Formal Concept Analysis of Higher Order

Ondrej Kr´ıdlo, Patrik Mihalˇcin, Stanislav Krajˇci and Lubom´ır Antoni University of Pavol Jozef ˇSaf´arik, Koˇsice, 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 asP 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 asF). The second one describes a situation about hotels and services they offer (denoted asH).

?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.

F Anna Bob Cyril David Erik

Anna • ◦ ◦

Bob ◦ • • ◦ ◦

Cyril ◦ • • • •

David ◦ • • •

Erik ◦ • • •

H TV W C R H1 • • • •

H2 • • •

H3 •

H4 • • •

All contexts are filled by truth degrees from the following set {•= true,◦= middle,” ” = false}. Computing ofL-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 fromF that are able to stay in any of a closed set of hotels fromH. The hotels of this closed set offer as much requirements fromP 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 Bˇelohl´avek in [3–6].

Definition 1. Complete residuated lattice is an algebra hL,∧,∨,0,1,⊗,→i, where

– hL,∧,∨,0,1iis a complete lattice with top1 and bottom0, – hL,⊗,1iis a commutative monoid,

– h⊗,→iis an adjoint pair, i.e.

a⊗b≤c is equivalent toa≤b→c for any a, b, c∈L.

Definition 2. L-fuzzy formal contextC is a triplehB, A, ri, wherer:B×A→ L is anL-fuzzy binary relation andL is the complete residuated lattice.

Definition 3. Let hB, A, ribe an L-fuzzy formal context. Lets define a pair of derivation operators h↑,↓iof the form ↑:L^B−→L^A and↓:L^A−→L^B, where

↑(f)(a) = ^

b∈B

(f(b)→r(b, a))for any f ∈L^B anda∈A,

↓(g)(b) = ^

a∈A

(g(a)→r(b, a))for anyg∈L^A andb∈B.

Lemma 1. Leth↑,↓ibe a pair of derivation operators defined on anL-fuzzy for- mal contexthB, A, ri. A pairh↑,↓i forms a Galois connection between complete lattices of allL-sets of objectsL^B and attributes L^A.

Definition 4. Let C =hB, A, ri be an L-fuzzy formal context. Formal concept is a pair of L-sets hf, gi ∈L^B ×L^A such that ↑ (f) = g and ↓ (g) = f. The set of allL-concepts ofC will be denoted by FCL(C). Object or attribute part of any concept is called extentor 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. LetCⁱ=hBi, Ai, riifori∈ {1,2}be twoL-fuzzy formal contexts.

Pair of L-multimappingsϕ=hϕL, ϕRisuch 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 andC². Set of all L-Chu correspondences between L-contexts C¹ and C² will be denoted by L-ChuCors(C¹,C²).

Definition 6. LetCⁱ=hBi, Ai, riifori∈ {1,2}be twoL-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 allL-bonds beyweenL-contexts C¹andC² will be denoted byL-Bonds(C¹,C²).

Lemma 2. Let Cⁱ =hBi, Ai, rii for i∈ {1,2} be twoL-fuzzy formal contexts.

Each set L-Bonds(C¹,C²) andL-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 twoL-fuzzy formal contextsCⁱ=hBi, Ai, rii for i ∈ {1,2} and let β ∈ L-Bonds(C¹,C²), then hϕβL, ϕβRisuch that for any (o1, a2)∈B1×A2

ϕβL(o1) =↓²(β(o1)) andϕβR(a2) =↑¹(β^t(a2)) is anL-Chu correspondence fromL-ChuCors(C¹,C²).

On the other hand, letϕ∈L-ChuCors(C¹,C²). Thenβϕdefined as βϕ(o1)(a2) =↓¹(ϕR(a2))(o1) =↑²(ϕL(o1))(a2) for any (o1, a2)∈B1×A2is anL-bond from L-Bonds(C¹,C²).

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 anyL-context C

2. Γ(ϕ) =hλ^ϕ_L, λ^ϕ_Rifor anyϕ∈L-ChuCors(C¹,C²)such thatλ^ϕ_L: FCL(C¹)−→

FCL(C²)and λ^ϕ_R: FCL(C²)−→FCL(C¹)

λ^ϕ_L(hf,↑¹(f)i) =h↓²↑²(ϕL+(f)),↑²(ϕL+(f))i λ^ϕ_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), where for any multifunction ω : X −→ L^Y is ω+ : L^X −→ L^Y defined as ω+(f)(y) =W

x∈Xf(x)⊗ω(x)(y)for any f ∈L^X andy∈Y.

In [10] is proved that Γ is the equivalence functor. Hence, it holds for the particular twoL-concepts that

hf,↑¹(f)i ¹λ^ϕ_R(h↓²(g), gi)

=

λ^ϕ_L(hf,↑¹(f)i)²h↓²(g), gi . Lemma 3. Consider two L-contexts Cⁱ = hBi, Ai, rii for i ∈ {1,2} and let ϕ ∈ L-ChuCors(C¹,C²). A functor Γ(ϕ) is a fuzzy Galois connection between hhFCL(C¹),≈¹i,¹iandhhFCL(C2^∗),≈²i,²i whereC2^∗=hA2, B2, r₂^ti.

Proof. Due to order reversing of dualL-context C2^∗ for any twoL-concepts we obtain

hf,↑¹(f)i ¹λ^ϕ_R(h↓²(g), gi)

=

h↓²(g), gi ²λ^ϕ_L(hf,↑¹(f)i) . ut

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. Consider two non-empty index setsI andJ and anL-fuzzy for- mal contexthS

i∈IBi,S

j∈JAj, ri, whereby – Bi1∩Bi2 =∅ for any i1, i2∈I, – Aj1∩Aj2 =∅ for any j1, j2∈J, – r:S

i∈IBi×S

j∈JAj−→L.

Moreover, consider two non-empty sets ofL-contexts notated – {Cⁱ=hBi, Ti, pii:i∈I}

– {D^j =hOj, Aj, qji:j∈J}.

Formal context of second order is a tuple D [

i∈I

Bi, {Cⁱ;i∈I}, [

j∈J

Aj, {D^j;j ∈J}, [

(i,j)∈I×J

ri,j

E,

where ri,j : Bi×Aj −→ L defined as ri,j(o, a) = r(o, a) for any o ∈ Bi and a∈Aj.

In what follows, consider the below used notation. Lets have an L-set f : Q

i∈IXi−→Lfor a non-empty universe setX =S

i∈IXi, whereXi1∩Xi2 =∅ for anyi1, i2∈I. Thenfⁱ :Xi−→Lis defined asfⁱ(x) =f(x) for an arbitrary x∈Xi andi∈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⇑,⇓ias follows

⇑:Y

i∈I

FCL(Cⁱ)−→ Y

j∈J

FCL(D^j) and ⇓:Y

j∈J

FCL(D^j)−→Y

i∈I

FCL(Cⁱ)

⇑(Φ)^j =^

i∈I

λijL(Φⁱ), for any Φ∈Y

i∈I

FCL(Cⁱ)

⇓(Ψ)ⁱ= ^

j∈J

λijR(Ψ^j), for any Ψ ∈Y

j∈J

FCL(D^j) such that λij =hλijL, λijRi=Γ(ϕρij), where

ρij=_

{β ∈L-Bonds(Cⁱ,D^j) : (∀(oi, aj)∈Bi×Aj)β(oi)(aj)≤rij(oi, aj)}. Lemma 4. Let {hf, gi} ∪ {hfk, gki:k ∈K} be a non-empty set of L-concepts of anyL-context and K be a non-empty index set. Then

hf, gi ^

k∈K

hfk, gki= ^

k∈K

(hf, gi hfk, gki).

Proof. Let theL-context be of the formhB, A, ri. Hence hf, gi ^

k∈K

hfk, gki= ^

o∈B

f(o)→ ^

k∈K

fk

(o)

= ^

o∈B

f(o)→ ^

k∈K

fk(o)

= ^

k∈K

^

o∈B

f(o)→fk(o)

= ^

k∈K

hf, gi hfk, gki .

u t

Lemma 5. Pair of mappingsh⇑,⇓iforms a Galois connection between complete latticeshQ

i∈IFCL(Cⁱ),v^IiandhQ

j∈JFCL(D^j),v^Ji.

Proof. Proof is provided in fuzzy ordering as a generalization of classical one.

Ψ v^J⇑(Φ) = ^

j∈J

(Ψ^{j j}⇑(Φ)^j) = ^

j∈J

Ψ^jj ^

i∈I

λijL(Φⁱ)

=^

i∈I

^

j∈J

(Ψ^{j j} λijL(Φⁱ)) =^

i∈I

^

j∈J

(Φ^{i i}λijR(Ψ^j))

=^

i∈I

Φⁱⁱ ^

j∈J

λijR(Ψ^j)

=^

i∈I

(Φⁱⁱ⇓(Ψ)ⁱ)

=Φv^I⇓(Ψ).

u t

3.1 Simplification

In this subsection will be presented a method that simplifies the previous con- sideration.

Definition 10. Let Cⁱ =hBi, Ai, riifor 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:L^B1−→L^A2 and↓^β:L^A2−→L^B1 such that

↑^β(f)(a) = ^

o∈B1

(f(o)→β(o)(a)), ↓^β(g)(o) = ^

a∈A2

(g(a)→β(o)(a)) for any f ∈L^B1 andg∈L^A2.

Lemma 6. Let Cⁱ =hBi, Ai, rii fori ∈ {1,2} be twoL-fuzzy contexts and let β be an arbitrary L-bond between C¹ and C². A pair h↑^β,↓^βi 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 thath↑^β,↓^βiforms a Galois connection betweenhL^B1,≤i and hL^A2,≤i is simple,h↑^β,↓^βi is a pair of derivation operators for L-context hB1, A2, β^ri, where binaryL-relationβ^r is defined asβ^r(o1, a2) =β(o1)(a2).

Now, we will show that h↑^β,↓^βi is a pair of mappings between complete lattices of extents and intents of C1 and C2, respectively. First, let f be an extent ofC1.

↑^β(f)(a) = ^

o∈B1

(f(o)→β(o)(a))

= ^

o∈B1

(f(o)→↑²(ϕβL(o))(a))

= ^

o∈B1

f(o)→ ^

b∈B2

(ϕβL(o)(b)→r2(b, a))

= ^

b∈B2

^

o∈B1

(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

(ϕβL+(f)(b)→r2(b, a)))

=↑²(ϕβL+(f))(a).

So↑^β(f) is an intent ofC2.

Proof of ↓^β (g) is an extent of C1 is easy to obtain similarly with equality

β(o)(a) =↓¹(ϕβR(o))(a). ut

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 K=D[

i∈I

Bi, {Cⁱ:i∈I}, [

j∈J

Aj, {D^j:j∈J}, [

(i,j)∈I×J

rij

E.

Lets define anL-context Kb Kb=D[

i∈I

Bi, [

j∈J

Aj, [

(i,j)∈I×J

ρij

E,

where ρij=_

{β ∈L-Bonds(Cⁱ,D^j) : (∀(oi, aj)∈Bi×Aj)β(oi)(aj)≤rij(oi, aj)}. Lemma 7. Concept lattices of K andKb are isomorphic.

Proof. LethΦ, Ψibe anL-concept ofKb ando∈Bi. Φⁱ(o) = (↓Kb(Ψ))ⁱ(o) = ^

j∈J

^

a∈Aj

(Ψ^j(a)→ρij(o)(a))

= ^

j∈J

^

a∈Aj

(Ψ^j(a)→↓ⁱ(ϕρijR(a))(o))

= ^

j∈J

↓ⁱ(ϕρijR+(Ψ^j))(o).

Φⁱ= (↓Kb(Ψ))ⁱ= ^

j∈J

↓ⁱ(ϕρijR+(Ψ^j)) = ^

j∈J

ext(λ^ϕ_R^ρij(Ψ^j))

= ext ^

j∈J

λ^ϕ_R^ρij(Ψ^j)

= ext(⇓(Ψ)ⁱ),

where Ψ^j =h↓^j (Ψ^j), Ψ^jifor any j ∈J. ThenΦ=⇓(Ψ) andhΦ, Ψiis a second

order concept ofK. ut

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,

wherebyrrepresents theL-relation fromP. Firstly, we find a bondρthat is the closest tor.

P TV W Ce R

Anna • • •

Bob ◦ ◦ • • Cyril ◦ • • ◦ David • • ◦ Erik • • ◦ •

ρ(P) TV W Ce R

Anna • • •

Bob ◦ ◦ • ◦ Cyril ◦ ◦ • ◦

David ◦

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 ofF concepts ofH {◦/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.

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 andAare non-empty sets,

– P ={hPb,a,≤^Pb,ai: (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,≤^Ubi : b ∈ B} and V = {hVa,≤^Vai : 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 andAwill be the index setsIandJ,

– complete latticesUiorVj for any (i, j)∈B×A=I×J will be the complete latticeshExt(Cⁱ),≤iandhInt(D^j),≤i,

– Pi,j will be a complete lattice of all fuzzy relations fromL^Bi×Aj, – any value of relationrwill be a binary relationr(i, j) =ri,j∈L^Bi×Aj, – operation◦^i,j: Ext(Cⁱ)×Int(D^j)−→L^Bi×Aj is defined as

(f ◦^i,jg)(b, a) =f(b)⊗g(a)

for anyf ∈Ext(Cⁱ) andg∈Int(D^j) 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 m ∈ L. It is equivalent to inequalityfk(b)≤g(a)→mfor allk∈K. Hence,W

k∈Kfk(b)≤g(a)→mand it is equivalent to

_

k∈K

fk◦g

(b, a) = _

k∈K

fk(b)⊗g(a)≤m.

Proof of left-continuity of the second argument is similar. ut

Definition 13. Lets define a pair of derivation operatorsh-,&iof the follow- ing form-:Q

i∈IExt(Cⁱ)→Q

j∈JInt(D^j)and&:Q

j∈JInt(D^j)→Q

i∈IExt(Cⁱ) defined for heterogeneous formal context mentioned above as follows:

-(Φ)^j =_

{g∈Int(D^j) : (∀i∈I)Φⁱ◦^i,jg≤ri,j}

&(Ψ)ⁱ=_

{f ∈Ext(Cⁱ) : (∀j∈J)f◦^i,jΨ^j≤ri,j} for any Φ∈Q

i∈IExt(Cⁱ)and any Ψ ∈Q

j∈JInt(D^j).

Lemma 9. Let K=hS

i∈IBi,{Ci;i∈I},S

j∈JAj,{Dj;j∈J},S

(i,j)∈I×Jri,ji be a second order formal context. Then

↑Kb (Φ)≤-(Φ)and ↓Kb(Ψ)≤&(Ψ) for any Φ∈Q

i∈IExt(Cⁱ)andΨ ∈Q

j∈JInt(D^j).

Proof. Letj∈J be arbitrary.

↑Kb(Φ)^j(a) =^

i∈I

^

o∈Bi

(Φⁱ(o)→ρij(o)(a))

=_

{g∈Int(D^j) : (∀i∈I)(∀o∈Bi)Φⁱ(o)⊗g(a)≤ρij(o)(a)}. Then

↑Kb(Φ)^j=_

{g∈Int(D^j) : (∀i∈I)(∀o∈Bi)Φⁱ◦^ijg≤ρij} because ofρij ≤rij

≤_

{g∈Int(D^j) : (∀i∈I)(∀o∈Bi)Φⁱ◦^ijg≤rij}

=-(Φ)^j.

Hence↑Kb(Φ)≤-(Φ). Similarly for ↓Kb and&. ut

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 valuek∈Lis an extent of⊥^x.

Proof. Letkbe an arbitrary value fromL.

↓↑(k)(x) = ^

m∈L

(↑(k)(m)→m) = ^

m∈L

( ^

x∈{x}

(k→m)→m)

= ^

m∈L:m≥k

(1→m)∧ ^

m∈L:m<k

((k→m)→m)

= ^

m∈L:m≥k

m∧ ^

m∈L:m<k

((k→m)→m) =?

^

m∈L:m≥k

m=k(Lis a complete lattice) (k→m)→m≥k(closure property)

^

m∈L:m<k

((k→m)→m)≥k

?=k

So↓↑(k) =kfor anyk∈L. ut

Lemma 11. Lets have an L-fuzzy formal contextC=hB, A, riwith derivation operatorsh↑,↓i. Lets have a second order formal context

K=D [

b∈B

{b},{⊥^b :b∈B}, [

a∈A

{a},{⊥^∗a:a∈A}, [

(b,a)∈B×A

r(b, a)E .

Then concept lattices ofC andK are isomorphic.

Proof. LetΦ∈Q

b∈BExt(⊥^b). By previous lemma is easy to see that Φ∈L^B. Moreoverr(b, a) for any (b, a)∈B×Aas an arbitrary value fromLis an extent of⊥^b and intent of⊥^∗a=hL,{a},^tλi. Hence anyr(b, a)∈L-Bonds(⊥^b,⊥^∗a).

↑Kb (Φ)(a) = ^

b∈B

↑^r(b,a)(Φ(o)) = ^

b∈B

(Φ(o)→r(o, a)) =↑(Φ)(a).

Similarly for↓Kb=↓. ut

6.2 6= connection

Moreover, an another connection is presented in this subsection. The connection is based on the fact that concept lattice ofhX, X,6=iis isomorphic toL^X in the case thatLis closed under double negation law.

Lemma 12. Lets have an L-fuzzy formal context X = hX, X,6=i, where L is closed under double negation law. Then anyL-set fromL^X is closed inX. Proof. Lets have an arbitraryL-setf ∈L^X.

↑(f)(x) = ^

y∈X

(f(y)→(y6=x))

= ^

y∈X:y6=x

(f(y)→1)∧(f(x)→(x6=x))

= 1∧ ¬f(x) =¬f(x)

Then↓↑(f)(x) =¬¬f(x) =f(x). ut

Lemma 13. Lets have an L-fuzzy formal context C = hB, A, ri, where L is closed under double negation law. Concept lattice ofC is isomorphic to a concept lattice of second order formal context K=

B,B, A,A, r

, whereB=hB, B,6=i andA=hA, A,6=i.

Proof. Index setsI andJ are singletons in this case. So due to previous lemma Φ ∈Ext(B) = L^B. ⇑(Φ) =↑^ρ (f) where ρ= W

{β ∈L-Bonds(B,A) :β ≤r}. Because of the previous lemma we know that any row and column orris closed inB andA, respectively. Hencer∈L-Bonds(B,A) andr=ρ.

Finally↑Kb=↑. Similarly for↓Kb=↓. ut

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