The The Basic Basic Theorem Theorem  On on Generalized Concept Lattice Generalized Concept Lattice Stanislav Krajči krajci@science.upjs.sk Stanislav Krajči Institute of Institute of Computer Computer Science, Science, Faculty Faculty of of Science, Science, UPJ Košice, Slovakia UPJŠ Košice, Slovakia krajci@science.upjs.sk Abstract. In [4] we have presented a new common platform for different types of fuzzification of a concept lattice. Now we show a pendant of the basic theorem on clasical concept lattices for this generalization which characterizes (complete) lattices isomorphic to this generalized concept lattice. 1 Introduction There are some attempts have arisen which try to fuzzify the classical crisp Ganter-Wille’s concept lattice, i.e. to consider the fuzzy values instead of 0’s/1’s in a matrix. One of them is the natural (and symmetric) fuzzification given by e.g. Bělohlávek ([1]) which consider (L-)fuzzy subsets of objects and (L-)fuzzy subsets of attributes. Another from author’s paper [4] is a ”bad boy”, be- cause it is not symmetric: it considers fuzzy subsets of attributes but ordi- nary/classical/crisp subsets of objects. Our paper [4] shows a relationship between these approaches by finding cer- tain common platform for both of them. The main idea of it is to differ between types of the fuzziness of objects and of the fuzziness of attributes as it is used in its special case in [3]. In this paper we formulate the extend version of the basic theorem on such generalized concept lattice which is the generalization of above-mentioned and we proved this added extension (the simple version of it was proved in [4]). This extension characterizes all lattices which are isomorphic to a generalized concept lattice. 2 A generalized fuzzy concept lattice Now we shortly repeat basic definitions and results from [4]: Let L be a poset,C and D be supremum-complete upper-semilattices (i.e. there exists sup X = X for each subset X of C or D). Let • : C × D → L be monotone and left-continuous in both their arguments, i.e.  Partially supported by grant 1/0385/03 of the Slovak grant agency VEGA. c V. Snášel, R. Bělohlávek (Eds.): CLA 2004, pp. 25–33, ISBN 80-248-0597-9. VŠB – Technical University of Ostrava, Dept. of Computer Science, 2004. 26 Stanislav Krajči 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 ≤  holds for d ∈ D,  ∈ L and for all c ∈ X ⊆ C, then sup X • d ≤ . 2b) If c • d ≤  holds for c ∈ C,  ∈ L and for all d ∈ Y ⊆ D, then c • sup Y ≤ . Let A and B be non-empty sets and let R be L-fuzzy relation on their Carte- sian product, i.e. R : A × B → L. Define the following mapping  : B D → A C: If g : B → D then (g) : A → C is defined in the following way: (g)(a) = sup{c ∈ C : (∀b ∈ B)c • g(b) ≤ R(a, b)}. Symmetrically we define the mapping  : A C → B D: If f : A → C then (f ) : B → D is defined in the following way: (f )(b) = sup{d ∈ D : (∀a ∈ A)f (a) • d ≤ R(a, b)}. We have shown in [4] that these two mappings form a Galois connection and they are generalization of classical crisp Ganter-Wille’s approach (see [2]), fuzzy Bělohlávek’s (or independently Pollandt’s) one (see [1], [5]) and our one-sided fuzzy one (see [3]). Hence we use our new mappings  and  for constructing a concept lattice following the original Ganter-Wille’s method from [2]. We assume from now that C and D are complete lattices. A concept is a pair g, f ∈ B D × A C s.t. (g) = f and (f ) = g. If g1 , f1 and g2 , f2 are concepts, we will write g1 , f1 ≤ g2 , f2 iff g1 ≤ g2 (or equivalently f1 ≥ f2 ). The set of all such concepts with the order ≤ is called a (generalized) concept lattice and denoted shortly by L. Theorem 1 (The Basic Theorem on Generalized Concept Lattice). 1) The generalized concept lattice L is a complete lattice in which        gi , fi = gi ,   fi i∈I i∈I i∈I and         gi , fi =   gi , fi . i∈I i∈I i∈I 2) Let moreover L have the least element 0L and 0C • d = 0L and c • 0D = 0L 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. The Basic Theorem on Generalized Concept Lattice 27 3) For every a ∈ A, b ∈ B, c ∈ C, d ∈ D α(a, c) ≥ β(b, d) if and only if c • d ≤ R(a, b). The proof of the part 1) is in [4], so we prove the second part now. We will need the following singleton functions: If T is an arbitrary set and U is a poset with the least element 0U , define ST,U t,u : T → U in the following way:  u, if x = t, ST,U t,u (x) = 0U , elsewhere. We show a few basic properties of these functions. Lemma 1. f = sup{ST,U t,f (t) : t ∈ T }. Proof. Let x ∈ T . Because of pointwise defined supremum of a set of functions we have sup{ST,U T,U t,f (t) : t ∈ T }(x) = sup{St,f (t) (x) : t ∈ T }. For all t = x we have ST,U T,U T,U t,f (t) (x) = 0U , hence sup{St,f (t) (x) : t ∈ T } = Sx,f (x) (x) = f (x). So we obtain f (x) = sup{ST,U t,f (t) : t ∈ T }(x) for all x ∈ T , q.e.d.. Lemma 2. a) For all a ∈ A, b ∈ B, c ∈ C a,c )(b) = sup{d ∈ D : c • d ≤ R(a, b)}. (SA,C b) For all a ∈ A, b ∈ B, d ∈ D (SB,D b,d )(a) = sup{c ∈ C : c • d ≤ R(a, b)}. Proof. By the definition (SA,C a,c )(b) = sup{d ∈ D : (∀x ∈ A)Sa,c (x) • d ≤ A,C a,c (x) • d = 0C • d = 0L for every x = a, we obtain (∀x ∈ R(x, b)}. Because SA,C A)Sa,c (x) • d ≤ R(x, b) iff SA,C A,C a,c (a) • d ≤ R(a, b), i.e. c • d ≤ R(a, b) for all d ∈ D. But it means that {d ∈ D : (∀a ∈ A)SA,C a,c (a) • d ≤ R(a, b)} = {d ∈ D : c • d ≤ R(a, b)} and (SA,C a,c )(b) = sup{d ∈ D : (∀x ∈ A)SA,C a,c (x) • d ≤ R(x, b)} = sup{d ∈ D : c • d ≤ R(a, b)}, q.e.d.. The proof of the part b) is analogous. Firstly we prove the theorem for the case V = L: Define αL (a, c) = (SA,C a,c ), ((Sa,c )) , A,C βL (b, d) = ((SB,D B,D b,d )), (Sb,d ) . Obviously both results are really concepts. Claim 1 a) αL is non-increasing in the second argument. 28 Stanislav Krajči b) βL is non-decreasing in the second argument. Proof. Let c1 ≤ c2 are from C, we want to prove that αL (a, c1 ) ≥ αL (a, c2 ) (for an arbitrary a ∈ A). It is enough to prove that (SA,Ca,c1 ) ≥ (Sa,c2 ), i.e. A,C (Sa,c1 )(b) ≥ (Sa,c2 )(b) for all b ∈ B, which (by the previous lemma) means A,C A,C sup{d ∈ D : c1 • d ≤ R(a, b)} ≥ sup{d ∈ D : c2 • d ≤ R(a, b)}. Because of monotoneity of ≤ in the first argument we have c1 • d ≤ c2 • d, so c2 • d ≤ R(a, b) implies c1 • d ≤ R(a, b). It means that {d ∈ D : c2 • d ≤ R(a, b)} ⊆ {d ∈ D : c1 • d ≤ R(a, b)}, which implies wanted sup{d ∈ D : c2 • d ≤ R(a, b)} ≤ sup{d ∈ D : c1 • d ≤ R(a, b)}, q.e.d.. The proof of the part b) is symmetrical. Claim 2 αL (a, c) ≥ βL (b, d) if and only if c • d ≤ R(a, b). Proof. If c • d ≤ R(a, b), then clearly c ∈ {k ∈ C : k • d ≤ R(a, b)}. It follows that B,D a,c (a) = c ≤ sup{k ∈ C : k • d ≤ R(a, b)} = (Sb,d )(a). For x = a we have SA,C B,D B,D obvious SA,Ca,c (x) = 0C ≤ (Sb,d )(x), hence Sa,c ≤ (Sb,d ). Because of the A,C B,D Galois connection between  and  it follows that (SA,C a,c ) ≥ ((Sb,d )), i.e. αL (a, c) ≥ βL (b, d). Inversely, if x ∈ A, clearly u ∈ {k ∈ C : k • d ≤ R(x, b)} iff u • d ≤ R(x, b). Hence (using left-continuousness of • in the second argument) (SB,D b,d )(x) • d = B,D sup{k ∈ C : k • d ≤ R(x, b)} • d ≤ R(x, b). So (∀x ∈ A)(Sb,d )(x) • d ≤ R(x, b), and this implies d ∈ {m ∈ D : (∀x ∈ A)(SB,D b,d )(x) • m ≤ R(x, b)}, hence d ≤ sup{m ∈ D : (∀x ∈ A)(Sb,d )(x) • m ≤ R(x, b)} = ((SB,D B,D b,d ))(b) (by the definition of ). Because αL (a, c) ≥ βL (b, d), i.e. (Sa,c ) ≥ ((SB,D A,C b,d )), we obtain d ≤ ((SB,D b,d ))(b) ≤ (SA,C a,c )(b). Because w ∈ {m ∈ D : c • m≤ R(a, b)} iff c • w ≤ R(a, b), by left-continuousness of • in the second argument we have c • sup{m ∈ D : c • m ≤ R(a, b)} ≤ R(a, b), i.e. (by the previous lemma) a,c )(b) ≤ R(a, b). Because of monotonity of • in the second argument c • (SA,C we obtain c • d ≤ c • (SA,C a,c )(b) ≤ R(a, b), q.e.d.. Claim 3 Let f : A → C and g : B → D and g, f be a concept. a) g, f = inf{αL (a, f (a)) : a ∈ A}. b) g, f = sup{βL (b, g(b)) : b ∈ B}. Proof. Let a ∈ A. Then SA,C A,C a,f (a) (a) = f (a) and Sa,f (a) (x) = 0C ≤ f (x) for the other x = a, hence SA,C a,f (a) ≤ f . Because  and  form a Galois connection and f is the intent of some concept, we obtain ((SA,C a,f (a) )) ≤ ((f )) = f , so g, f ≤ (SA,C A,C a,f (a) ), ((Sa,f (a) )) = αL (a, f (a)). This is true for all a ∈ A, so g, f ≤ inf{αL (a, f (a)) : a ∈ A}. The Basic Theorem on Generalized Concept Lattice 29 Conversely, by the part 1) of the theorem we have inf{αL (a, f (a)) : a ∈ A} = A,C A,C a∈A αL (a, f (a)) = a∈A (Sa,f (a) ), ((Sa,f (a) )) =  a∈A (SA,C a,f (a) ),  A,C (( a∈A ((Sa,f (a) )))) . On the other hand, using the property of com- position  and  (following from a Galois connection between them) we have SA,C A,C A,C a,f (a) ≤ ((Sa,f (a) )) = sup{((Sx,f (x) )) : x ∈ A} for all a ∈ A. It means (using lemma 1) that f = sup{SA,C A,C a,f (a) : a ∈ A} ≤ sup{((Sx,f (x) )) : x ∈  A} ≤ ((sup{((SA,C A,C x,f (x) )) : x ∈ A})) = (( x∈A ((Sx,f (x) )))), i.e.  f ≤ (( a∈A ((SA,C a,f (a) )))). Hence g, f ≥ inf{αL (a, f (a)) : a ∈ A}, q.e.d.. The proof of the part b) is analogous. Claim 4 a) αL [A × C] is infimum-dense in L. b) βL [B × D] is supremum-dense in L. Proof. They are obvious consequences of the previous claim. We have proven the second part of the basic theorem for the special case V = L. Now we will prove it for another cases. First we assume that V is a complete lattice isomorphic to our L. Let ϕ : L → V is a witness isomorphism. Define α(a, c) = ϕ(αL (a, c)), β(b, d) = ϕ(βL (b, d)). We prove that these mapping fulfill appropriate properties: Claim 5 a) α is non-increasing in the second argument. b) β is non-decreasing in the second argument. Proof. If c1 ≤ c2 , we have (by the claim 1) αL (a, c1 ) ≥ αL (c, a2 ). Because ϕ is an isomorphism, we obtain ϕ(αL (a, c1 )) ≥ ϕ(αL (c, a2 )), i.e. α(a, c1 ) ≥ α(c, a2 ), q.e.d.. The proof of the part b) is analogous. Claim 6 a) α[A × C] is infimum-dense. b) β[B × D] is supremum-dense. Proof. If v ∈ V , let g, f ∈ L s.t. ϕ(g, f ) = v (such v exists, because ϕ is a bijection). Then inf{α(a, f (a)) : a ∈ A} = inf{ϕ(αL (a, f (a))) : a ∈ A} = ϕ(inf{αL (a, f (a)) : a ∈ A}) (because ϕ is an isomorphism), = ϕ(g, f ) (by claim 2) = v. Hence we can express an arbitrary element of V as the infimum of some subset of α[A × C], i.e. α[A × C] is infimum-dense, q.e.d.. The fact that β[B × D] is supremum-dense can be proven analogously. 30 Stanislav Krajči Claim 7 α(a, c) ≥ β(b, d) if and only if c • d ≤ R(a, b). Proof. α(a, c) ≥ β(b, d) i.e. ϕ(αL (a, c)) ≥ ϕ(βL (b, d)) iff αL (a, c) ≥ βL (b, d) (because ϕ is an isomorphism), iff c • d ≤ R(a, b) (we have proven it above), q.e.d.. Now we have proven the first direction, we are going to show the opposite one. Let α and β have appropriate properties. Claim 8 a) For all a ∈ A, c ∈ C and v ∈ V α(a, c) ≥ v iff (∀b ∈ B)(∀d ∈ D)(β(b, d) ≤ v → α(a, c) ≥ β(b, d)). b) For all b ∈ B, d ∈ D and v ∈ V β(b, d) ≤ v iff (∀a ∈ A)(∀c ∈ C)(α(a, c) ≥ v → α(a, c) ≥ β(b, d)). Proof. If α(a, c) ≥ v, then clearly β(b, d) ≤ v implies α(a, c) ≥ β(b, d) what means c • d ≤ R(a, b) by assumption on α and β. Conversely, using supremum-density of β[B × D] our v can be expressed in a form v = sup{β(bi , di ) : i ∈ I} for some set of pairs {bi , di : i ∈ I}. This implies that β(bi , di ) ≤ v for all i ∈ I, hence (by our assumption) α(a, c) ≥ β(bi , di ) for all i ∈ I. But it means that α(a, c) ≥ sup{β(bi , di ) : i ∈ I} = v, q.e.d.. The proof of the part b) is symmetrical. Now we can define the following function ϕ : L → V in the following way: ϕ(g, f ) = inf{α(a, f (a)) : a ∈ A} and show that it is a wanted isomorphism. Claim 9 ϕ is order-preserving. Proof. g1 , f1 ≤ g2 , f2 iff f1 ≥ f2 iff f1 (a) ≥ f2 (a) for all a ∈ A. Because α is non-increasing in the second argument, it implies α(a, f1 (a)) ≤ α(a, f2 (a)). So we have ϕ(g1 , f1 ) = inf{α(x, f1 (x)) : x ∈ A} ≤ α(a, f2 (a)) for all a ∈ A and (using the definition of an infimum) ϕ(g1 , f1 ) ≤ inf{α(a, f2 (a)) : a ∈ A} = ϕ(g2 , f2 ), q.e.d.. Now we define the function ψ : L → V by ψ(v) = gv , fv such that fv (a) = sup{c ∈ C : α(a, c) ≥ v}, gv (b) = sup{d ∈ D : β(b, d) ≤ v} −1 and show that ψ = ϕ . First we prove that gv , fv is really concept: The Basic Theorem on Generalized Concept Lattice 31 Claim 10 a) (gv ) = fv . b) (fv ) = gv . Proof. α(a, c) ≥ v is (by previous lemma) equivalent to (∀b ∈ B)(∀d ∈ D)(β(b, d) ≤ v → α(a, c) ≥ β(b, d)) and because of the relation between α, β and R, this is equivalent to (∀b ∈ B)(∀d ∈ D)(β(b, d) ≤ v → c • d ≤ R(a, b)). This can be replaced equivalently by (∀b ∈ B)c • sup{d ∈ D : β(b, d) ≤ v} ≤ R(a, b) (one implication follows from the definition of a supremum, the second one from left-continuity of •), i.e. (∀b ∈ B)c • gv (b) ≤ R(a, b). This gives the equality sup{c ∈ C : α(a, c) ≥ v} = sup{c ∈ C : (∀b ∈ B)c • gv (b) ≤ R(a, b)}, i.e. fv (a) = (gv )(a) (by the definitions of fv and ) for all a ∈ A. It means that fv = (gv ), q.e.d.. The proof of the part b) is symmetrical. Claim 11 ψ is order-preserving. Proof. If v1 ≤ v2 then clearly α(a, c) ≥ v2 implies α(a, c) ≥ v1 . It follows that {c ∈ C : α(a, c) ≥ v2 } ⊆ {c ∈ C : α(a, c) ≥ v1 }, i.e. fv2 (a) = sup{c ∈ C : α(a, c) ≥ v2 } ≤ sup{c ∈ C : α(a, c) ≥ v1 } = fv1 (a) for all a ∈ A. This means that fv1 ≥ fv2 , i.e. ψ(v1 ) = gv1 , fv1 ≤ gv2 , fv2 = ψ(v2 ), q.e.d.. Claim 12 ϕ(ψ(v)) = v. Proof. Using supremum-density of β[B × D] our v can be expressed in a form v = sup{β(bi , di ) : i ∈ I} for some set of pairs {bi , di : i ∈ I}. This implies that β(bi , di ) ≤ v for all i ∈ I. If a ∈ A and c ∈ C are such that α(a, c) ≥ v, we have β(bi , di ) ≤ α(a, c) for all i ∈ I. Hence c • di ≤ R(a, bi ), and using the left-continuousness of • in the first argument we obtain sup{c ∈ C : α(a, c) ≤ v}•di ≤ R(a, bi ), i.e. (by the definition of fv ) fv (a)•di ≤ R(a, bi ) for all i ∈ I and a ∈ A. But it means that α(a, fv (a)) ≥ β(bi , di ) and this implies α(a, fv (a)) ≥ sup{β(bi , di ) : i ∈ I} = v. Hence ψ(gv , fv ) = inf{α(a, fv (a)) : a ∈ A} ≥ v. Conversely, using supremum-density of α[A × C] our v can be expressed in a form v = inf{α(ai , ci ) : i ∈ I} for some set of pairs {ai , ci : i ∈ I}. Clearly α(ai , ci ) ≥ v, therefore ci ∈ {c ∈ C : α(ai , c) ≥ v}, which implies ci ≤ sup{c ∈ C : α(ai , c) ≥ v} = fv (ai ), for all i ∈ I. Because α is non- increasing in the second argument, we obtain α(ai , ci ) ≥ α(ai , fv (ai )) for all i ∈ I. And because α(ai , fv (ai )) ∈ {α(a, fv (a)) : a ∈ A}, we have α(ai , ci ) ≥ α(ai , fv (ai )) ≥ inf{α(a, fv (a)) : a ∈ A} = ψ(gv , fv ), for all i ∈ I. This implies v = inf{α(ai , ci ) : i ∈ I} ≥ ψ(gv , fv ), q.e.d.. Claim 13 ψ(ϕ(g, f )) = g, f . Proof. Let v = ϕ(g, f ) = inf{α(a, f (a)) : a ∈ A}, it is enough to prove that g = gv . Because (∀a ∈ A)(f (a) • d ≤ R(a, b)) iff (∀a ∈ A)(β(b, d) ≤ α(a, f (a))) (it is a property of α and β), and this is (by the definition of a infimum) equivalent to β(b, d) ≤ inf{α(a, f (a)) : a ∈ A} = v, we obtain {d ∈ D : (∀a ∈ A)f (a) • d ≤ 32 Stanislav Krajči R(a, b)} = {d ∈ D : β(b, d) ≤ v}, which implies (using the definitions of  and gv ) (f )(b) = sup{d ∈ D : (∀a ∈ A)f (a) • d ≤ R(a, b)} = sup{d ∈ D : β(b, d) ≤ v} = gv (b). It is true for all b ∈ B, so g = (f ) = gv , q.e.d.. To conclude, we prove that both ϕ and ψ are order-preserving and both ϕ ◦ ψ and ψ ◦ ϕ are identities. All this means that really ψ = ϕ−1 and both ϕ : V → L and psi : L → V are isomorphisms. Quod erat demonstrandum. As a bonus we prove next property of our mappings: Claim 14 a) α(a, sup{ci : i ∈ I}) = inf{α(a, ci ) : i ∈ I}. b) β(b, sup{di : i ∈ I}) = sup{α(b, di ) : i ∈ I}. Proof. β[B × D] is supremum-dense, so α(a, sup{ci : i ∈ I} = sup{β(bj , dj ) : j ∈ J} for some {bj , dj : j ∈ J}. This implies α(a, sup{ci : i ∈ I} ≥ β(bj , dj ), i.e. (by the definition) sup{ci : i ∈ I} • dj ≤ R(a, bj ) for all j ∈ J. It follows that ci • dj ≤ R(a, bj ), i.e. α(a, ci ) ≥ β(bj , dj ) for all i ∈ I, j ∈ J. By the definitions of a infimum and a supremum we obtain inf{α(a, ci ) : i ∈ I} ≥ β(bj , dj ) and inf{α(a, ci ) : i ∈ I} ≥ sup{β(bj , dj ) : j ∈ J} = α(a, sup{ci : i ∈ I}. Conversely again from the supremum-density of β[B×D], we have inf{α(a, ci ) : i ∈ I} = sup{β(bj , dj ) : j ∈ J} for some {bj , dj : j ∈ J}. It follows that α(a, ci ) ≥ sup{β(bj , dj ) : j ∈ J} for all j ∈ J and α(a, ci ) ≥ β(bj , dj ), i.e. ci • dj ≤ R(a, bj ) for all i ∈ I, j ∈ J. Again, it is equivalent sup{ci : i ∈ I} • dj ≤ R(a, bj ), i.e. α(a, sup{ci : i ∈ I}) ≥ β(bj , dj ) for all j ∈ J, hence α(a, sup{ci : i ∈ I}) ≥ sup{β(bj , dj ) : j ∈ J} = inf{α(a, ci ) : i ∈ I}, q.e.d.. The proof of the part b) is symmetrical, we use the infinum-density of the set α[A × C]. Yet one note on an asymmetry in the definition of ϕ. If we define the function ϕ : L → V in the following way (symmetrically to ϕ): ϕ (g, f ) = sup{β(b, g(b)) : b ∈ B}, it can be proven (analogously to the claim 12) that ϕ (ψ(v)) = v. So for arbitrary concept g, f we take v = ψ −1 (g, f ) and we obtain ϕ (g, f ) = ϕ (ψ(v)) = v = ϕ(ψ(v)) = ϕ(g, f ), i.e. ϕ = ϕ. Hence our asymmetry in the definition of ϕ is only seeming and inf{α(a, f (a)) : a ∈ A} = sup{β(b, g(b)) : b ∈ B}. 3 Conclusions In this paper we prove the extended version of the basic theorem on a generalized concept lattice, which is a common platform for till now known fuzzifications of a classical crisp concept lattice. The main idea of it is to use two different complete lattices, one for objects and maybe another for attributes. The Basic Theorem on Generalized Concept Lattice 33 References 1. R. Bělohlávek: Concept Lattices and Order in Fuzzy Logic, Annals of Pure and Applied Logic, to appear 2. B. Ganter, R. Wille: Formal Concept Analysis, Mathematical Foundation, Springer Verlag 1999, ISBN 3-540-62771-5 3. S. Krajči: Cluster based efficient generation of fuzzy concepts, Neural Network World 13,5 (2003) 521-530 4. S. Krajči: A Generalized Concept Lattice, submitted to ERCIM workshop on soft computing, Vienna, July 2004 5. S. Pollandt: Datenanalyse mit Fuzzy-Begriffen, in: G. Stumme, R. Wille: Begrif- fliche Wissensverarbeitung. Methoden und Anwendungen, Springer, Heidelberg 2000, 72-98