Decomposition integrals

International Journal of Approximate Reasoning 54 (2013) 1252–1259

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Decomposition integrals a,∗ Radko Mesiar a,b , Andrea Stupnanová ˇ a b

Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Slovak University of Technology, Bratislava, Slovakia Centre of Excellence IT4 Innovations, Division University of Ostrava, IRAFM, Ostrava, Czech Republic

ARTICLE INFO

ABSTRACT

Article history: Received 15 November 2012 Received in revised form 4 February 2013 Accepted 4 February 2013 Available online 24 February 2013

Decomposition integrals recently proposed by Even and Lehrer are deeply studied and discussed. Characterization of integrals recognizing and distinguishing the underlying measures is given. As a by-product, a graded class of integrals varying from Shilkret integral to Choquet integral is proposed. © 2013 Elsevier Inc. All rights reserved.

Keywords: Monotone measure Choquet integral Shilkret integral Lehrer integral Universal integral

1. Introduction Several distinguished integrals are substantially linked to special set systems. So, for example, the classical Lebesgue integral is based on disjoint set systems (partitions), Choquet integral [1] deals with chains of sets, Shilkret integral [11] considers only single sets. Lehrer in [8] has proposed to deal with arbitrary set systems. All above mentioned integrals for non-negative functions exploit the common addition and multiplication of reals. As a common framework for all of them, Even and Lehrer [3] have proposed a new concept of decomposition integral, see also [12]. Decomposition integrals can be seen as solution of optimization problems under given constraints on the allowed set systems. The nature and properties of decomposable integrals can be sufficiently enlighted when considering finite spaces only, what is the case of this contribution, too. Our aim is to clarify some properties and relationships between decomposition integrals in the language of the corresponding set systems. The paper is organized as follows. In the next section, decomposition integrals from [3] are recalled, and some relationships and properties from [12] are stressed. Section 3 is devoted to the study of links between properties of decomposition set systems and the corresponding decomposition integrals. In Section 4, we introduce a new graded family of universal integrals (for this concept see [6]) varying from Shilkret integral to the Choquet integral. Note that our restriction on finite spaces follows the ideas of several other generalizations of the probability theory, such as Dempster–Shafer evidence theory, see [5,10], for example. Moreover, it is also linked to another root of our framework multicriteria decision support [2]. 2. Decomposition integrals Throughout this paper, if not stating explicitly the form of X, we will consider a fixed finite space X = {1, 2, . . . , n}. A mapping m : 2X → [0, ∞] which is monotone, A ⊆ B ⊆ X implies m(A) ≤ m(B), and satisfies the boundary conditions ∗ Corresponding author. E-mail addresses: [email protected] (R. Mesiar), [email protected] (A. Stupnanová). ˇ 0888-613X/$ - see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.ijar.2013.02.001

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m(∅) = 0 and m(X ) > 0, is called a monotone measure. The class of all monotone measures on X will be denoted by M, will be denoted by F. To define decomposition integrals, we deal with systems H and the class of all X → [0, ∞] functions  X 22 \{∅} of set systems, H ∈ X ≡ 2 {∅}. Definition 2.1 [3]. Let H ∈ X be fixed. A mapping IH : M × F → [0, ∞] given by ⎧ ⎨  IH (m, f ) = sup ai · m(Ai ) (Ai )i∈J ∈ H, ai ≥ 0 for each i ∈ J , ai 1Ai ≤ ⎩ i∈J

i∈J

⎫ ⎬ f

(1)

⎭

is called a H-decomposition integral. Observe that the basic idea of H-decomposition integral is to find the “best” decomposition (or subdecomposition) of f ∈ F into the sum of basic functions attaining at most two values. It is obvious that if ∅ = H ⊂ G ∈ X, then also H ∈ X and IH ≤ IG in the sense of partial order described by IH (m, f ) ≤ IG (m, f ) for all (m, f ) ∈ M × F . Moreover, formula (1) shows the positive homogeneity of decomposition integrals, i.e., for each H ∈ X, m ∈ M, f ∈ F and c > 0 it holds IH (m, c f ) = c IH (m, f ). In the next example we recall some distinguished systems from X and the related integrals. Example 2.2. (i) H1 = {{A} | ∅ = A ⊂ X }. All considered set systems in H1 are singletons, and the corresponding decomposition integral coincide with Shilkret integral [11], IH 1 (m, f )

= Sh(m, f ) = sup{a · m(A)| a · 1A  f }.

(ii) H2 = {B |B is a partition

n of X }. For additive measures, H2 was used by Lebesgue [7] to define (discrete) Lebesgue integral, L(m, f ) = i=1 f (i) m({i}). For a general monotone measure m ∈ M, we recognize the PAN-integral introduced in [14], ⎧ ⎫ ⎨ ⎬  IH 2 (m, f ) = PAN (m, f ) = sup ai · m(Ai ) (Ai )i∈J is a partition of X , ai · 1Ai ≤ f . ⎩ ⎭ i∈J i∈J (iii) H3

= {B|B ⊂ 2X \ {∅} is a chain}. The chain-based approach corresponds to the Choquet integral [1], IH 3 (m, f )

= Ch(m, f ) =

 ∞ 0

m(f

≥ t )dt .

(iv) H4 = 22 \{∅} is the maximal system in X, and hence IH 4 is the greatest decomposition integral. This integral (called also concave integral) was introduced by Lehrer [8], ⎧ ⎫ ⎨ ⎬  ai · m(Ai ) ai ≥ 0 for all i ∈ J , (Ai )i∈J ⊂ 2X \ {∅}, ai · 1Ai ≤ f . IH 4 (m, f ) = Co(m, f ) = sup ⎩ ⎭ X

i∈J

Fig. 1. Hasse diagram of decomposition integrals from Example 2.2.

i∈J

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⎧ All four integrals given in Example 2.2 reduce to the Lebesgue integral once m is a Dirac measure, m = δi , δi (A) = ⎪ ⎨1 if i ∈ A , and L(δi , f ) = f (i). ⎪ ⎩0 elsewhere Moreover, all  four introduced decomposition  integrals coincide  if and only if m is the minimum of some Dirac measures, m = δB = i∈B δi , and then IH j (δB , f ) = i∈B IH j (δi , f ) = i∈B f (i), j = 1, 2, 3, 4. For more details and discussion about further relationships of the mentioned 4 integrals we recommend [12]. In the next Fig. 1 we exemplify the ordering relation between these 4 integrals. 3. Decomposition set systems and decomposition integrals Recall that a system A for all i, j ∈ J , i = j.

= (Ai )i∈J ⊆ 2X  {∅} is said to be logically independent if and only if Ai ∩ Aj = ∅ and Ai = Aj

Lemma 3.1. A system A = {A1 , . . . , Ak } ⊆ 2X 

a1 , . . . , ak ≥ 0 such that ki=1 ai · 1Ai ≤ 1A it holds k  i=1

ai m(Ai )

{∅} is logically independent if and only if for any m ∈ M, A ⊆ X and

≤ m(A).



Proof. Suppose that A is a logically independent system, i.e., ki=1 Ai = B = ∅. Then ki=1 ai 1Ai (x) ≤ 1A for A such that

k B \ A = ∅ implies a1 = · · · = ak = 0, while if B \ A = ∅ then i=1 ai ≤ 1. Moreover, if Ai \ A = ∅ then necessarily ai = 0,

k

i.e., ai can be positive only if Ai ⊆ A. Thus ki=1 ai m(Ai ) = ai =0 ai m(Ai ) ≤ ai =0 ai m(A) = i=1 ai m(A) ≤ m(A), independently of m ∈ M. k On the other hand, if A = {A1 , . . . , Ak } is not logically independent, i.e., if i=1 Ai = ∅, consider m∗ ∈ M, m∗ (B) = ⎧ ⎪ ⎨1 if B = ∅ 

1 1 Denote A = ki=1 Ai and observe that necessarily k > 1. Then ki=1 k− 1 ≤ 1A but ki=1 k− m∗ (Ai ) = 1 Ai 1 ⎪ ⎩0 else. k k−1

> 1 = m∗ (A), proving the necessity. 

Not all systems H ∈ X requires the complete knowledge of a monotone measure m Consider, for example, H = {{X }}. Then IH (m, f )

∈ M for determination of IH (m, f ).

= sup{a · m(X )| a · 1X ≤ f } = m(X ) · min{f (1), . . . , f (n)}.

Systems exploiting the complete knowledge of considered monotone measures will be called the complete systems. Definition 3.2. A system H A ∈ A.

∈ X is called complete whenever for each ∅ = A ⊆ X there is a system A ∈ H such that

Observe that all systems H1 , H2 , H3 and H4 introduced in Example 2.2 are complete. Lemma 3.3. A system H ∈

X is complete if and only if IH (m, 1A ) ≥ m(A) for each m ∈ M, A ⊆ X.

Proof. Suppose that H is a complete system. It is a matter of Definition 2.1 only to see that for⎧each m ∈ M, A ⊆ X it holds ⎪ ⎨1 if Ai = A

IH (m, 1A ) ≥ i∈J ai m(Ai ) = m(A), where (Ai )i∈J ∈ H is a system containing A, and ai = . Suppose that H ⎪ ⎩0 else is not complete, i.e., there is ∅ = A ⊆ X such that it is not contained in any A = (Ai )i∈J ∈ H. Consider the measure δA

introduced in Section 2. Then if i∈J ai 1Ai (x) ≤ 1A , clearly for each i ∈ J, either ai = 0 or Ai  A, and thus i∈J ai m(Ai ) = 0. Consequently, IH (δA , 1A ) = 0 = 1 = δA (A).  In [6], the concept of universal integrals was introduced. In this concept, when considering the standard product, it should hold I (m, c · 1A ) = c · m(A) for each m ∈ M, A ⊆ X , c ∈ [0, ∞[. Due to the positive homogeneity of decomposition integrals, such integral can be considered universal only if IH (m, 1A ) = m(A) for each m ∈ M and A ⊆ X.

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Proposition 3.4. Let H ∈ X. Then the H-decomposition integral gives back the underlying monotone measure, i.e., IH (m, 1A ) = m(A) for each m ∈ M and A ⊆ X, if and only if H is a complete system and each system A = (Ai )i∈J ∈ H is logically independent. Proof. The result follows from Lemmas 3.1 and 3.3.  Coming back to Example 2.2, systems H1 and H3 satisfy the requirements of Proposition 3.4 (indeed, both of them are universal integrals). On the other hand, H2 and H4 contain systems which are not logically independent. We have already mentioned that for H, G ∈ X such that H ⊂ G it holds IH ≤ IG . However, the subsethood H ⊂ G is not necessary to ensure IH ≤ IG . Proposition 3.5. Consider two systems H, G denote the later relation as H  G.

∈ X. Then IH ≤ IG whenever for each A ∈ H there is B ∈ G so that A ⊆ B. We

Proof. Suppose H  G. For each m ∈ M, f ∈ F and A ∈ H, if such that A ⊆ B it holds Bj ∈B bj 1Bj ≤ f , where bj = ai if Bj

Ai ∈A ai m(Ai ) = Bj ∈B bj m(Bj ) and thus IH (m, f ) ≤ IG (m, f ). 

Ai ∈A ai 1Ai ≤ f then also for corresponding B ∈ G = Ai , and bj = 0 if Bj ∈ B  A. Obviously, then

It is transparent that H ⊆ G ⇒ H  G , but not vice-versa, i.e.,  is an ordering on systems from X refining the standard inclusion ordering. Moreover, the relation H ≈ G (i.e., H  G and G  H) is an equivalence relation and thus the space X of decomposition systems can be partitioned into equivalence classes [H]≈ . Obviously, for any G ∈ [H]≈ it holds IG = IH . Due to the finiteness of X, each system H ∈ X is finite. Thus in each equivalence class [H]≈ we can introduce a system H which is a member of [H]≈ with minimal cardinality. Such system will be called a minimal system. Example 3.6. For systems introduced in Example 2.2 it holds: H1 is a maximal chain in 2X \ {∅}}, H4 = {2X \ {∅}}.

=

H1 , H2

=

H2 , H3

=

{B| B

We have also the next result. Lemma 3.7. A system H ∈

X is complete if and only if H1  H, where H1 was introduced in Example 2.2 i).

Proof. The result follows directly from Definition 3.2.  Remark 3.8. Note that the problem when IH ≤ IG was completely solved in [3, Proposition 3]. Indeed, IH ≤ IG if and only if for any system A ∈ H there is a system B ∈ G such that when C ⊆ A, C = {C1 , . . . , Ck }, is related to algebraically independent functions 1C1 , . . . , 1Ck , then C ⊆ B. 4. Decomposition integrals as universal integrals In Proposition 3.4 we have characterized decomposition systems H yielding a decomposition integral IH satisfying one of the axioms of universal integrals introduced in [6], namely IH gives back the considered monotone measure m ∈ M if and only if H is complete and each system A ∈ H is logically independent. Note that each chain is logically independent, i.e., if H1  H ⊆ H3 then IH (m, 1A ) = m(A) for each m ∈ M and A ⊆ X. The monotonicity of decomposition integrals is obvious, and thus we have to check the third axiom characterizing universal integrals, namely the equality IH (m, f ) = IH (v, g ) for all integral equivalent pairs (m, f ), (v, g ) ∈ M × F , i.e., pairs satisfying m({f ≥ t }) = v({g ≥ t }) for all t ∈]0, ∞[. Example 4.1 (i) For n = 2, i.e., X = {1, 2}, let H = {{1}, {{2}, {1, 2}}}. Then H ∈ H1  H ⊆ H3 . Then for any pair (m, f ) in M × F holds ⎧ ⎪ ⎨Ch(m, f ) if f (1) ≤ f (2) IH (m, f ) = ⎪ ⎩ Sh(m, f ) else,

X is a complete logically independent system, and

i.e., this decomposition integral acts on a part of domain as the Choquet integral, and on the reminder as the Shilkret integral. Obviously, it is not a universal integral. Consider two integral equivalent pairs (m1 , f1 ) and (m2 , f2 ) such that m1 ({1}) = m2 ({2}) = 0.3, m1 ({2}) = m2 ({1}) = 0.7, m1 ({1, 2}) = m2 ({1, 2}) = 1, and f1 (1) = f2 (2) = 0.4, f1 (2) = f2 (1) = 0.8. Then IH (m1 , f1 )

= Ch(m1 , f1 ) = 0.68 = 0.56 = Sh(m2 , f2 ) = IH (m2 , f2 ).

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(ii) For n

= 3, i.e., X = {1, 2, 3}, let H=

{{{1}, {1, 2}, {1, 2, 3}} , {{2}, {2, 3}} , {{3}, {1, 3}}} .

Then H ∈ X and H1  H ⊆ H3 . Consider a monotone measure m ∈ M given by m({1}) = m({3}) = 0, m({2}) = m({1, 2}) = m({2, 3}) = 0.5 and m({1, 3}) = m({1, 2, 3}) = 1 (obviously, m(∅) = 0). For functions f , g : X → [0, ∞] given by f (1) = 1, f (2) = 0.6, f (3) = 0.4 and g (1) = 0.4, g (2) = 0.6, g (3) = 1, we have ⎧ ⎪ 1 if t ∈ ]0, 0.4] ⎪ ⎪ ⎪ ⎪ ⎨ m({f ≥ t }) = m({g ≥ t }) = 0.5 if t ∈ ]0.4, 0.6] ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 0 if t ∈ ]0.6, ∞[, i.e., (m, f ) and (m, g ) are integral equivalent. However, IH (m, f ) integral.

= 0.5 while IH (m, g ) = 0.4, i.e., IH is not a universal

We have the following sufficient condition ensuring that a decomposition integral is also a universal integral on X in the sense of [6]. Proposition 4.2. For a fixed i (i)

H

Then H(i)

∈ X = {1, . . . , n}, let

= {B| B is a chain in 2X \ {∅} with cardinality i}.

∈ X and IH (i) is a universal integral.

Proof. It is evident that H(i)  H(j) whenever i ≤ j, and that H(1) = H1 , H(n) = H3 (see Example 3.6). Thus H1  H(i) ⊆ H3 and hence IH (i) gives back the underlying monotone measure. × F be integral equivalent. For any {A1 , . . . , Ai } ∈ H(i) , A1 ⊃ A2 ⊃ · · · ⊃ Ai , and On the other side, let (m, f ), (v, g ) ∈

M i constants a1 , . . . , ai ≥ 0 such that j=1 aj 1Aj ≤ f necessarily also ij=1 aj 1Bj ≤ f , where Bj = {f ≥ a1 + · · · + aj } ⊇ Aj ,

and ij=1 aj m(Aj ) ≤ ij=1 aj m(Bj ) = ij=1 aj v(Cj ), where Cj = {g ≥ a1 + · · · + aj }.

Obviously, C1 ⊇ C2 ⊇ · · · ⊇ Ci . If card{C1 , . . . , Ci } = i, {C1 , . . . , Ci } ∈ H(i) . If card{C1 , . . . , Ci } = k < i then

{C1 , . . . , Ci } = {Cj1 , . . . , Cjk } with Cj1  Cj2  · · ·  Cjk . Putting bjr = Cj =Cjr aj , we get kr=1 bjr v(Cjr ) = ij=1 aj v(Cj ). Now, it is enough to complete the chain {Cj1 , . . . , Cjk } with i − k sets so that the new chain {D1 , . . . , Di } ∈ H(i) with

i

Djr = Cjr , r = 1, . . . , k and putting bj = 0 if j ∈ {j1 , . . . , jr }, we have ij=1 aj m(Aj ) ≤ j=1 bj v(Dj ), and thus IH (i) (m, f ) ≤ IH (i) (v, g ). Now exchanging the role of f and g, and using the same arguments, we can shown that IH (i) (v, g ) ≤ IH (i) (m, f ), and thus IH (i) (m, f ) = IH (i) (v, g ). Consequently, IH (i) is a universal integral.  Based on Proposition 4.2 and results of [3] we have the following important result. Theorem 4.3. For a fixed X decomposition integrals.

= {1, . . . , n}, the functionals IH (i) , i = 1, . . . , n, are the only universal integrals which are also

Proof. We have to show the necessity only. For each fixed m ∈ M, the decomposition integral IH (m, ·) can be considered as a universal integral only if IH (m, f ) = IH (m, g ) for all f , g ∈ F such that the pairs (m, f ) and (m, g ) are integral equivalent. However, then due to Theorem 2 of [3], necessarily IH = IH (i) for some i ∈ {1, . . . , n}.  Observe that, in general, IH is simultaneously a decomposition and a universal integral if and only if H ≈ H(i) for some i ∈ {1, . . . , n}, i.e., if H contains all chains of length i (of subsets of X) and possibly some shorter chains. Obviously, then IH = IH (i) . Based on the proof of the Proposition 4.2, we have the next formula: ⎧ ⎫ i ⎨ ⎬ IH (i) (m, f ) = max f (σ (jk )) − f (σ (jk−1 ))) · m ({f ≥ f (σ (jk ))}) , ( ⎭ 1≤j1 <···
where σ : {1, . . . , n} → {1, . . . , n} is a permutation such that f (σ (1)) is an i-element set of natural numbers, with convention f (σ (j0 )) = 0.

≤ f (σ (2)) ≤ · · · ≤ f (σ (n)) and {j1 , . . . , ji } ⊆ X

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Recall that a monotone measure m in M is symmetric whenever m(A) depends on the cardinality of A only, i.e., m(A) = vcard(A) , 0 = v0 ≤ v1 ≤ · · · ≤ vn . If vn = 1 (i.e., m is a capacity), then the Choquet integral Ch(m, f ) with respect to a symmetric capacity m was shown to be an OWA operator, see [4,13], Ch(m, f )

=

n  j=1

f (σ (j)) · (vn−j+1

− vn−j )

Our new types of integrals introduced in Proposition 4.2, when applied to symmetric capacities, can be seen as generalizations of OWA operators, and for symmetric capacity m in M it holds ⎧ ⎫ i ⎨ ⎬ IH (i) (m, f ) = max f (σ (jk )) · (vn−jk +1 − vn−jk+1 +1 ) | 1 ≤ j1 < · · · < ji ≤ n . ⎩ ⎭ k=1

It is also evident that H(1)  H(2)  · · ·  H(n) , and hence IH (1) ≤ IH (2) ≤ · · · ≤ IH (n) , with extremal cases IH (1) = IH 1 = Sh, i.e., the Shilkret integral, and IH (n) = IH 3 = Ch, i.e., Choquet integral. Thus we have introduced a graded family of universal and decomposition integrals, connecting the weakest universal integral Sh linked to the product, and the Choquet integral. Recall that the Shilkret and the Choquet integral are defined on any abstract measurable space (X , A). We can formally define also integrals I (i) = IH (i) to act on any abstract space, omitting the constraint X = {1, . . . , n} and A = 2X . Due to the arbitrariness of n ∈ N, this leads to a new family of universal integrals (which are also decomposition integrals) {I (1) , I (2) , . . . , I (n) , . . .} acting on an arbitrary measurable space (X , A) given as follows: Definition 4.4. Let n

∈ N be fixed. The mapping I (n) :







× F (X ,A ) → [0, ∞], where S is the class is the set of all monotone measures on (X , A) and F (X ,A ) is the set of all measurable

of all measurable spaces, M(X ,A ) functions f : X → [0, ∞], is given by ⎧ n ⎨ (n) I (m, f ) = sup ai · m(Ai ) a1 , . . . , an ⎩ i=1

(X ,A ) (X ,A )∈S M

≥ 0, {A1 , . . . , An } ∈ A is a chain,

n  i=1

ai 1Ai

≤f

⎫ ⎬ ⎭

.

Using arguments similar to those from the proof of Proposition 4.2, we have the next result. Theorem 4.5. For each n ∈ N the mapping I (n) is a universal integral in the sense of [6]. Moreover, it holds ⎧ ⎫ n ⎨ ⎬ (n) I (m, f ) = sup ai · m({f ≥ a1 + · · · + ai }) |a1 , . . . , an ≥ 0 . ⎩ ⎭ i=1

Remark 4.6. It is not difficult to check that for X = {1, . . . , n}, I (i) |(X , 2X ) = IH (i) , i = 1, . . . , n. Moreover I (1) = Sh is the Shilkret integral acting on any (X , A) ∈ S. On the other side, if X = {1, . . . , n} is finite, I (n) |(X , 2X ) = I (n+j) |(X , 2X ) = Ch is the Choquet integral for all j = 1, 2, . . .. Finally, if X is infinite, then Ch

= sup{I (n) , n ∈ N} = lim I (n) . n→∞

Example 4.7. Let X = [0, 1], A = B([0, 1]) (the σ -algebra of all Borel subsets of [0, 1]). Consider m : A → [0, ∞], m(A) = λ(A), where λ is the standard Lebesgue measure on B([0, 1]), and f : X → [0, ∞] ⎧ the identity function, f (x) = x. ⎪ ⎨1 − t if t ∈ ]0, 1] Then the function h :]0, ∞] → [0, ∞] defined by h(t ) = m({f ≥ t }) is given by h(t ) = ⎪ ⎩ 0 if t > 1. It holds: I (1) (m, f )

= sup {a(1 − a)| a ∈ ]0, 1]} =

I (2) (m, f ) for n

1 4

,

= sup {a(1 − a) + b(1 − a − b)| a, b, a + b ∈ ]0, 1]} =

∈ N, I (n) (m, f ) =

n 2(n + 1)

, and Ch(m, f ) =

1 2

= lim

n→∞

1 3

, n

2(n + 1)

.

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Fig. 2. Shaded areas correspond to integrals I (1) , I (2) and Ch.

Observe that the value 2(nn+1) for I (n) (m, f ) is attained putting a1 i = 1, . . . , n.

= · · · = an =

1 , n+1

and Ai

= {f ≥

i } n+1

=





i ,1 n+1

,

From the geometric point of view, the Shilkret integral Sh = I (1) is just the maximal area of a rectangle with sides parallel to axis which is inside the hypergraph H = {(x, y) ∈ [0, ∞]2 | y ≤ h(x)}. For I (2) , we look for the maximal sum of two areas of such non-overlapping rectangles, for I (n) we consider n such non-overlapping rectangles. Finally, the Choquet integral is just the area of the hypergraph H. As an illustration, see Fig. 2 related to I (1) , I (2) and Ch based on Example 4.7 and f from this example. 5. Concluding remarks We have discussed several relationships between decomposition systems and decomposition integrals. We have introduced a new class of universal integrals which are also decomposition integrals, and which connect the classical Shilkret and Choquet integrals. Though our considerations were restricted to the finite spaces X, the extension of the introduced integrals I (n) to arbitrary monotone measure spaces (X , A, m) is obvious, and thus we obtain (possibly as a limit case) also the standard Choquet and Lebesgue integrals on abstract spaces. Note that several questions remains still open. For example, for a fixed H ∈ X, is there an algorithm how to compute IH ? Note that all integrals we have discussed are based on the standard arithmetical operations + and · on [0, ∞]. However, several types of integrals with respect to monotone measures are built by means of some pseudo-arithmetical operations ⊕ and  on [0, ∞]. For example, Sugeno integral is based on ⊕ = ∨ (max) and  = ∧ (min). As an interesting topic for ⊕, the next investigations we propose to study pseudo-decomposition integrals IH given by ⎧ ⎫ ⎨ ⎬  ⊕, (m, f ) = sup ai  m(Ai ) (Ai )i∈J ∈ H, ai ≥ 0, i ∈ J , and b(ai , Ai ) ≤ f , IH ⎩ ⎭ i∈J i∈J where b(ai , Ai )(x) ∨,∧

=

⎧ ⎪ ⎨ai if x

∈ Ai

⎪ ⎩ 0 else.

Obviously, IH = Su is the Sugeno integral for any complete decomposition system H. For interested readers we recommend recent paper [9] discussing the above proposed topic in the specific case H4

22 {∅} , which in the case ⊕ = + and  = · yields the concave integral Co case, based on H2 ( all partitions of X), was treated in [15,9]. X

= = IH 4 introduced by Lehrer [8]. Another special

Acknowledgment The authors express their gratitude to Prof. E. Lehrer for his valuable comments to the preliminary version of this paper. The support of the grants European Regional Development Fund in the IT4 Innovations Centre of Excellence Project (CZ.1.05/1.1.00/02.0070) , APVV-0073-10 and VEGA 1/0184/12 is kindly announced. References [1] G. Choquet, Theory of capacities, Annales de l’Institut Fourier 5 (1953/54) 131–295. [2] M. Ehrgott, J. Figueira, S. Greco, Trends in Multiple Criteria Decision Analysis, Springer Verlag, Berlin, 2010. [3] Y. Even, E. Lehrer, Decomposition-Integral: Unifying Choquet and the Concave http://www.math.tau.ac.il/∼lehrer/Papers/decomposition.pdf .

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submitted

for

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