The paper presents a generalization of the main theorem of fuzzy concept lattices. The theorem is investigated from the point of view of fuzzy logic. There are various fuzzy order types which differ by incorporated relation of antisymmetry. This paper focuses on fuzzy order which uses fuzzy antisymmetry defined by means of multiplication operation and fuzzy equality.
A notion of fuzzy order has been derived from the classical one by fuzzification
the three underlying relations. This led to various versions, at the beginning
versions utilizing the classical relation of equality (see e.g. [
In [
Definition 1. A residuated lattice is an algebra L = hL, ∧, ∨, ⊗, →, 0, 1i such that – hL, ∧, ∨, 0, 1i is a lattice with the least element 0 and the greatest element 1, – hL, ⊗, 1i is a commutative monoid, – ⊗ and → form so-called adjoint pair, i.e. a ⊗ b ≤ c iff a ≤ b → c holds for all a, b, c ∈ L.
Residuated lattice L is called complete if hL, ∧, ∨i is a complete lattice.
Throughout the paper, L will denote a complete residuated lattice. An Lset (or fuzzy set) A in a universe set X is any mapping A : X → L, A(x) being interpreted as the truth degree of the fact that “x belongs to A”. By LX we denote the set of all L-sets in X. A binary L-relation is defined obviously. Operations on L extend pointwise to LX , e.g. (A ∨ B)(x) = A(x) ∨ B(x) for any A, B ∈ LX . As is usual, we write A ∪ B instead of A ∨ B, etc.
L-equality (or fuzzy equality) is a binary L-relation ≈ ∈ LX×X such that
(x ≈ x) = 1 (reflexivity), (x ≈ y) = (y ≈ x) (symmetry), (x ≈ y) ⊗ (y ≈ z) ≤
(x ≈ z) (transitivity), and (x ≈ y) = 1 implies x = y. We say that a binary
L-relation R ∈ LX×Y is compatible with respect to ≈X and ≈Y if R(x, y) ⊗
(x ≈X x0) ⊗ (y ≈Y y0) ≤ R(x0, y0) for any x, x0 ∈ X, y, y0 ∈ Y . Analogously
an L-set A ∈ LX is compatible with respect to ≈X if A(x)⊗(x ≈X x0) ≤ A(x0) for
any x, x0 ∈ X. Given A, B ∈ LX , in agreement with [
Definition 2. An L-order on a set X with an L-equality ≈ is a binary L-relation which is compatible with respect to ≈ and satisfies the following conditions for all x, y, z ∈ X: (x
x) = 1
(x
(x
y) ⊗ (y
y) ⊗ (y
x) ≤ (x ≈ y)
z) ≤ (x
(reflexivity),
(antisymmetry),
z)
(transitivity).
(Cf. T-E-ordering from [
Note that (x y) ⊗ (y x) ≤ (x ≈ y) ≤ (x y) ∧ (y x).
(1)
Indeed, the first inequality represents antisymmetry and the second one follows
from compatibility (see proof of Lemma 4 in [
The inequalities (1) represent the fact that L-equality must satisfy the interval confinement as follows: (x ≈ y) ∈ [(x y) ⊗ (y x), (x y) ∧ (y x)] .
On Generalization of Fuzzy Concept Lattices Based on Change of 209
Underlying Fuzzy Order
On the other hand, the definition of L-order by [
(x ≈ y) = (x [
Now, we can interpret the relationship between L-order and L-equality defined
either in [
This will play an important role during generalizing results achieved in [
If is an L-order on a set X with an L-equality ≈, we call the pair X =
hhX, ≈i, i an L-ordered set. In agreement with [
Note that in case of firm binding of L-equality and L-order by [
In this section, we describe some notions and properties related to fuzzy ordered
sets which represent appropriate generalizations of notions and facts known from
classical (partial) ordered sets. These generalizations were introduced mainly
in [
Definition 3. For an L-ordered set hhX, ≈i, i and A ∈ LX we define the L-sets L(A) and U (A) in X by x0∈X x00∈X L(A)(x) = ^ (A(x0) → (x
x0)) for all x ∈ X, U (A)(x) = ^ (A(x00) → (x00 x)) for all x ∈ X.
L(A) and U (A) are called the lower cone and upper cone of A, respectively. These L-sets can be described as the L-sets of elements which are smaller (greater) than all elements of A. We will abbreviate U (L(A)) by U L(A), L(U (A)) by LU (A) etc. Definition 4. For an L-ordered set hhX, ≈i, i and A ∈ LX we define the L-sets inf(A) and sup(A) in X by (inf(A))(x) = (L(A))(x) ∧ (U L(A))(x) for all x ∈ X, (sup(A))(x) = (U (A))(x) ∧ (LU (A))(x) for all x ∈ X. inf(A) and sup(A) are called the infimum and supremum of A, respectively. Lemma 1. Let hhX, ≈i, i be an L-ordered set, A ∈ LX . If (inf(A))(x) = 1 and (inf(A))(y) = 1 then x = y (and similarly for sup(A)).
Proof. The proof is almost verbatim repetition of the proof of Lemma 9 in [
Proof. The proof can be found in [
Definition 5. For a set X with an L-equality ≈, an L-set A ∈ LX is called an S-singleton if it is compatible with respect to ≈ and there is some x0 ∈ X such that A(x0) = 1 and A(x) < 1 for x 6= x0.
Remark 1. There are various definitions of fuzzy singletons (see e.g. [
Proof. The proof is analogous to the proof of Theorem 13 in [
We remind some basic facts about concept lattices in fuzzy setting. A formal
L-context is a tripple hX, Y, Ii where I is an L-relation between the sets X and Y
(with elements called objects and attributes, respectively). For any L-context we
can generalize notions introduced in Definition 3 as follows. Let X, Y be sets with
L-equalities ≈X , ≈Y , respectively; I ∈ LX×Y be an L-relation compatible with
respect to ≈X and ≈Y . For any A ∈ LX , B ∈ LY , we define A↑ ∈ LY , B↓ ∈ LX
(see e.g. [
^ (A(x) → I(x, y)) for all y ∈ Y, B↓(x) =
^ (B(y) → I(x, y)) for all x ∈ X.
Clearly, A↑(y) describes the truth degree, to which “for each x from A, x and y
are in I”, and similarly B↓(x). We will abbreviate (A↑)↓ by A↑↓, (B↓)↑ by B↓↑
etc. The equation A↑ = A↑↓↑ holds true for all A ∈ LX (see e.g. [
We denote by B(X, Y, I) the set of all L-concepts given by an L-context hX, Y, Ii, i.e.
B(X, Y, I) = {hA, Bi ∈ LX × LY ; A↑ = B, B↓ = A}.
For any B(X, Y, I), we put (see [
(hA1, B1i S hA2, B2i) = S(A1, A2) for all hA1, B1i, hA2, B2i ∈ B(X, Y, I). The L-relation S obviously satisfies the conditions of reflexivity and transitivity. As to the antisymmetry, we need an L-equality. Therefore, consider an arbitrary L-equality ≈ on the set B(X, Y, I) such that S is compatible with respect to ≈ and the inequality
(hA1, B1i S hA2, B2i) ⊗ (hA2, B2i S hA1, B1i) ≤ (hA1, B1i ≈ hA2, B2i)
holds true for every hAi, Bii ∈ B(X, Y, I), i ∈ {1, 2}. (Existence of such an
Lequality is demonstrated e.g. by (hA1, B1i ≈ hA2, B2i) = S(A1, A2)∧S(A2, A1) in
[
The next theorem characterizing L-concept lattices needs further denotation.
As usual, for an L-set A in U and a ∈ L, we denote by a ⊗ A the L-set such that
A∈Y
(a ⊗ A)(u) = a ⊗ A(u) for all u ∈ U . If M is an L-set in Y and each y ∈ Y is
an L-set in X, we define the L-set S M in X (see [
([ M)(x) = _ M(A) ⊗ A(x) for all x ∈ X.
Clearly, S M represents a generalization of a union of a system of sets. For an
L-set M in B(X, Y, I), we put SX M = S prX (M), SY M = S prY (M) where
prX (M) is an L-set in the set {A ∈ LX ; A = A↑↓} of all extents of B(X, Y, I)
defined by (prX M)(A) = M(A, A↑) and, similarly, prY (M) is an L-set in the
set {B ∈ LY ; B = B↓↑} of all intents of B(X, Y, I) defined by (prY M)(B) =
M(B↓, B). Hence, SX M is the “union of all extents from M” and SY M is
the “union of all intents from M” (see [
Theorem 2. Let hX, Y, Ii be an L-context. An L-ordered set hhB(X, Y, I), ≈i, Si is completely lattice L-ordered set in which infima and suprema can be described as follows: for an L-set M in B(X, Y, I) we have 1inf(M) = 1sup(M) = (* (* ([
Y ([
X
M)↓, ([
M)↓↑
Y M)↑↓, ([
M)↑
X
+)
+)
(2)
(3)
Proof. The proof of (2) and (3) is analogous to the proof of part (i) of Theorem 14
in [
For any completely lattice L-ordered set X = hhX, ≈i, i, a subset K ⊆ X
is called {0, 1}-infimally dense ({0, 1}-supremally dense) in X (cf. [
Theorem 3. Let hX, Y, Ii be an L-context. A completely lattice L-ordered set V = hhV, ≈V i, i is isomorphic to an L-ordered set hhB(X, Y, I), ≈i, Si iff there are mappings γ : X × L → V, μ : Y × L → V , such that (i) γ(X × L) is {0, 1}-supremally dense in V, (ii) μ(Y × L) is {0, 1}-infimally dense in V, (iii) ((a ⊗ b) → I(x, y)) = (γ(x, a)
μ(y, b)) for all x ∈ X, y ∈ Y, a, b ∈ L. (iv) (hA1, B1i ≈ hA2, B2i) =
W γ(x, A1(x)) ≈V x∈X
W γ(x, A2(x)) x∈X for all hA1, B1i, hA2, B2i ∈ B(X, Y, I).
On Generalization of Fuzzy Concept Lattices Based on Change of 213
Underlying Fuzzy Order
Proof. Let γ and μ with the above properties exist. If we define the mapping
ϕ : B(X, Y, I) → V by ϕ(A, B) = W γ(x, A(x)) for all hA, Bi ∈ B(X, Y, I),
x∈X
then by the proof of Part (ii) of Theorem 14 in [
Conversely, let V and hhB(X, Y, I), ≈i, Si be isomorphic. Similarly to [
x∈X x∈X Since we consider identity on hhB(X, Y, I), ≈i, Si, we have ≈ = ≈V and it suffices to prove that W γ(x, A(x)) = hA, Bi for all hA, Bi ∈ B(X, Y, I).
x∈X
We start with proof of the equation A = Sx∈X On the one hand we get for each x0 ∈ X:
S nA(x)/xo↑↓ (x0) = W nA(x)/xo↑↓ x∈X x∈X (x0) =
nA(x)/xo↑↓ for any A = A↑↓.
" # = W V nA(x)/xo↑(y) → I(x0, y) = x∈X y∈Y
" = W V V nA(x)/xo(x˜) → I(x˜, y) x∈X y∈Y x˜∈X # → I(x0, y) = " " " # # On the other hand we have:
S nA(x)/xo↑↓ (x0) = W V (A(x) → I(x, y)) → I(x0, y) ≤ x∈X x∈X y∈Y Using also the definition of γ and Theorem 2, we obtain W γ(x, A(x)) = W x∈X x∈X * nA(x)/xo↑↓, nA(x)/xo↑
= =
S nA(x)/xo↑↓ ↑↓ x∈X ,
S nA(x)/x
x∈X
= hA↑↓, A↑i = hA, Bi.
o↑↓ ↑+
=
Remark 2. Note that the essential difference between Theorem 3 in this paper
and Theorem 14, part (ii) in [
There is an interesting proposition which deals with a completely lattice L-ordered set hhB(V, V, ), ≈Si, Si such that • is an L-order on V , Proposition 1. A completely lattice L-ordered set V = hhV, ≈V i, i is isomorphic to hhB(V, V, ), ≈Si, Si.
The proposition represents a corollary of Theorem 3, but an elegant proof of this fact is a matter of further investigations. 6
The author wishes to thank the anonymous reviewers for their criticisms and useful comments.