Fuzzy Formal Analysis

Abner M. Brito a,b,c La´ecioC. de Barros a Estev˜aoE. Laureano a F´abioM. Bertato b Marcelo E. Coniglio b,c

aDepartment of Applied - IMECC State University of Campinas (UNICAMP), Brazil bCentre for , , and the History of Science (CLE) State University of Campinas (UNICAMP), Brazil cDepartment of - IFCH State University of Campinas (UNICAMP), Brazil Emails: [email protected]; [email protected]; [email protected]; fabio [email protected]; [email protected]

Abstract Formal Context Analysis is a mathematical theory that enables us tofind from a given set of objects, a set of attributes and a relation on them. There is a hierarchy of such concepts, from which a complete can be made. In this paper we present a generalization of these ideas using fuzzy and fuzzy implications defined from lower semicontinuous t-norms which, under suitable conditions, also results in a . Keywords: - FCA - fuzzy formal concept analysis - fuzzy attributes - concept lattice - lattice

1 Introduction

Formal Concept Analysis (FCA) constitutes a powerful tool for acquisition and representation of knowledge as well as for conceptual data analysis, based on notions from general lattice theory. In FCA, data is represented as a conceptual hierarchy, organized as a concept lattice that relates objects and their properties (see Ganter and Wille [GW99]). FCA has applications in several fields. According to Hardy-Vall´ee,a concept is “a general knowledge that [...] represents a category of objects, events or situations.”1 For example, the concept “Library” represents each individual library. One such “general knowledge” (“Library”) abstracts attributes (e.g. having a catalogue of its books) common to all objects (libraries). A concept in FCA is defined by a set of objects and a corresponding set of attributes. Nevertheless, real life knowledge is seldom precise. For instance, an automobile manufacturer may construct “concepts” that relate car features (objects) and consumer profiles (attributes). These concepts would be useful if they are interested for example in selling more cars to young women. Because “youngness” is a linguistic imprecise idea, a good strategy would be to use fuzzy FCA.

1“une connaissance g´en´eralequi [...] repr´esente une cat´egoried’objets, d’´ev´enements ou de situations.” See https://www.researchgate.net/profile/Benoit Hardy-Vallee/publication/228799196

1 Fuzzy FCA has been previously proposed in the literature and there exists a vast body of literature on thefield, a fair portion of which has been surveyed by Poelmans et al. ([PIKD14]). Most authors, such as Belohl´avek and Vychodil ([BV07]), consider left continuous t-norms in order to define fuzzy implications underlying the fuzzy FCA notions. In this paper we propose the use of fuzzy implications defined from lower semicontinuous t-norms. It should be observed that, due to monotonicity (and commutativity), both notions of continuity are equivalent for t-norms. However, the techniques used in the proofs are different.

2 Formal Concept Analysis

The definitions and theorems presented in this section follow those presented by Ganter and Wille ([GW99]) with a different notation and slightly different proofs.

Definition 1.A formal context is an ordered tripleC := , ,I , in which and are �O A � O A non-empty sets, andI is a . ⊆O×A The elements of are called objects, and the elements of attributes. We say that, in the O A contextC, objecto has attributeaiff oIa.Afinite context can be represented as a table, indexing rows by objects and columns by attributes, and marking cell (o, a) iff oIa, as in Table 1.

Table 1: A formal context of animals ❍ ❍ ❍❍A Vertebrate Lay eggs Carnivorous Has wings Flies Quadruped Crawls O ❍ Eagle X X X X X Snake X X X X Goose X X X X Swan X X X X Lion X X X

Definition 2. LetC= , ,I be a formal context. We define two maps, ∗ : 2O 2 A and �O A � → ∧ : 2A 2 O (and writeO ∗ andA ∧) as follows: →

O∗ := a : oIa for allo O (1) { ∈A ∈ } A∧ := o : oIa for alla A . (2) { ∈O ∈ } Finally, we define the central object of FCA:

Definition 3. LetC= , ,I be a formal context. A formal concept (or simply concept) of �O A � C is an ordered pairC= O,A such thatO ,A ,O ∗ =A andA ∧ =O. The setsO andA are called the extent� and� intent of the⊆O concept⊆A C respectively. Thus,C= O,A is a concept whenO is precisely the of objects that has all the attributes ofA� in common.� Example 4. The following are formal concepts of the context presented in Table 1:

C = Lion , Quadruped, Carnivorous, Vertebrate , (3) Lion �{ } { }� C = Eagle, Snake, Lion , Carnivorous, Vertebrate . (4) Carnivorous �{ } { }�

2 Notice that there is an inverse relation between the numbers of elements in the intent and the extent of a concept. If we increase the number of elements in the extent (we added “Eagle” and “Snake” to it), the number of elements in the intent is reduced (“Quadrupede” is not in the intent ofC Carnivorous). In fact, the following useful properties hold: Theorem 5. LetO,O ,O andA,A ,A . Then 1 2 ⊆O 1 2 ⊆A

1. IfO O thenO ∗ O ∗ 1’. IfA A thenA ∧ A ∧ 1 ⊆ 2 2 ⊆ 1 1 ⊆ 2 2 ⊆ 1 2.O O ∗∧ 2’.A A ∧∗ ⊆ ⊆ 3.O ∗ =O ∗∧∗ 3’.A ∧ =A ∧∗∧ 4.O A ∧ iffA O ∗ iffO A I ⊆ ⊆ × ⊆ Proof. We shall prove items 1., 2., 3. and 4.. Items with a prime can be proved analogously.

1. Leta O ∗. Then oIa for allo O . In particular, oIa for allo O . Thus,a O ∗. ∈ 2 ∈ 2 ∈ 1 ∈ 1 2. Leto O. Then oIa for alla O ∗, by definition ofO ∗. Thus, by definition ofO ∗∧, we ∈ ∈ haveo O ∗∧. ∈ 3. From 2’. withA=O ∗ we already know thatO ∗ O ∗∧∗. Leta O ∗∧∗. Then ⊆ ∈

(i) oIa for allo O ∗∧, ∈

by definition ofO ∗∧∗. Now let ˜o O∗∧ befixed. For all ˜a , ∈ ∈A

(ii) if ˜oI˜a then ˜a ∗O, ∈

by definition ofO ∗∧. From (i) we have ˜oIa. Thus, using (ii) we conclude thata O ∗. ∈ 4. SupposeO A ∧. By 1.,A ∧∗ O ∗. Using 2’. and transitivity of , we haveA O ∗. ⊆ ⊆ ⊆ ⊆ Now supposeA O ∗. By 1’. and 2., we haveO O ∗∧ A ∧. ⊆ ⊆ ⊆ AssumingA O ∗, leto O anda A. By definition ofO ∗, oI˜a for all ˜a O∗. By ⊆ ∈ ∈ ∈ hypothesis,a A O ∗. Thus, oIa. Sinceo O anda A are arbitrary,O A I. ∈ ⊆ ∈ ∈ × ⊆ Hence,A O ∗ O A I. ⊆ ⇒ × ⊆ Finally, supposeO A I. Leto O. By hypothesis, for alla A we have oIa. By × ⊆ ∈ ∈ definition ofO ∗, if oIa thena O ∗. Thus ifa A thena O ∗. This completes the proof. ∈ ∈ ∈

From properties 3. and 3’. we see that, givenO andA , O ∗∧,O∗ and A ∧,A∧∗ are concepts. On the other hand ifC= O,A is a concept⊆O then⊆A by� definition � � � � �

C= A ∧,A = (O ∗)∧,O∗ . � � � �

Consequently every concept has either form O ∗∧,O∗ or A ∧,A∧∗ , whereO andA are arbitrary. � � � � ⊆O ⊆A This gives us a procedure forfinding concepts. Choose a subset of objects (or attributes), apply ∗ to get an intent, and then apply ∧ to get an extent (in fact, the smallest extent containing the original object subset). Tofind all the concepts of a given context, simply list all the subsets of objects (or attributes), and then apply the maps ∗ and ∧ respectively (or ∧ and ∗).

3 Example 6. Notice that

Eagle ∗∧ = Vertebrate, Lay eggs, Carnivorous, Has wings, Flies ∧ = Eagle . { } { } { } Thus, the following is a concept: C = Eagle , Flies, Has Wings, Lay eggs, Carnivorous, Vertebrate . (5) Eagle �{ } { }� There are also interesting properties regarding intersections of extents and intents. For in- stance, the intersection of the intents of (3) and (5) is the intent of (4). This is a particular case of a more general fact, presented in the following proposition. Proposition 7. LetJ be an index set and, for eachα J, letO and letA . Then ∈ α ⊆O α ⊆A ∗ Oα∗ = Oα , (6) α J �α J � �∈ �∈ ∧ Aα∧ = Aα . (7) α J �α J � �∈ �∈ Proof. We shall prove only (6). The proof of (7) is analogous.

a O∗ iffa O ∗ for allα J ∈ α ∈ α ∈ α J �∈ iff oIa for allo O , for allα J ∈ α ∈ iff oIa for allo O ∈ α α J �∈ ∗ iffa O . ∈ α �α J � �∈

Given two setsY X and a partial order onX, the infimum ofY (if it exists) is the elementi such that⊆i y for anyy Y and≤ ifx y for ally Y thenx i . Replacing Y y ≤ ∈ ≤ ∈ ≤ y ≤ by andi Y bys Y we get the definition of the supremums Y ofY. = X, is called a lattice if every≥ finite subset ofX has both an infimum and a supremum.L If every� ≤� subset ofX has an infimum and a supremum, then is said to be complete. From Proposition 7 we can showL that, with an order induced by set inclusion, the set of all formal concepts of any context constitutes a complete lattice. Theorem 8. LetC be a formal context. LetB(C) be the set of all concepts ofC. Define the 2 relation onB(C) by O 1,A1 O2,A2 iffO 1 O 2. Then is an order onB(C). If C C ≤we say thatC �is a subconcept� ≤ � ofC� . Correspondingly,⊆ C≤ is a superconcept ofC . 1 ≤ 2 1 2 2 1 Furthermore, := B(C), is a complete lattice, called the concept lattice ofC. L C � ≤� IfJ is an index set andC α = O α,Aα B(C) for eachα J then � � ∈ ∈ ∗ inf Cα = Oα, Oα (8) α J ∈ �α J �α J � � �∈ �∈ ∧ sup Cα = Aα , Aα . (9) α J ∈ ��α J � α J � �∈ �∈

4 Proof. That is an order is clear from the fact that is an order. Now we prove (9). For each ≤ ⊆ α J we haveA =O ∗ . Thus, by (6), ∈ α α ∗ Aα = Oα∗ = Oα α J α J �α J � �∈ �∈ �∈ is the intent of a concept (applying ∧ gives the concept’s extent). Hence, by properties of set intersection, α J Aα is the greatest intent smaller than all theC α. Using 1’. of Theorem 5, ∩ ∈ ( α J Aα)∧ is the smallest extent greater than all theO α, and so the supremum of is as stated. ∩ ∈ ≤ The proof of (8) is similar to that of (9), only working with the extents of theC α rather than their intents, and applying (7) instead of (6).

Theorem 8 allows us to use lattice theory forfinding out many properties that come from a formal context. In particular, afinite concept lattice has an easy visual representation (see Example 9 below). In order to interpret the concept lattice from the diagram, one may write, for each concept on the diagram, the elements of its intent and extent. However, from the order of the concept lattice a tidier manner of presenting the diagram can be devised: for a given concept,≤ instead of writing every element of its extent (or intent), we write only those objects (attributes) that did not appear below (above) in the concept lattice. Example 9. The concept lattice of the context presented in Table 1 is shown in Fig. 9. Each concept is represented by a circle. Here animals are represented by numbers 1-5 in the order they appear in Table 1. Attributes are represented by letters a-g, also in the order they appear in the table. The extent (intent) of a given conceptC has an object (attribute) iff that object (attribute) appears near a concept C˜ such that there is a descending (ascending) path fromC to C˜. a

b C1 c

d,e C2 C3 C4 C5 g 3,4 f C6 C7 2 C8 1 C9 5

Take, for instance, the conceptC 3. It has ascending paths to conceptsC 1 andC 3, and descending paths to conceptsC 8,C5,C6,C7 andC 9. On the other hand,C 3 has no (strictly ascending or descending) paths toC 2 orC 4. Thus,C 3 = 1,2,5 , a, c . Concept lattices are visual tools that allow us tofind out�{ relations} { } on� attributes and objects (for example, any animal with propertye also has propertyb). However, limitations may arise on the theory, as, for example, the definition of formal context allows us only to work only with precise relations. For example, a chicken canfly for short distances, but this information could not be expressed on a formal context as defined earlier. In the next section we generalize these ideas to allow fuzzy objects, attributes and relations.

3 Fuzzy Formal Concept Analysis

Wefirst state some basic definitions of and then proceed to a fuzzy generalization of FCA. Definition of triangular norm corresponds to that presented by Klement et al. ([KMP00]), whereas definitions of fuzzy implication, fuzzy subset and fuzzy (binary) relation correspond to definitions by Barros et al. ([dBBL17]).

5 Definition 10.A triangular norm (or t-norm) is a : [0, 1] 2 [0, 1] satisfying, for all � → x, y, z [0, 1]: ∈ 1.x�y=y�x (commutativity)

2.x�(y�z)=(x�y)�z (associativity) 3. Ify z thenx y x z (monotonicity) ≤ � ≤ � 4.x�1=x (boundary condition) Definition 11.A fuzzy implication is a map : [0, 1] 2 [0, 1] such that for all x, y, z [0, 1]: ⇒ → ∈ 1. (0 0) = 1, (1 0) = 0, (boundary conditions) ⇒ ⇒ (0 1) = 1 and (1 1) = 1 ⇒ ⇒ 2. Ify x then(x z) (y z) (monotonicity in thefirst component) ≤ ⇒ ≤ ⇒ 3. Ifx y then(z x) (z y) (monotonicity in the second component) ≤ ⇒ ≤ ⇒ Definition 12.A R-implication is a fuzzy implication defined by ⇒ R (x y)= z [0, 1] :x z y , (10) ⇒ R { ∈ � ≤ } � where is a t-norm and stands for supremum. We say that is induced by . � ⇒ R � Definition 13. LetU be� a (classical) set. A fuzzy subsetF ofU is defined by a function φ :U [0, 1], called the membership function ofF. F → Given two fuzzy subsetsF 1,F2 ofU, we say thatF 1 is a (fuzzy) subset ofF 2 ifφ F1 (u) φ (u) for allu U, and in this case we writeF F . ≤ F2 ∈ 1 ⊆ 2 If F α α J is a family of fuzzy subsets ofU, then their union and intersection are defined respectively� � ∈ by the membership functions

φ α J Fα (u) = φα(u) (11) ∪ ∈ α J �∈

φ α J Fα (u) = φα(u), (12) ∩ ∈ α J �∈ where and stand for supremum and infimum respectively.

Definition� 14.�A fuzzy (binary) relation on a pair of (classical) setsU 1,U2 is a fuzzy subset of U U . In particular, ifA ,A are fuzzy subsets ofU ,U respectively, and is a t-norm, then 1 × 2 1 2 1 2 � the (fuzzy) cartesian product ofA 1,A2 induced by� is the fuzzy relationA 1 � A2 onU 1 U 2 defined by the membership function × ×

φA1 A2 (x1, x2) =φ 1(x1)�φ 2(x2). (13) ×� Now we can start defining the objects of Fuzzy Formal Concept Analysis (fFCA).

Definition 15.A fuzzy formal context is an ordered tripleC f := , ,I f , in which and �O A � O A are non-empty (classical) sets, andI is a fuzzy binary relation. f ⊆O×A

6 Notice that, in (1) and (2), the characteristic functions ofO ∗ andA ∧ can be expressed respectively by χ (˜a) = o (o O oI˜a), O∗ ∀ ∈O ∈ −→ χ (˜o) = a (a A ˜oIa). A∧ ∀ ∈A ∈ −→ Since an application of the universal quantifier, , returns the smallest truth value a predicate assumes ( xP(x) = 1 iffP(u) = 1 for eachu U), we∀ generalize as an infimum. ∀ ∈ ∀ Definition 16. LetC f = , ,I f be a fuzzy formal context. Let be a fuzzy implication. �O A � ⇒ Then we define, for all fuzzy subsetsO andA , the fuzzy subsetsO ∗ andA ∧ by their membership functions defined⊆ asO ⊆A ⊆A ⊆O

φO (a) = inf [φO(o) φ I (o, a)], (14) ∗ o ⇒ f ∈O φA (o) = inf φA(a) φ I (o, a) . (15) ∧ a ⇒ f ∈A � � With these maps defined we canfinally generalize the idea of formal concept.

Definition 17. LetC f = , ,I f be a fuzzy formal context and let be a fuzzy implication. �O A � ⇒ LetO,A be fuzzy subsets of and , respectively. We say thatC = O,A is a fuzzy formal O A f � � concept ofC f ifO ∗ =A andA ∧ =O. Our goal is to show that, as in the classical case, it is possible to produce a complete lattice of fuzzy formal concepts. Our definitions of semicontinuous functions are proven by Bourbaki ([Bou66]) to be equivalent with the definitions of semicontinuity he presents. The upper semi- continuous R-implication induced by a lower semicontinuous t-norm is precisely what allows us to have a complete lattice of fuzzy concepts. In what follows, a sequence (x1, x2, ...) is denoted by (xn), and sox 0 is not an element of the sequence (xn). Definition 18. LetX,Y IR 2 be non-empty (classical) sets. Letf:X Y be a function. We ⊆ → say thatf is upper semicontinuous atx X if, for any sequence(x ) converging tox , 0 ∈ n 0

inf sup f(x m) =: lim sup f(x n) f(x 0). (16) n>0 m n n ≤ � ≥ � →∞

Iff is upper semicontinuous at everyx 0 X thenf is upper semicontinuous. Similarly,f is lower semicontinuous at∈x X if for any sequence(x ) converging tox , 0 ∈ n 0

sup inf f(x m) =: lim inf f(x n) f(x 0). (17) m n n ≥ n>0 � ≥ � →∞ Iff is lower semicontinuous at everyx X thenf is lower semicontinuous. 0 ∈ Iff is both upper and lower semicontinuous atx 0 thenf is continuous atx 0, and

lim sup f(x n) =f(x 0) = lim inf f(x n). (18) n n →∞ →∞ Iff is continuous at everyx X then it is continuous. 0 ∈ We now derive a useful expression concerning lower semicontinuous t-norms in Lemma 20, and for its proof we use Proposition 19. In the following, we shall writex n x 0 meaning that the increasing sequence (xn) converges tox . Similarly,x x means that� the decreasing sequence (x ) converges tox . 0 n � 0 n 0

7 Proposition 19. Let be a lower semicontinuous t-norm. Let x, y [0, 1]. Define � ∈ S= z [0, 1] :x z y . { ∈ � ≤ } Then supS S. ∈ Proof. Letz = supS, and let (z ) be a sequence on [0, 1] such thatz z . Then 0 n n � 0 x z y for allm>0, � m ≤ asz z for eachm> 0. Thus, for eachn>0, m ≤ 0 inf (x�z m) x�z n y. m n ≤ ≤ ≥ Taking the supremum forn> 0 on the left-hand side we have, by lower semicontinuity of�,

x�z 0 lim inf(x�z n) = sup inf (x�z m) y. ≤ n m n ≤ →∞ n>0 � ≥ � Therefore,z A. 0 ∈ Lemma 20. Let be a lower semicontinuous t-norm and let be the R-implication induced � ⇒ by . Then, for all x, y [0, 1] we havex [(x y) y]. � ∈ ≤ ⇒ ⇒ Proof. Let x, y [0, 1]. By Proposition 19,x�(x y) y. Using commutativity of�, it is clear thatx S :=∈ z [0, 1] : (x y) z y . Hence⇒ ≤ ∈ { ∈ ⇒ � ≤ } x supS = [(x y) y]. ≤ ⇒ ⇒

Now we have what is necessary to generalize Theorem 5, establishing dual relations for the maps ∗ and ∧ in the fuzzy case.

Theorem 21. LetC f = , ,I f be a fuzzy context. LetO,O 1,O2 be fuzzy subsets of and �O A � 2 O A, A1,A2 be fuzzy subsets of . Let�: [0, 1] [0, 1] be a lower semicontinuous t-norm, and let be the R-implication inducedA by . Then: → ⇒ � 1. IfO O thenO ∗ O ∗ 1’. IfA A thenA ∧ A ∧ 1 ⊆ 2 2 ⊆ 1 1 ⊆ 2 2 ⊆ 1 2.O O ∗∧ 2’.A A ∧∗ ⊆ ⊆ 3.O ∗ =O ∗∧∗ 3’.A ∧ =A ∧∗∧ 4.O A ∧ iffA O ∗ iffO A I ⊆ ⊆ × � ⊆ f Proof. We shall prove items 1., 2., 3. and 4.. Items with a prime can be proved analogously.

1. Suppose thatO 1 O 2. Leta . Leto . By hypothesis,φ O1 (o) φ O2 (o). Since is decreasing in its⊆first component∈A we have∈O ≤ ⇒

(φ (o) φ (o, a)) (φ (o) φ (o, a)). O2 ⇒ If ≤ O1 ⇒ If Taking the infimum overo on both sides, we see that

φO (a) φ O (a) 2∗ ≤ 1∗

by definition of the map ∗. Buta is arbitrary. ThusO ∗ O ∗. ∈A 2 ⊆ 1

8 2. Leto . By hypothesis,� is lower semicontinuous and so Lemma 20 holds. Using monotonicity∈O of in itsfirst component we get ⇒ φ (o) [(φ (o) φ (o, a)) φ (o, a)] O ≤ O ⇒ If ⇒ If

inf (φO(˜o) φI (˜o,a)) φI (o, a) ≤ ˜o ⇒ f ⇒ f � ∈O � = [φ (a) φ (o, a)]. O∗ ⇒ If Taking the infimum overa on the right-hand side, we get

φ (o) φ (o). O ≤ O∗∧ Sinceo is arbitrary,O O ∗∧. ∈O ⊆ 3. From 2., we haveO O ∗∧. Thus, using 1.,O ∗∧∗ O ∗. On the other hand, usingA=O ∗ ⊆ ⊆ in 2’., we haveO ∗ O ∗∧∗. Hence,O ∗ =O ∗∧∗. ⊆ 4. We shallfirst prove that ifA O ∗ thenO A ∧; then we show thatO A ∧ implies ⊆ ⊆ ⊆ O � A I f ; andfinally we prove that ifO � A I f thenA O ∗, concluding that the three× properties⊆ are equivalent. × ⊆ ⊆

(a) Suppose thatA O ∗. By 1’.,O ∗∧ A ∧. Using 2. and transitivity of , we have ⊆ ⊆ ⊆ O A ∧. ⊆ (b) Suppose thatO A ∧. Leto anda . Then ⊆ ∈O ∈A φO(o) φ A (o) = inf [φA(˜a) φI (o,˜a))] ≤ ∧ ˜a ⇒ f ∈A [φ (a) φ (o, a)] ≤ A ⇒ If = sup z [0, 1] :φ (a) z φ (o, a) . { ∈ A � ≤ If } By lower semicontinuity of�, Proposition 19 holds and so φ (o) z [0, 1] :φ (a) z φ (o, a) . O ∈{ ∈ A � ≤ If } Using commutativity of�, we have

φO A(o, a) =φ O(o)�φ A(a) φ If (o, a). ×� ≤ Buto anda are arbitrary and soO A I . ∈O ∈A × � ⊆ f (c) Suppose thatO A I . Then for allo and for alla , × � ⊆ f ∈O ∈A φ (a) z [0, 1] :φ (o) z φ (o, a) , A ∈{ ∈ O � ≤ If } whence

φ (a) sup z [0, 1] :φ (o) z φ (o, a) =[φ (o) φ (o, a)]. A ≤ { ∈ O � ≤ If } O ⇒ If Taking the infimum overo on the right-hand side, we see that

φ (a) φ (a). A ≤ O∗ Buta is arbitrary, and soA O ∗. ∈A ⊆

9 As stated earlier, we want to prove that a lattice of fuzzy formal concepts is complete. As we shall see, the fuzzy implication we use when defining ∗ and has to be upper semicontinuous. It turns out that for R-implicationsm this follows from∧ lower semicontinuity of the t-norm, as we show in Proposition 23. Before that we make an intermediate step. Lemma 22. LetX,Y IR. Letf:X Y be an increasing function. Let(x ) be a sequence ⊆ → n onX such thatx n x X. Let(y n) be a sequence onY such that, for alln>0,f(x n) y n. Then � ∈ ≤ f(x) lim inf yn . (19) n ≤ →∞

Proof. By monotonicity of (xn), we have supm n xm =x n0 for alln 0 > 0, whence for each ≥ 0 n0 >0fixed, inf n>0 supm n xm x n0 . But (xn) converges tox, and so for alln 0 > 0 we have ≥ ≤ x = lim supn xn x n0 . Becausef is increasing we have, for eachn 0 > 0,f(x) f(x n0 ) →∞ ≤� � ≤ ≤ yn0 . Now, taking the limit inferior overn 0 on the right-hand side and changing the variablen 0 ton, we complete the proof.

Proposition 23. Let be a lower semicontinuous t-norm, and let be the R-implication � ⇒ induced by�. Then for allx 0, y0 [0, 1]fixed, the mapsx (x y 0) andy (x 0 y) are upper semicontinuous. ∈ �→ ⇒ �→ ⇒

Proof. Letx �, y� [0, 1]. We want to show thatx (x y ) andy (x y) are upper semi- ∈ �→ ⇒ 0 �→ 0 ⇒ continuous atx � andy � respectively. Let (xn),(y n) be sequences on [0, 1] converging respectively tox � andy �. For eachn> 0, consider the following definitions: z(1) = (x y ), z (2) = (x y ), (20) n n ⇒ 0 n 0 ⇒ n (1) (2) ˜x =x � , ˜x =x 0 , (21) (1) (2) ˜xn = inf xm , ˜xn =x 0 , (22) m n ≥ (1) (2) ˜y =y 0 , ˜y =y � , (23) (1) (2) ˜ny =y 0 , ˜ny = sup ym , (24) m n ≥ ˜(1)z = ˜x(1) ˜y(1) ,˜z (2) = ˜x(2) ˜y(2) . (25) n n ⇒ n n n ⇒ n � � � � Notice that, fori=1, 2 the following hold:

(i) (i) (1) (2) 1. ˜xn ˜x , as the ˜xn are infima of decreasing sets and ˜xn is constant; � � � (i) (i) (1) (2) 2. ˜ny ˜y , because ˜ny is constant and the ˜xn are suprema of decreasing sets; � � � (i) (i) (i) (i) 3. For alln> 0 monotonicity of yieldsz n ˜zn , whence lim supn zn lim supn ˜nz . ⇒ ≤ →∞ ≤ →∞ (i) 4. The sequence (˜nz ) is decreasing, fori=1, 2.

Now letn 0 >0befixed, and letn>n 0. By Proposition 19 and (25), we have

(i) (i) (i) (i) (i) ˜xn �˜zn ˜xn �˜zn ˜yn . 0 ≤ ≤ Thus, (19) yields (i) (i) (i) (i) ˜xn0 � lim sup ˜nz lim inf ˜ny = ˜y . n ≤ n � →∞ � →∞

10 Taking the limit inferior overn 0 on the left-hand side and using lower semicontinuity of�,

(i) (i) (i) (i) (i) ˜x � lim sup ˜nz lim inf ˜xn0 � lim sup ˜nz ˜y . n ≤ n0 n ≤ � →∞ � →∞ � � →∞ �� Hence by definition of ˜x(i) ˜y(i) and 3. above, ⇒ � (i) � (i) (i) (i) lim sup zn lim sup ˜nz ˜x ˜y . n ≤ n ≤ ⇒ →∞ →∞ � � Fori = 1 this meansx (x y ) is upper semicontinuous atx �, and fori = 2,y (x y) �→ ⇒ 0 �→ 0 ⇒ is upper semicontinuous aty �. Butx �, y� [0, 1] are arbitrary. Therefore, both the maps x (x y ) andy (x y) are upper semicontinuous.∈ �→ ⇒ 0 �→ 0 ⇒

Proposition 24. LetC f = , ,I f be a fuzzy context. Let be a lower semicontinuous t- �O A � � norm, and let be the implication induced by it. LetJ be an index set and, for eachα J, let O andA⇒ . Then ∈ α ⊆O α ⊆A ∗ Oα∗ = Oα , (26) α J �α J � �∈ �∈ ∧ Aα∧ = Aα . (27) α J �α J � �∈ �∈ Proof. We prove (26). The proof of (27) is analogous. Leta . For eachα J we have ∈A 0 ∈

φ( α J Oα)∗ (a) = inf [φ α J Oα (o) φ If (o, a)] ∪ ∈ o ∪ ∈ ⇒ ∈O

= inf sup φOα (o) φ If (o, a) o α J ⇒ ∈O �� ∈ � �

sup φOα (o) φ If (o, a) ≤ α J ⇒ �� ∈ � � φO (o) φ I (o, a) , ≤ α0 ⇒ f as is decreasing in thefirst component.� Applying the infi�mum overo and then the ⇒ ∈O infimum overα J on the right-hand side yieldsφ ( α J Oα)∗ (a) φ α J Oα∗ (a). Sincea is ∈ ∪ ∈ ≤ ∩ ∈ ∈A ∗ arbitrary, α J Oα α J Oα∗ . On the other∈ hand,⊆ suppose∈ for the sake of that for someo anda , �� � � ∈O ∈A

κ := inf inf (φOα (˜o) φIf (˜o,a)) > sup φOα (o) φ If (o, a) . α J ˜o ⇒ α J ⇒ ∈ � ∈O � �� ∈ � �

Definex 0 = supα J φOα (o). Let (αn) be a sequence onJ such that φOαn (o) converges tox 0 and, for eachn> 0,∈ definex =φ (o). By hypothesis, for eachn> 0, n Oαn � �

(x0 φ I (o, a))< inf inf (φO (˜o) φI (˜o,a)) (=κ) ⇒ f α J ˜o α ⇒ f ∈ � ∈O � inf φO (˜o) φI (˜o,a) ≤ ˜o αn ⇒ f ∈O φO � (o) φ I (o, a) � ≤ αn ⇒ f = (x φ (o, a)). � n ⇒ If �

11 Thus,

(x0 φ If (o, a))<κ lim sup (xn φ If (o, a)), ⇒ ≤ n ⇒ →∞

and sox (x φ If (o, a)) is not upper semicontinuous, contradicting Proposition 23. Hence, �→ for⇒ allo and for alla , ∈O ∈A φ (a) =κ φ (o) φ (o, a) . α J Oα∗ α J Oα If ∩ ∈ ≤ ∈ ⇒ �� � � � Taking the infimum overo on the right-hand side, and sincea is arbitrary, we conclude that ∗ ∈A α J Oα∗ α J Oα . ∈ ⊆ ∈ Theorem� 25.�� LetC f� = , ,I f be a fuzzy formal context. Using a lower semicontinuous �O A � t-norm and the R-implication induced by it to define the maps ∗ and ∧, define the order on ≤ the setB f (Cf ) of all the fuzzy formal concepts ofC f by

O ,A O ,A iffO O (iffA =O ∗ O ∗ =A ). (28) � 1 1� ≤ � 2 2� 1 ⊆ 2 2 2 ⊆ 1 1

Then f := B f (Cf ), is a complete lattice, called the fuzzy concept lattice ofC f . L C � ≤� Proof. Similar to that of Theorem 8, using Proposition 24 rather than Proposition 7.

4 Final Remarks

As mentioned in the Introduction, there exists a vast body of literature on fuzzy Formal Context Analysis. Burusco and Fuentes-Gonz´alez([FGB94]) introduced fuzzy concept lattices. Belohl´avek and Vychodil ([BV07]) have shown that it is possible to define a complete lattice of fuzzy concepts, by means of fuzzy implications obtained from left continuous t-norms. These t- norms are equivalent to lower semicontinuous t-norms. In this paper we recast some basic notions and results from fuzzy FCA in terms of lower semicontinuous t-norms. Given the difference between both definitions for t-norms (left continuous and lower semicontinuous), proving the results requires somewhat different techniques. Within our framework it is shown that the set of fuzzy concepts is a complete lattice.

Acknowledgement

Brito wasfinancially supported by a scolarship from CAPES (Brazil). Barros was supported by a individual research grant from CNPq (Brazil) n. 306546/2017-5. Coniglio was supported by a individual research grant from CNPq (Brazil) n. 308524/2014-4.

References

[Bou66] Nicolas Bourbaki. Elements of Mathematics: General Topology, volume Part 1. Addison-Wesley, Dordrecht, 1966. [BV07] Radim Belohl´avek and Vil´emVychodil. Fuzzy concept lattices constrained by hedges. In JACIII, volume 11, pages 536–545. 01 2007. [dBBL17] La´ecioCarvalho de Barros, Rodney Carlos Bassanezi, and Weldon A. Lodwick.A First Course in Fuzzy Logic, Fuzzy Dynamical Systems, and Biomathematics: Theory and Applications. Springer, Berlin, 2017.

12 [FGB94] Ram´onFuentes-Gonz´alezand Ana Burusco. The study of the L-fuzzy concept lattice. 1, 01 1994. [GW99] Bernhard Ganter and Rudolf Wille. Formal Concept Analysis: Mathematical Foun- dations. Springer, Berlin, 1999. [KMP00] Erich P. Klement, Radko Mesiar, and Endre Pap. Triangular Norms. Kluwer Academic Publishers, Dordrecht, 2000. [PIKD14] Jonas Poelmans, Dmitry I. Ignatov, Sergei O. Kuznetsov, and Guido Dedene. Fuzzy and rough formal concept analysis: a survey. International Journal of General Sys- tems, 43(2):105–134, 2014.

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