J ^ ^ ϕρ Φi = (↓Kb (Ψ ))i = ↓i (ϕρij R+ (Ψ j )) = ext(λR ij (Ψ j )) j∈J j∈J ^ ϕ  ρ = ext λR ij (Ψ j ) = ext(⇓ (Ψ )i ), j∈J 124 Ondrej Krı́dlo, Patrik Mihalčin, Stanislav Krajči and Lubomı́r Antoni j where Ψ = h↓j (Ψ j ), Ψ j i for any j ∈ J. Then Φ =⇓ (Ψ ) and hΦ, Ψ i is a second order concept of K. t u 4 Motivation example – solution The motivation example introduced in Section 1 can be considered as the second order formal context h{Anna,Bob,Cyril,David,Eva}, F, {TV,W,Ce,R}, H, ri, whereby r represents the L-relation from P. Firstly, we find a bond ρ that is the closest to r. P TV W Ce R ρ(P) TV W Ce R Anna • • • Anna • • • Bob ◦ ◦ • • Bob ◦ ◦ • ◦ Cyril ◦ • • ◦ Cyril ◦ ◦ • ◦ David • • ◦ David ◦ Erik • • ◦ • Erik ◦ There are just six (instead of twenty-eight L-concepts of P) L-concepts of ρ(P) such that we can easily convert into the form of second order concepts. The following table contains the list of the all second order concepts. concepts of F concepts of H {◦/A, •/B, •/C, ◦/D, ◦/E} {◦/TV, ◦/W, •/Ce, ◦/R} {◦/A, •/B, •/C, ◦/D, ◦/E} {•/H1, /H2, ◦/H3, ◦/H4} {•/A, •/B, •/C, •/D, •/E} { /TV, /W, ◦/Ce, /R} { /A, ◦/B, ◦/C, /D, /E} {•/H1, ◦/H2, •/H3, •/H4} {•/A, •/B, •/C, ◦/D, ◦/E} {◦/TV, /W, •/Ce, ◦/R} {◦/A, ◦/B, ◦/C, /D, /E} {•/H1, /H2, ◦/H3, •/H4} {•/A, ◦/B, ◦/C, /D, /E} {•/TV, /W, •/Ce, •/R} {•/A, ◦/B, ◦/C, /D, /E} {•/H1, /H2, /H3, •/H4} {◦/A, ◦/B, ◦/C, /D, /E} {•/TV, ◦/W, •/Ce, •/R} {•/A, •/B, •/C, ◦/D, ◦/E} {•/H1, /H2, /H3, ◦/H4} { /A, ◦/B, ◦/C, /D, /E} {•/TV, •/W, •/Ce, •/R} {•/A, •/B, •/C, •/D, •/E} {•/H1, /H2, /H3, /H4} The first concept can be interpreted as follows. Friends Bob and Cyril with their common requirements should stay in hotel H1. They should stay also in H3 and H4 with a little relaxation of their requirements. The fourth concept is saying that Anna as a very lonely person should stay in H1 or H4. The second concept includes the whole group of co-workers who have very poor common requirements. Thus, all people should stay in an arbitrary hotel together. Formal Concept Analysis of higher order 125 5 Connection to heterogeneous formal contexts The fruitful idea is to view the second order formal context in terms of a het- erogeneous formal context proposed in [2]. The corresponding notions of the underlying structures are introduced now. Definition 12. Heterogeneous formal context is a tuple hB, A, P, R, U, V, i, where – B and A are non-empty sets, – P = {hPb,a , ≤Pb,a i : (b, a) ∈ B × A} is a system of posets, – R is a mapping from B × A such that R(b, a) ∈ Pb,a for any b ∈ B and a ∈ A, – U = {hUb , ≤Ub i : b ∈ B} and V = {hVa , ≤Va i : a ∈ A} are systems of complete latices, – = {◦b,a : (b, a) ∈ B × A} is a system of isotone and left-continuous mappings ◦b,a : Ub × Va −→ Pb,a . Lets describe our situation in terms of heterogeneous formal contexts. Below is the translation: – B and A will be the index sets I and J, – complete lattices Ui or Vj for any (i, j) ∈ B × A = I × J will be the complete lattices hExt(Ci ), ≤i and hInt(Dj ), ≤i, – Pi,j will be a complete lattice of all fuzzy relations from LBi ×Aj , – any value of relation r will be a binary relation r(i, j) = ri,j ∈ LBi ×Aj , – operation ◦i,j : Ext(Ci ) × Int(Dj ) −→ LBi ×Aj is defined as (f ◦i,j g)(b, a) = f (b) ⊗ g(a) for any f ∈ Ext(Ci ) and g ∈ Int(Dj ) and any (b, a) ∈ Bi × Aj . The mapping ◦i,j is isotone due to isotonicity of ⊗. Lemma 8. The mapping ◦i,j is left-continuous. Proof. Let (fk ◦ g)(b, a) = fk (b) ⊗ g(a) ≤ m for all k ∈ K and for some (b, a) ∈ B × A and Wm ∈ L. It is equivalent to inequality fk (b) ≤ g(a) → m for all k ∈ K. Hence, k∈K fk (b) ≤ g(a) → m and it is equivalent to _  _ fk ◦ g (b, a) = fk (b) ⊗ g(a) ≤ m. k∈K k∈K Proof of left-continuity of the second argument is similar. t u 126 Ondrej Krı́dlo, Patrik Mihalčin, Stanislav Krajči and Lubomı́r Antoni Definition 13. Q Lets define aQpair of derivation operators Q h-, &i ofQthe follow- ing form -: i∈I Ext(Ci ) → j∈J Int(Dj ) and &: j∈J Int(Dj ) → i∈I Ext(Ci ) defined for heterogeneous formal context mentioned above as follows: _ - (Φ)j = {g ∈ Int(Dj ) : (∀i ∈ I)Φi ◦i,j g ≤ ri,j } _ & (Ψ )i = {f ∈ Ext(Ci ) : (∀j ∈ J)f ◦i,j Ψ j ≤ ri,j } Q Q for any Φ ∈ i∈I Ext(Ci ) and any Ψ ∈ j∈J Int(Dj ). S S S Lemma 9. Let K = h i∈I Bi , {Ci ; i ∈ I}, j∈J Aj , {Dj ; j ∈ J}, (i,j)∈I×J ri,j i be a second order formal context. Then ↑Kb (Φ) ≤- (Φ) and ↓Kb (Ψ ) ≤& (Ψ ) Q Q for any Φ ∈ i∈I Ext(Ci ) and Ψ ∈ j∈J Int(Dj ). Proof. Let j ∈ J be arbitrary. ^ ^ ↑Kb (Φ)j (a) = (Φi (o) → ρij (o)(a)) i∈I o∈Bi _ = {g ∈ Int(Dj ) : (∀i ∈ I)(∀o ∈ Bi )Φi (o) ⊗ g(a) ≤ ρij (o)(a)}. Then _ ↑Kb (Φ)j = {g ∈ Int(Dj ) : (∀i ∈ I)(∀o ∈ Bi )Φi ◦ij g ≤ ρij } because of ρij ≤ rij _ ≤ {g ∈ Int(Dj ) : (∀i ∈ I)(∀o ∈ Bi )Φi ◦ij g ≤ rij } =- (Φ)j . Hence ↑Kb (Φ) ≤- (Φ). Similarly for ↓Kb and &. t u 6 Connections to standard homogenic fuzzy operators In this part, we focus on an appropriate generalization of the standard homogenic fuzzy formal concept derivation operators in two different ways. 6.1 Singleton connection Lemma 10. Lets have an L-fuzzy formal context h{x}, L, λi, where for an ar- bitrary k ∈ L is ⊥x = λ(x, k) = k. Any value k ∈ L is an extent of ⊥x . Proof. Let k be an arbitrary value from L. ^ ^ ^ ↓↑ (k)(x) = (↑ (k)(m) → m) = ( (k → m) → m) m∈L m∈L x∈{x} Formal Concept Analysis of higher order 127 ^ ^ = (1 → m) ∧ ((k → m) → m) m∈L:m≥k m∈L:m