=Paper=
{{Paper
|id=Vol-162/paper-4
|storemode=property
|title=What is a fuzzy concept lattice?
|pdfUrl=https://ceur-ws.org/Vol-162/paper4.pdf
|volume=Vol-162
|dblpUrl=https://dblp.org/rec/conf/cla/BelohlavekV05
}}
==What is a fuzzy concept lattice?==
https://ceur-ws.org/Vol-162/paper4.pdf
What is
What is aa fuzzy
fuzzy concept lattice???
concept lattice?
Radim Bělohlávek and Vilém Vychodil
Radim Bělohlávek and Vilém Vychodil
Department of Computer Science, Palacky University, Olomouc
Department of Computer
Tomkova 40, CZ-779Science, Palacký
00 Olomouc, University,
Czech RepublicOlomouc
Tomkova 40, CZ-779 00vilem.vychodil}@upol.cz
{radim.belohlavek, Olomouc, Czech Republic
{radim.belohlavek, vilem.vychodil}@upol.cz
Abstract. The paper is an overview of several approaches to the notion
of a concept lattice from the point of view of fuzzy logic. The main aim
is to clarify relationships between the various approaches.
Keywords: formal concept analysis, fuzzy logic, fuzzy attribute
1 Introduction
Formal concept analysis (FCA) deals with a particular kind of analysis of data
which, in the basic setting, has the form of a table with rows corresponding
to objects, columns corresponding to attributes, and table entries containing
1’s and 0’s depending on whether an object has or does not have an attribute
(we assume basic familiarity with FCA and refer to [24] for information). The
basic setting is well-suited for attributes which are crisp, i.e. each object of the
domain of applicability of the attribute either has (1) or does not have (0) the
attribute. Many attributes are fuzzy rather than crisp. That is to say, it is a
matter of degree to which an object has a (fuzzy) attribute. For instance, when
asking whether a man with a height of 182 cm is tall, one probably gets an
answer like “not completely tall but almost tall” or “to a high degree tall”, etc.
A natural idea, developed in fuzzy logic [12, 28, 31], is to assign to an object a
truth degree to which the object has a (fuzzy) attribute. Degrees are taken from
an appropriate scale L of truth degrees. A favorite choice of L is the real unit
interval [0, 1] or some subset of [0, 1]. Then, we can say that a man with a height
of 182 cm is tall to a degree, say, 0.8. Doing so, the entries of a table describing
objects and attributes become degrees from L instead of values from {0, 1} as is
the case of the basic setting of FCA.
FCA provides means to process tables with degrees from a scale L. Namely,
one can consider a table with degrees from L a many-valued context and use a
so-called conceptual scaling [24]. There is, however, another way to process such
data tables. The way is based on considering the table entries as truth degrees
in fuzzy logic and proceed analogously as we do in the basic setting of FCA, just
“replacing classical (bivalent) logic with fuzzy logic”. Recently, the second way
?
Supported by grant No. 1ET101370417 of GA AV ČR and by institutional support,
research plan MSM 6198959214.
Radim Bělohlávek, Václav Snášel (Eds.): CLA 2005, pp. 34–45, ISBN 80–248–0863–3.
What is a fuzzy concept lattice? 35
gained a considerable interest. The structures which result this way are called
fuzzy concept lattices, fuzzy Galois connections, etc. However, the particular
approaches, and hence the resulting structures, differ.
The basic aim of this paper is to compare them and to show some of the
basic relationships between them. Due to a limited scope, we omit some technical
details (mostly proofs). These are left for a full version of this paper.
Note that we do not study mathematical or any other (like epistemic) re-
lationships of the discussed “fuzzy logic approaches” to the above-mentioned
possibility of using conceptual scaling. This, again, is left for a forthcoming pa-
per.
2 Why fuzzy logic in formal concept analysis and a basic
standpoint
2.1 Why fuzzy logic in formal concept analysis?
The main reason is the following. FCA is based on working with formulas like
“an object belongs to (a set) A iff it has all attributes from (a set) B”. In the
basic setting (i.e., table entries contain 0’s and 1’s), this is done in classical logic.
When the truth degrees in a data table are taken from scale L, do the same as
in the basic setting of FCA, only “replace classical logic by fuzzy logic”. Doing
so, the results of FCA in a fuzzy setting will have the same verbal interpretation
as those in the classical setting. Namely, the underlying formulas are the same
in both the fuzzy setting and the classical setting and the difference is only in
the interpretation of the formulas (many-valued interpretation in fuzzy setting
vs. bivalent interpretation in classical setting).
2.2 A basic standpoint: first-order fuzzy logic
However, “replace classical logic by fuzzy logic” is ambiguous. Namely, there are
several ways to perform the “replacement”. Nevertheless, there is a way which
might be considered direct and simple. Namely, one can use a framework of
first-order fuzzy logic (and fuzzy relations) the same way as one uses first-order
classical logic (and classical relations) in the basic setting of FCA. Since the
standpoint of a first-order logic (and relations) is simple and easy to understand,
we take it for a basic standpoint in our paper. It is the one to which we compare
the other approaches. In the rest of this section, we recall some basic concepts
we need. We refer to [12, 28, 30] for further information.
As a set of truth degrees equipped with suitable operations (truth functions
of logical connectives) we use a complete residuated lattice i.e. an algebra L =
hL, ∧, ∨, ⊗, →, 0, 1i such that hL, ∧, ∨, 0, 1i is a complete lattice with 0 and 1
being the least and greatest element of L, respectively; hL, ⊗, 1i is a commutative
monoid (i.e. ⊗ is commutative, associative, and a⊗1 = 1⊗a = a for each a ∈ L);
⊗ and → satisfy so-called adjointness property: a ⊗ b ≤ c iff a ≤ b → c; for each
a, b, c ∈ L. Elements a of L are called truth degrees (usually, L ⊆ [0, 1]). ⊗ and
36 Radim Bělohlávek, Vilém Vychodil
→ are (truth functions of) “fuzzy conjunction” and “fuzzy implication”. By LU
(or LU ) we denote the collection of all fuzzy sets in a universe U , i.e. mappings
A of U to L. For A ∈ LU and a ∈ L, a set a A = {u ∈ U | A(u) ≥ a} is called an
a-cut of A.
2.3 What else than a first-order fuzzy logic?
Other approaches can be based on working with the truth degrees from L in
the data table in any way which, if taking L = {0, 1}, yields the same what we
have in the basic setting of FCA (we will see some of them later). Of course,
this requirement follows from a requirement saying that we want to have a
generalization of the basic setting of FCA. If the requirement is satisfied, we can
consider the results (over a general scale L of truth degrees) as “having the same
meaning as in the basic setting of FCA”.
3 Overview and comparison of existing approaches to
fuzzy concept lattices
A common point of all of the presented approaches is the notion of a fuzzy con-
text, i.e. the input data. Let L be a scale of truth degrees. L might be a support
set of some structure L = hL, . . .i of truth degrees like a complete residuated
lattice described above. Then, a fuzzy context (L-context, or L-context) is a
triplet hX, Y, Ii where X and Y are sets of objects and attributes, respectively,
and I : X × Y → L is a fuzzy relation (L-relation, L-relation) between X and Y .
A degree I(x, y) ∈ L is interpreted as a degree to which object x has attribute
y. Note that if one takes L = {0, 1} then the notion of a fuzzy context coincides
in an obvious way with the notion of an (ordinary) context.
3.1 Basic setting of FCA
To fix notation for further reference, we recall the following [24]. A formal context
(formal counterpart to a data table) is a triplet hX, Y, Ii where X and Y are
sets of objects and attributes, respectively, and I ⊆ X × Y is a binary relation
“to have” (hx, yi ∈ I means that object x has attribute y). A formal concept in
hX, Y, Ii is a pair hA, Bi of a set A ⊆ X of objects and a set B ⊆ Y of attributes
such that
B is the collection of all attributes shared by all objects from A (1)
A is the collection of all objects sharing all the attributes from B. (2)
For A ⊆ X and B ⊆ Y , put
A↑ = {y | for each x ∈ A : hx, yi ∈ I}, (3)
B ↓ = {x | for each y ∈ B : hx, yi ∈ I}. (4)
What is a fuzzy concept lattice? 37
This says that, A↑ is the set of all attributes from Y shared by all objects
from A, and B ↓ is the set of all objects from X sharing all attributes from
B. Therefore, a formal concept is a pair hA, Bi for which A↑ = B and B ↓ =
A. For a formal concept hA, Bi, A is called an extent (collection of objects
covered by hA, Bi), B is called an intent (collection of attributes covered by
hA, Bi). Denote by B (X, Y, I) the set of all formal concepts in hX, Y, Ii, i.e.
B (X, Y, I) = {hA, Bi | A↑ = B, B ↓ = A}. Introduce a relation ≤ on B (X, Y, I)
by hA1 , B1 i ≤ hA2 , B2 i iff A1 ⊆ A2 (or, equivalently B2 ⊆ B1 ). The structure
hB (X, Y, I), ≤i, i.e. set B (X, Y, I) equipped with ≤, is called a concept lattice of
hX, Y, Ii. Basic structure of concept lattices is described by the so-called main
theorem of concept lattices [24, 39].
3.2 The approach by Burusco and Fuentes-Gonzáles
Burusco and Fuentes-Gonzáles are the authors of the first paper on FCA in a
fuzzy setting. Namely, in [21], they proceed as follows. Let L = hL, ≤, 0 , ⊕, 0, 1i
be a structure such that hL, ≤, 0, 1i is a complete lattice bounded by 0 and 1, 0 be
a unary operation of complementation, and ⊕ be a t-conorm on L (i.e. a binary
operation which is associative, commutative, and has 0 as its neutral element).
For an L-context hX, Y, Ii, define mappings ↑ : LX → LY and ↓ : LY → LX by
A↑ (y) = x∈X (A(x)0 ⊕ I(x, y)),
V
(5)
↓ V 0
B (x) = y∈Y (B(y) ⊕ I(x, y)), (6)
for A ∈ LX and B ∈ LY . Furthermore, put B(X, Y, I) = {hA, Bi ∈ LX ×
LY | A↑ = B, B ↓ = A} and define a partial order ≤ on B(X, Y, I) by hA1 , B1 i ≤
hA2 , B2 i iff A1 ⊆ A2 (iff B2 ⊆ B1 ).
Note first that for L = {0, 1}, 0 being a classical negation (i.e. 00 = 1 and
0
1 = 0), and ⊕ being a classical disjunction (i.e. a ⊕ b = max(a, b)), B(X, Y, I)
equipped with ≤ coincides in an obvious way with the ordinary concept lattice.
In [21], the authors show that B(X, Y, I) equipped with ≤ is a complete
lattice and show some of the basic properties of ↑ and ↓ . However, ↑ and ↓
do not satisfy some useful properties which hold true in the classical case, like
A ⊆ A↑↓ a B ⊆ B ↓↑ . The authors extended (generalized) their approach later to
include so-called implication operators [22]. Note also that the authors discussed
other extensions of FCA in their setting, see e.g. [23]. But since some useful
properties are missing in their approach, we stop our visit to this approach here
and leave a more detailed discussion to a full paper.
3.3 The approach by Pollandt and Bělohlávek
Pollandt [36] and, independently, Bělohlávek ([3] is the first publication, an
overview can be found in [12]), elaborated the following approach which turned
out to be a feasible way to develop FCA and related structures in a fuzzy setting.
Let L be a complete residuated lattice (see above). The choice of a residuated
38 Radim Bělohlávek, Vilém Vychodil
lattice structure, and particularly the adjointness property of conjunction and
implication turns out to be a crucial property. Let hX, Y, Ii be an L-context, i.e.
I : X × Y → L. For fuzzy sets A ∈ LX and B ∈ LY , consider fuzzy sets A↑ ∈ LY
and B ↓ ∈ LX (denoted also A↑I and B ↓I ) defined by
A↑ (y) = x∈X (A(x) → I(x, y)),
V
(7)
↓ V
B (x) = y∈Y (B(y) → I(x, y)). (8)
Using basic rules of predicate fuzzy logic [12, 28], one can easily see that A↑ (y)
is the truth degree of “y is shared by all objects from A” and B ↓ (x) is the truth
degree of “x has all attributes from B”. That is, (7) and (8) properly generalize
(3) and (4). Putting
B (X, Y, I) = {hA, Bi | A↑ = B, B ↓ = A},
B (X, Y, I) is the set of all pairs hA, Bi such that (a) A is the collection of all
objects that have all the attributes of (the intent) B and (b) B is the collection
of all attributes that are shared by all the objects of (the extent) A. Elements of
B (X, Y, I) are called formal concepts of hX, Y, Ii; B (X, Y, I) is called a (fuzzy)
concept lattice (L-concept lattice) of hX, Y, Ii. Note that as in the case of Bu-
rusco and Fuentes-Gonzáles, both the extent A and the intent B of a formal
concept hA, Bi are in general fuzzy sets. This corresponds to the fact that in
general, concepts apply to objects and attributes to various intermediate de-
grees, not only 0 and 1. Putting
hA1 , B1 i ≤ hA2 , B2 i iff A1 ⊆ A2 (iff B1 ⊇ B2 ) (9)
for hA1 , B1 i, hA2 , B2 i ∈ B (X, Y, I), ≤ models the subconcept-superconcept hi-
erarchy in B (X, Y, I).
Note that if L is a two-element Boolean algebra, i.e. L = {0, 1}, the notions
of an L-context, a formal concept, and an L-concept lattice coincide with the
ordinary notions [24].
For this approach, several issues of FCA have been generalized:
– Main theorem of L-concept lattices has two versions. The first one deals
with the above defined ordinary partial order on formal concepts (a proof
via reduction to the ordinary case can be found in [36] and in [9], a direct
proof can be found in [8]). The second one deals with a fuzzy order on formal
concepts, see [14]. In particular, B (X, Y, I) is a complete lattice w.r.t. ≤.
– Related structures like Galois connections and closure operators are studied
in [5, 10, 11].
– A direct algorithm for generating all formal concepts based on Ganter’s
algorithm [24] is presented in [13].
– Issues which are degenerate in the ordinary case have been studied, see e.g. [7,
6].
– Several further issues like many-valued contexts, relationship to approximate
reasoning, attribute implications are studied in [36]. Attribute implications
are studied also in [16, 17, 19, 20].
What is a fuzzy concept lattice? 39
3.4 The aproach by Yahia et al. and Krajči
Yahia [40] and, independently, Krajči [32] proposed the following approach,
called “one-sided fuzzy approach” by Krajči. Note first that the approaches are
not the same but the only difference is that for an L-context hX, Y, Ii, Yahia’s
definitions yield what Krajči’s definitions yield for hY, X, I −1 i with I −1 ∈ LY ×X
defined by I −1 (y, x) = I(x, y), i.e. the approaches are the same up to the roles of
objects and attributes. Note also that both Yahia and Krajči consider L = [0, 1].
For an L-context hX, Y, Ii, define mappings f : 2X → LY (assigning a fuzzy
set f (A) ∈ LY of attributes to a set A ⊆ X of objects) and h : LY → 2X
(assigning a set h(B) ⊆ of objects to a fuzzy set B ∈ LY of attributes) by
V
f (A)(y) = x∈A I(x, y) (10)
and
h(B) = {x ∈ X | for each y ∈ Y : B(y) ≤ I(x, y)}. (11)
Then, put
Bf,h (X, Y, I) = {hA, Bi ∈ 2X × LY | f (A) = B, h(B) = A}.
The set Bf,h (X, Y, I), equipped with a partial order ≤ defined by (9) is what
Yahia and Krajči call a (one-sided) fuzzy concept lattice (it is, indeed, a complete
lattice [40, 32]). Note that extents of concepts from Bf,h (X, Y, I) are crisp sets
while the intents are fuzzy sets. Needless to say, one can modify this approach
and have fuzzy extents and crisp intents.
The following theorem (partly contained in [32, Observation 2]) shows a basic
hint to the relationship of f and h, and ↑ and ↓ defined by (7) and (8).
Theorem 1. For A ⊆ X and B ∈ LY and ↑ and ↓ defined by (7) and (8), we
have
f (A) = A↑ , and h(B) = 1 (B ↓ ).
Proof. Easy, omitted due to lack of space.
↑
Here, A↑ for A means A0 where A0 is a fuzzy set in X corresponding to A,
i.e. A0 (x) = 1 for x ∈ A and A0 (x) = 0 for x 6∈ A. Therefore, f and h can be
expressed by ↑ and ↓ . In fact, Bf,h (X, Y, I) is isomorphic to (and, in fact, almost
the same as) a substructure of Pollandt’s and Bělohlávek’s L-concept lattice
B (X, Y, I). The details will be shown in the next section.
3.5 Crisply generated fuzzy concepts
In [15], when dealing with the problem of a possibly large number of formal
concepts in Pollandt’s and Bělohlávek’s L-concept lattice B (X, Y, I), the authors
proposed the following approach. Instead of considering the whole B (X, Y, I),
they consider only its part Bc (X, Y, I) ⊆ B (X, Y, I) consisting of what they
call crisply generated fuzzy concepts: hA, Bi ∈ B (X, Y, I) is called crisply if
40 Radim Bělohlávek, Vilém Vychodil
there is a crisp (i.e., ordinary) set Bc ⊆ Y of attributes such that A = Bc↓
(and thus B = Bc↓↑ ). Then, Bc (X, Y, I) = {hA, Bi ∈ B (X, Y, I) | there is
Bc ⊆ Y : A = Bc↓ }. Restricting oneself to crisply generated fuzzy concepts, one
eliminates “strange” concepts like concepts hA, Bi such that for any x ∈ X and
y ∈ Y we have A(x) = 1/2 and B(y) = 1/2 which might be difficult to interpret.
Note that in the modified approach of Yahia and Krajči, extents are fuzzy
sets and intents are crisp sets while in case of crisply generated concepts, both
extents and intents are in general fuzzy sets. As the next theorem shows, this
is only a negligible difference since there is a natural isomorphism between the
lattice of “one-sided fuzzy concepts” and the lattice of crisply generated fuzzy
concepts:
Theorem 2. For an L-context hX, Y, Ii, the “one-sided fuzzy concept lattice”
Bf,h (X, Y, I) with fuzzy extents and crisp intents (equipped with the above-defined
partial order) is a complete lattice which is isomorphic to Bc (X, Y, I) (equipped
with a partial order inherited from B (X, Y, I)). Moreover, there exists an iso-
morphism such that for the corresponding concepts hA, Bi ∈ Bf,h (X, Y, I) and
hC, Di ∈ Bc (X, Y, I) we have A = C and B = 1 D.
Proof. Omitted due to lack of space (use definitions and Theorem 1).
For further properties of crisply generated fuzzy concepts we refer to [15].
3.6 The approach of Snášel, Vojtáš et al.
Another approach appeared in [38] and in a series of related papers. The basic
idea is to consider, for a given L-context hX, Y, Ii, formal contexts hX, Y, a Ii
for a ∈ K for a suitable subset K ⊆ L. That is, one takes a-cuts a I =
{hx, yi | I(x, y) ≥ a} of the original fuzzy relation I for each a ∈ K. K contains
truth degrees which are considered important, relevant, sufficiently covering L,
etc. Since a I is an ordinary relation, hX, Y, a Ii is an ordinary formal context.
Therefore, one can apply (ordinary) formal concept analysis to each hX, Y, a Ii.
a a
Particularly, one can form concept F lattices Ba(X, Y, I). All B (X, Y, I)’s are
then “merged” into a structure a∈K B (X, Y, I) defined by
a a
F
a∈K B (X, Y, I) = {hA, Bi | A ∈ Ext( I) for some a ∈ K;
B is a multiset in Y with B(y) = |{a ∈ K; A ∈ Ext(a I), y ∈ A↑a I }|}.
a
F
a∈K B (X, Y, I) then represents the resulting structure derived from the fuzzy
context hX, Y, Ii. Here, Ext(a I) = {A ⊆ X | hA, A↑a I i ∈ B (X, Y, a I)} denotes
the set of all extents of formal concepts from B (X, Y, a I). Recall that a mul-
tiset may contain multiple occurrences of elements. Therefore, a multiset B in
a universe set Y may be understood as a function assigning to each y ∈ Y
a nonnegative integer B(y) (the number of occurrences of y in B). That is,
a
F
a∈K B (X, Y, I) contains pairs hA, Bi where A is the extent of some formal
a
concept of I, and B is a multiset of attributes with B(y) being the number of
cut-levels a ∈ K for which A is an extent of hX, Y, a Ii and y belongs to A↑a I .
What is a fuzzy concept lattice? 41
Although this approach is very different from the other ones, there is still the
following relationship:
Theorem 3. For a linearly ordered complete residuated lattice L of truth values
with a ⊗ b = min(a, b) and any truth degree a ∈ L we have
Ext(a I) = {a A | hA, Bi ∈ Bc (X, Y, I)}.
Proof. Omitted due to lack of space.
Therefore, extents of concepts from a∈K B (X, Y, a I) are just a-cuts of
F
(crisply generated) concepts from Bc (X, Y, I). A more detailed discussion is left
to a full paper.
3.7 Concept lattices and Galois connections with hedges
As a generalization of both fuzzy concept lattcies of Pollandt and Bělohlávek and
lattices of crisply generated fuzzy concepts, the authors proposed the following
approach in [18]. The approach consists in emplyoing, as parameters, two unary
functions on L, called hedges. For a complete residuated lattice L, a (truth-
stressing) hedge is a unary function ∗ satisfying (i) 1∗ = 1, (ii) a∗ ≤ a, (iii)
(a → b)∗ ≤ a∗ → b∗ , (iv) a∗∗ = a∗ , for all a, b ∈ L. A hedge ∗ is a (truth function
of) logical connective “very true” [29]. The largest hedge (by pointwise ordering)
is identity, the least hedge is globalization which is defined by a∗ = 1 for a = 1
and a∗ = 0 for a < 1.
Given an L-context hX, Y, Ii, one can proceed as follows. Let ∗X and ∗Y be
hedges (their meaning will become apparent later). For L-sets A ∈ LX (L-set
of objects), B ∈ LY (L-set of attributes) we define L-sets A↑ ∈ LY (L-set of
attributes), B ↓ ∈ LX (L-set of objects) by
A↑ (y) = x∈X A(x)∗X → I(x, y) ,
V
B ↓ (x) = y∈Y B(y)∗Y → I(x, y) .
V
We put B(X ∗X , Y ∗Y , I) = {hA, Bi ∈ LX × LY | A↑ = B, B ↓ = A}. For
hA1 , B1 i, hA2 , B2 i ∈ B(X ∗X , Y ∗Y , I), put hA1 , B1 i ≤ hA2 , B2 i iff A1 ⊆ A2 (or,
iff B2 ⊆ B1 ; both ways are equivalent). Operators ↓ , ↑ form a so-called Galois
connection with hedges [18]. hB(X ∗X , Y ∗Y , I), ≤i is called a concept lattice with
hedges induced by hX, Y, Ii (it is, indeed, a complete lattice [18]). The following
is the basic relationship to the above described approaches.
Theorem 4. For ∗X and ∗Y being identities, B (X ∗X , Y ∗Y , I) coincides with
B (X, Y, I) (fuzzy concept lattice of Pollandt and Bělohlávek). For ∗X being iden-
tity and ∗Y being globalization, B (X ∗X , Y ∗Y , I) coincides with Bc (X, Y, I) (set
of all crisply generated fuzzy concepts).
Note, however, that there are other choices of ∗X and ∗Y possible. As demon-
strated in [18], ∗X and ∗Y can be seen as parameters controlling the size of the
resulting B (X ∗X , Y ∗Y , I).
42 Radim Bělohlávek, Vilém Vychodil
3.8 Krajči’s generalized concept lattice
In [33–35], Krajči studies a so-called generalized concept lattice. In the previous
approaches to concept lattices, formal concepts are defined as certain pairs hA, Bi
where for A and B we can have the following possibilities: both A and B are
crisp sets, both A and B are fuzzy sets, A is crisp and B is fuzzy, A is fuzzy
and B is crisp. Krajči suggests to consider three sets of truth degrees in general,
namely, a set LX (for objects), LY (for attributes), and L (for degrees to which
objects have attributes, i.e. table entries).
Given sets X and Y of objects and attributes, one can consider a fuzzy
context as a triplet hX, Y, Ii where I is an L-relation between X and Y , i.e.
I ∈ LX×Y . Furthermore, one can consider LX -sets A of objects and LY -sets B
of attributes, i.e. A ∈ LX Y
X and B ∈ LY . Furthermore, Krajči assumes that LX
and LY are complete lattices and L is a partially ordered set. For convenience,
we denote all the partial orders (on LX , LY , and L) by ≤.VLikewise, W infima and
suprema in both hLX , ≤i and hLY , ≤i will be denoted by and .
In order to define arrow-operators, Krajči assumes that there is an operation
⊗ : LX × LY → L satisfying
a1 ≤ a2 implies a1 ⊗ b ≤ a2 ⊗ b, (12)
b1 ≤ b2 implies a ⊗ b1 ≤ a ⊗ b2 , (13)
W
if aj ⊗ b ≤ c for each j ∈ J then ( j∈J aj ) ⊗ b ≤ c, (14)
W
if a ⊗ bj ≤ c for each j ∈ J then a ⊗ ( j∈J bj ) ≤ c, (15)
for each index set J and all a, aj ∈ LX , b, bj ∈ LY , and c ∈ L. That is, we
have a three-sorted structure hL1 , L2 , L, ⊗, ≤, . . .i of truth degrees. Call such a
structure hL1 , L2 , L, ⊗, ≤, . . .i satisfying (12)–(15) Krajči’s structure.
Then, Krajči introduces mappings % : LX Y
X → LY and
.
: LYY → LXX by
% W
A (y) = {b ∈ LY | for each x ∈ X : A(x) ⊗ b ≤ I(x, y)}, (16)
. W
B (x) = {a ∈ LX | for each y ∈ Y : a ⊗ B(y) ≤ I(x, y)}, (17)
and defines a formal concept in hX, Y, Ii as a pair hA, Bi ∈ LX Y
X × LY satisfying
% .
A = B and B = A.
At first sight, it looks like there is no adjointness involved. Also, compared
to (7) and (8), (16) and (17) seem a bit complicated. However, the following
theorem shows that Krajči’s structure hL1 , L2 , L, ⊗, ≤, . . .i can be looked at as
a three-sorted structure with two implication connectives satisfying adjointness-
like conditions.
Theorem 5 (residuated structure for generalized concept lattices). Let
L1 , L2 be complete lattices and L be a partially ordered set (all partial orders
are denoted by ≤).
(1) For a Krajči’s structure hL1 , L2 , L, ⊗, ≤, . . .i define for c ∈ L1 , d ∈ L2 , and
p ∈ L,
W
c →1 p = {d ∈ L2 | c ⊗ d ≤ p}, (18)
W
d →2 p = {c ∈ L1 | c ⊗ d ≤ p}. (19)
What is a fuzzy concept lattice? 43
Then we have the following form of adjointness:
c⊗d≤p iff c ≤ d →2 p iff d ≤ c →1 p. (20)
(2) If hL1 , L2 , L, ⊗, →1 , →2 , . . .i satisfies (20), then hL1 , L2 , L, ⊗, ≤, . . .i is a
Krajči’s structure.
Proof. Omitted due to lack of space.
Then, arrow operators have the form similar to (7) and (8).
Theorem 6 (arrows via residuum for generalized concept lattices). We
have
A% (y) = x∈X (A(x) →1 I(x, y)),
V
(21)
. V
B (x) = y∈Y (B(y) →2 I(x, y)). (22)
Proof. Omitted due to lack of space.
Due to Theorem 5, a structure hL1 , L2 , L, ⊗, →1 , →2 , . . .i satisfying (20) can
be called a three-sorted complete residuated lattice. Theorems 5 and 6 show that
Krajči’s approach can be seen as another approach (a very general one) in the
framework of residuated structures of truth degrees.
A main theorem for a generalized concept lattice is presented in [33]. In par-
ticular, each generalized concept lattice is a complete lattice. Note furthermore
that in a recent paper [35], Krajči shows that each concept lattice with hedges
is isomorphic to some generalized concept lattice. The question of whether each
generalized concept lattice is isomorphic to some concept lattice with hedges
seems to be an open problem.
4 Concluding remarks
– As mentioned above, we did not discuss the relationship of the approaches
presented here with so-called conceptual scaling of fuzzy contexts considered
as many-valued contexts. This issue will be discussed in a forthcoming paper.
– Proofs of the results presented here, more detailed comments, further results,
and examples will be presented in a full version of this paper. In particular,
we did not discuss concept lattices and related structures over structures of
truth degrees equipped with non-commutative conjunction. This has been
developed in [25, 26]. A discussion of relationships to this approach will also
be included in the full paper.
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