We show the relationship between two different types of common platforms of till known fuzzifications of a concept lattice, namely that the notion of a concept lattice with hedges is a special case of our generalized concept lattice. There are some approaches to fuzzify (i. e. generalize for fuzzy case too) the classical Ganter-Wille construction of concept lattice. If we omit the (maybe slightly naive) attempt done by Burusco & Fuentes-Gonzalez ([6]), the first approach, which was theoretically and practically well developed, was given by Bˇelohla´vek ([1]) and Pollandt ([11]). It use (L-)fuzzy subsets of objects and (L)fuzzy subsets of attributes. The another approach, so-called one-sided fuzzy concept lattice was invented independently by Ben Yahia & Jaoua ([5]), by Bˇelohla´vek et al. ([4]) and by the author ([8]). It considers fuzzy subsets of attributes but ordinary/classical/crisp subsets of objects (or vice versa). Because there is no inclusion between these two fuzzy approaches the natural asking for some common platform of both had arose. We know about two such generalizations. One of them was shown by Bˇelohl´avek et al. ([3] and partially in [2]) and it uses so-called hedges (or truth-stressers) (details below). The second one was given by author ([9]) and its idea is to separate between the ranges of fuzzy sets of objects and fuzzy sets of attributes (again details below). Till now it seemed that these two generalizing approaches are not compatible, but in this paper we try to show that the first is contained in the second.
Introduction
A generalized concept lattice
Let us shortly recall a notion of a generalized concept lattice given by the author.
All these results are proven in [
Let P be a poset, C and D be complete lattices. Let • : C × D → P be monotone and left-continuous in both their arguments, i.e.
? Partially supported by grant 1/0385/03 of the Slovak grant agency VEGA. 1a) c1 ≤ c2 implies c1 • d ≤ c2 • d for all c1, c2 ∈ C and d ∈ D. 1b) d1 ≤ d2 implies c • d1 ≤ c • d2 for all c ∈ C and d1, d2 ∈ D. 2a) If c • d ≤ p holds for d ∈ D, p ∈ P and for all c ∈ X ⊆ C, then 2b) If c • d ≤ p holds for c ∈ C, p ∈ P and for all d ∈ Y ⊆ D, then sup X • d ≤ p.
c • sup Y ≤ p.
Let A and B be non-empty sets and let R be P -relation on their Cartesian product, i.e. R : A × B → P .
Define the following mapping % : BD → AC (by S T we understand the set of all mappings from the set S to the set T ):
If g : B → D then %(g) : A → C is defined as follows:
%(g)(a) = sup{c ∈ C : (∀b ∈ B)c • g(b) ≤ R(a, b)}.
Symmetrically we define the mapping . : AC → BD: If f : A → C then .(f ) : B → D is defined as follows:
.(f )(b) = sup{d ∈ D : (∀a ∈ A)f (a) • d ≤ R(a, b)}.
Because these two mappings . and % form a Galois connection, it can be
repeated a classical construction of concept lattice. The result of this
construction is called a generalized concept lattice and the following basic theorem on
generalized concept lattice holds (the proofs are in [
. i∈I _ fi i∈I !!+ and _hgi, fii = . % _ gi i∈I i∈I i∈I 2) Let moreover P have the least element 0P and 0C • d = 0P and c • 0D = 0P for every c ∈ C and d ∈ D. Then a complete lattice V is isomorphic to L if and only if there are mappings α : A × C → V and β : B × D → V s.t. 1a) α is non-increasing in the second argument. 1b) β is non-decreasing in the second argument. 2a) α[A × C] is infimum-dense. 2b) β[B × D] is supremum-dense. 3) For every a ∈ A, b ∈ B, c ∈ C, d ∈ D !!
+ , ^ fi . α(a, c) ≥ β(b, d)
This approach is really a generalization of Bˇelohl´avek’s fuzzy concept lattice and of one-sided fuzzy concept lattice (and, of course, of a classical crisp case).
A concept lattice with hedges Bˇelohla´vek consider a (complete) residuated lattice L = hL, ∨, ∧, ⊗, →, 0, 1i, where ⊗ and → are connectives on L which form an adjoint pair, i.e. x ⊗ y ≤ z iff x ≤ y → z. ⊗ is isotone in both their arguments, → is antitone in the first argument and isotone in the second one, ⊗ is commutative and x⊗1 = 1⊗x = x.
Moreover he has sets X and Y and an incidence relation I : X × Y → L. Then he defines the mappings 0 : LX → LY and 00 : LY → LX as follows: If A ∈ LX and B ∈ LY then
A0(y) =
^ (A(x) → I(x, y))
The new idea of Bˇelohl´avek (et al.)’s is to modify these definitions in this way: and and x∈X y∈Y x∈X y∈Y B00(x) = ^ (B(y) → I(x, y)).
A↑ (y) =
^ (A(x)∗X → I(x, y)) B↓ (x) = ^ (B(y)∗Y → I(x, y)), 1∗ = 1, a∗ ≤ a, (a → b)∗ ≤ a∗ → b∗, a∗∗ = a∗.
∗ ◦ ∗ = ∗ where ∗X and ∗Y are so-called hedges on L. A hedge is a function ∗ on L which fulfills these properties: (Note that the last one can be rewrite in the form where ◦ is the composition of mappings.) Moreover they work with the following functions: – For arbitrary A : U → L (U is some universe) define: – For arbitrary B ⊆ U × L take the function dBe : U → L defined by: bAc = {hu, ai ∈ U × L : a ≤ A(u)}.
dBe(u) = _{a ∈ L : hu, ai ∈ B}. – For arbitrary A : U → L and ∗ : L → L define the function A∗ given pointwise by: – For arbitrary B ⊆ U × L and ∗ : L → L define:
A∗(u) = (A(u))∗.
B∗ = {hx, a∗i : hx, ai ∈ B}.
Then they have taken the set
B(X∗X , Y ∗Y , I) = {hA, Bi : A↑ = B, B↓ = A} of all fixpoints of the pair h↑, ↓i and showed that this structure is isomorphic to the ordinary concept lattice B(X × ∗X [L], Y × ∗Y [L], Ihf,gi) where and and relation Ihf,gi is given by
Ag = bdAe↑c∗X
Bf = bdBe↓c∗Y , hhx, ai, hy, bii ∈ Ihf,gi iff a ⊗ b ≤ I(x, y).
Finally, they have proven this basic theorem for concept lattice with hedges (we present here only its second part): Theorem 2. An arbitrary complete lattice hV, ≤i is isomorphic to the complete lattice B(X∗X , Y ∗Y , I) iff there are mappings γ : X × ∗X [L] → V and μ : Y × ∗Y [L] → V s. t. 1a) μ[Y × ∗Y [L]] is infimum-dense. 1b) γ[X × ∗X [L]] is supremum-dense. 2) γ(x, a) ≤ μ(y, b) Proof. By properties of a hedge we obtain: a ≤ b iff 1 ⊗ a ≤ b iff 1 ≤ a → b iff 1 = a → b, which implies 1 = (a → b)∗ ≤ a∗ → b∗, and then a∗ = 1 ⊗ a∗ ≤ b∗. Lemma 2. For every hedge ∗ if A ⊆ ∗[L] then sup A ∈ ∗[L].
Proof. For every a ∈ A we know a ≤ sup A, and then (from the previous lemma) a = a∗ = (sup A)∗. It means that (sup A)∗ is an upper bound of A, which implies sup A ≤ (sup A)∗. But it follows sup A = (sup A)∗, i. e. sup A ∈ ∗[L]. Lemma 3. The function b·c is monotonous.
Proof. If A1, A2 : U → L and A1 ≤ A2, then obviously bA1c = {hu, ai ∈ U × L : a ≤ A1(u)} ⊆ {hu, ai ∈ U × L : a ≤ A2(u)} = bA2c.
Lemma 4. The function d·e is monotonous.
Proof. If B1 ⊆ B2 ⊆ U × L, then for all u ∈ U we have clearly dB1e(u) = W{a ∈ L : hu, ai ∈ B1} = W{a ∈ L : hu, ai ∈ B2} = dB2e(u).
Lemma 5. For arbitrary A : U → L it is true that dbAce = A.
Proof. dbAce(u) = W{a ∈ L : hu, ai ∈ bAc} = W{a ∈ L : a ≤ A(u)} = A(u). 5
Relationship between these two approaches We can see that the basic theorem for concept lattice with hedges and the basic theorem for our generalized concept lattice are very similar. And it is suspicious. So take such special case of our generalized concept lattice given by the following table: general
P A B C D special
L Y
X ∗Y [L] ∗X [L] • ⊗
R : A × B → P I : X × Y → L Then our definitions can be rewritten in this form: If g : X → ∗X [L] then %(g) : Y → ∗Y [L] is defined as follows:
%(g)(y) = sup{c ∈ ∗Y [L] : (∀x ∈ X)c ⊗ g(x) ≤ I(x, y)}, and if f : Y → ∗Y [L] then .(f ) : X → ∗X [L] is defined by:
.(f )(x) = sup{d ∈ ∗X [L] : (∀y ∈ Y )f (y) ⊗ d ≤ I(x, y)}.
And now we will try to prove that such special case of generalized concept
lattice is isomorphic to the lattice from [
It is enough to prove the following six claims: Claim 1 If hg,fi ∈ L then Ψ(hg,fi) ∈ B.
Proof. Let hg,fi ∈ L, i. e. g = .(f) and f = %(g), we want to prove bgc = bfcg = bdbfce↓c∗ = bf↓c∗ (because from the lemma 5 we have dbfce = f) and g f = bdbgce↑c∗ = bg↑c∗ (because again dbgce = g), which will mean bfc = b c Φ(hg,fi) = hbgc,bfci ∈ B, what we want to show.
By above definitions we obtain: bf↓c∗ = {hx,d∗Xi ∈ X × ∗X[L] : hx,di ∈ bf↓c}, = {hx,di ∈ X × ∗X[L] : hx,d∗Xi ∈ bf↓c}, (because ∗X ◦ ∗X = ∗X, we have d∗X = d for all d ∈ ∗X[L]), = {hx,di ∈ X × ∗X[L] : d ≤ f↓(x)}, = {hx,di ∈ X × ∗X[L] : d ≤ Vy∈Y (f(y)∗Y → I(x,y))}, = {hx,di ∈ X × ∗X[L] : (∀y ∈ Y )(f(y)∗Y ⊗ d ≤ I(x,y))}, = {hx,di ∈ X×∗X[L] : (∀y ∈ Y )(f(y)⊗d ≤ I(x,y))} (because ∗Y ◦∗Y = ∗Y , we have c∗Y = c for all c ∈ ∗Y [L], especially f(y) ∈ ∗Y [L]), = {hx,di ∈ X × ∗X[L] : d ≤ sup{e ∈ ∗X[L] : (∀y ∈ Y )f(y) ⊗ e ≤ I(x,y)}}, = {hx,di ∈ X × ∗X[L] : d ≤ .(f)(x)}, = {hx,di ∈ X × ∗X[L] : d ≤ g(x)}, = bgc, So we have bf↓c∗ = bgc, and the equality bg↑c∗ = bfc can be proven symmetrically.
Claim 2 If hS,Ti ∈ B then Φ(hS,Ti) ∈ L.
Proof. Let hS,Ti ∈ B, i. e. S = Tf = bdTe↓c∗ and T = Sg = bdSe↑c∗, we want to prove %(dSe) = dTe and .(dTe) = dSe which will mean Ψ(hS,Ti) = hdSe,dTei ∈ L, what we want to show.
Firstly we show that for all y ∈ Y we have dTe(y) ∈ ∗Y [L]: dTe(y) = dbdSe↑c∗e(y), = W{a ∈ L : hy,ai ∈ bdSe↑c∗}, = W{a ∈ ∗Y [L] : hy,ai ∈ bdSe↑c∗}, ∈ ∗Y [L] (because of lemma 2).
Hence by above definitions we obtain for every x ∈ X: .(dTe)(x) = sup{d ∈ ∗X[L] : (∀y ∈ Y )dTe(y) ⊗ d ≤ I(x,y)}, = sup{d ∈ ∗X[L] : (∀y ∈ Y )dTe(y)∗Y ⊗d ≤ I(x,y)} (because ∗Y ◦∗Y = ∗Y , we have c∗Y = c for all c ∈ ∗Y [L], especially for dTe(y) as we have shown above), = sup{d ∈ ∗X[L] : d ≤ Vy∈Y (dTe(y)∗Y → I(x,y))},
Isomorphism of Concept Lattice With Hedges 7 = sup{d ∈ ∗X[L] : d ≤ dTe↓(x)}, = sup{d ∈ ∗X[L] : hx,di ∈ bdTe↓c}, = sup{d ∈ ∗X[L] : hx,d∗Xi ∈ bdTe↓c∗}, = sup{d ∈ ∗X[L] : hx,di ∈ bdTe↓c∗} (because ∗X ◦ ∗X = ∗X, we have d∗X = d for all d ∈ ∗X[L]), = sup{d ∈ L : hx,di ∈ bdTe↓c∗} (the (stronger) condition d ∈ ∗X[L] is covered by the condition hx,di ∈ bdTe↓c∗), = sup{d ∈ L : hx,di ∈ S}, = W{d ∈ L : hx,di ∈ S}, = dSe(x).
So we have .(dTe) = dSe, and the equality %(dSe) = dTe can be proven symmetrically.
Φ(Ψ(hg,fi)) = hg,fi.
Proof. Φ(Ψ(hg,fi)) = hdbgce,dbfcei = hg,fi because of lemma 5. (You can see that assumption hg,fi ∈ L is only formal, we do not need it.)
Ψ(Φ(hS,Ti)) = hS,Ti.
Proof. By definition we obtain Ψ(Φ(hS,Ti)) = hbdSec,bdTeci, so we want to prove that bdSec = S and bdTec = T.
For one inclusion we will need the following observation: Because hS,Ti ∈ B, we have S = Tf = bdTe↓c∗, i. e. S = bgc∗ for some g : X → ∗X[L].
Using definitions we obtain: hx,di ∈ bdSec iff d ≤ dSe(x), iff d ≤ W{e ∈ L : hx,ei ∈ S}, iff d ≤ W{e ∈ L : hx,ei ∈ bgc∗}, iff d ≤ W{e ∈ ∗X[L] : hx,e∗Xi ∈ bgc∗} (because if hx,ei ∈ bgc∗ then e ∈ ∗X[L] and e = e∗X), iff d ≤ W{e ∈ ∗X[L] : hx,ei ∈ bgc}, iff d ≤ W{e ∈ ∗X[L] : e ≤ g(x)} = g(x), iff hx,di ∈ bgc and d ∈ ∗X[L], iff hx,d∗Xi ∈ b c
g ∗ and d ∈ ∗X[L], iff hx,di ∈ bgc∗ (because ∗X ◦ ∗X = ∗X, so d = d∗X for all d ∈ ∗X[L]), iff hx,di ∈ S.
So we have bdSec = S, and the second equality bdTec = T can be proven symmetrically.
Claim 5 Φ is order-preserving. Proof. Let hg1, f1i, hg2, f2i ∈ L and hg1, f1i ≤ hg2, f2i. This inequality means that g1 ≤ g2 and f1 ≥ f2 (pointwise) and by monotonicity of b·c we obtain bg1c ⊆ bg2c and bf1c ⊇ bf2c, which implies Φ(hg1, f1i) = hbg1c, bf1ci ≤ hbg2c, bf2ci = Φ(hg2, f2i).
Claim 6 Ψ is order-preserving.
Proof. Let hS1, T1i, hS2, T2i ∈ B and hS1, T1i ≤ hS2, T2i. This inequality means that S1 ⊆ S2 and T1 ⊇ T2 and by monotonicity of d·e we obtain dS1e ≤ dS2e and dT1e ≥ dT2e, which implies Ψ (hS1, T1i) = hdS1e, dT1ei ≤ hdS2e, dT2ei = Ψ (hS2, T2i).
Now we have proven that the lattices B and L are isomorphic. Hence we have this: Corrolary 1 The lattices L(X (∗X [L]), Y (∗Y [L]), I) and B(X∗X , Y ∗Y , I) are isomorphic.
Or we can say it in another words, that every concept lattice with hedges is isomorphic to some generalized concept lattice. 6
Conclusions
In this paper we showed that the notion of a generalized concept lattice defined
in [
generalized CL L-fuzzy CL with hedge L-fuzzy CL
one-sided fuzzy CL
The inverse question arises how big is distinction between these classes, e. g. what additional conditions are needed for a generalized concept lattices to be isomorphic to some fuzzy concept lattice with hedges. Or are these notions the same?. . .