Fuzzy Integrals over Complete Residuated Lattices

IFSA-EUSFLAT 2009

Fuzzy Integrals over Complete Residuated Lattices Anton´ın Dvoˇra´ k, Michal Holˇcapek Institute for Research and Applications of Fuzzy Modeling, University of Ostrava, 30. dubna 22, 701 03 Ostrava 1, Czech Republic Email: {antonin.dvorak, michal.holcapek}@osu.cz

Abstract— The aim of this paper is to introduce two new types of fuzzy integrals, namely, ⊗-fuzzy integral and →-fuzzy integral, where ⊗ and → are the multiplication and residuum of a complete residuated lattice, respectively. The first integral is based on a fuzzy measure of L-fuzzy sets and the second one on a complementary fuzzy measure of L-fuzzy sets, where L is a complete residuated lattice. Some of their properties and a relation to the fuzzy (Sugeno) integral are investigated. Keywords— fuzzy measure, fuzzy integral, fuzzy quantifier.

where P(T ) is the power set of T and ψ(Y ) := “the set Y is a big subset of T ”.

Note that formula ϕ(i, j) may be defined using the formula (Qk ∈ T )ϕ(i, j, k) with a fuzzy quantifier Q like for nearly all or many etc. Some types of fuzzy quantifiers could be determined by the interpretations of the formula ψ (cf. [1] and also see [2]). For example, if we consider the Łukasiewicz algebra as the structure of truth values for our logic, the truth value of the formula ϕ(i, j, k) is defined as R(tik , tjk ) from 1 Introduction (1), i.e., as the degree that tik is better than tjk , and the truth Let us consider two time series t1 = (t1k )k∈T , t2 = (t2k )k∈T value of the formula ψ(Y ) is interpreted by the value µ(Y ), displayed on Fig. 1 and suppose that our goal is to compare where µ : P(T ) → [0, 1] is a fuzzy measure (see Definithem and to find the “better” time series, where a greater value tion 3.2), then the evaluation of the formula ϕ(i, j) is given of time series at some time point k means a better value. To by 
t2 0.9 (R(tik , tjk ) ⊗ µ(Y )), (2) ||ϕ(i, j)|| = 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0























Y ∈P(T )\{∅} k∈Y

t1






























0

1

2

3

4

5

6

7

8

9

Figure 1: Time series.

10

11

12

t

where ⊗ is the operation of multiplication in the Łukasiewicz algebra (see Example 2.1) interpreting the logical connective &. Finally, we can conclude that, for example, the time series t1 is better than t2 , if ||ϕ(1, 2)|| > ||ϕ(2, 1)||. Let us define a mapping Iµ : F(T ) → [0, 1] by 
(A(k) ⊗ µ(Y )), (3) Iµ (A) = Y ∈P(T )\{∅} k∈Y

solve this task it is reasonable, firstly, to determine degrees of where F(T ) denotes the set of all fuzzy sets over T and truth saying how much formulas µ : P(T ) → [0, 1] is a fuzzy measure. One could simply veϕ(i, j, k) := “the value tik is better than the value tjk ” rify that if c ∈ [0, 1] is a constant and A(k) = c for any k ∈ T , then Iµ (A) = c, and if A(k) ≤ B(k), then Iµ (A) ≤ Iµ (B). are true, where, first, i = 1 and j = 2, and then i = 2 and Hence, I is a fuzzy measure (in the sense of Definition 3.2) µ j = 1. Obviously, the degrees of truth of formula ϕ(i, j, k) which could be understood, according to Mesiar [3], as an may be modeled by a fuzzy relation R : [0, 1]2 → [0, 1], e.g. example of (fuzzy) integral. Putting A(k) = R(tik , tjk ) for any k ∈ T , we can write ||ϕ(i, j)|| = Iµ (A) or also R(a, b) = max(a − b, 0), (1) ||(Qk ∈ T )ϕ(i, j, k)|| = Iµ (A) and, hence, we can see that where R(a, b) determines the degree that a is better than b. fuzzy integrals may be used to model some types of fuzzy Note that R(a, b) = ¬(a → b), where a → b is the oper- quantifiers. This idea is not new (see e.g. [4] or [5]), but some ation of residuum and ¬ is the operation of negation in the disadvantage of proposed approaches is that fuzzy quantifiers Łukasiewicz algebra (see Example 2.1 and consider ¬a = are defined as mappings from the set of all measurable (fuzzy) 1 − a). If we know how one value of time series is better than sets over a set M to [0, 1], although, the classical definition the other one for each time k, then we have to solve a task how introduces fuzzy quantifiers as mappings from the set of all to aggregate the obtained values to find a degree in which one (fuzzy) sets over a set M to [0, 1] (see e.g. [6] or [7]). It will times series is better than the second one. One of the natural be clear that this drawback vanishes when fuzzy quantifiers approaches could be to evaluate the following formula of the are modeled by fuzzy integrals in a form similar to (3). The aim of this contribution is to generalize the fuzzy intesecond order logic gral defined in (3), namely, to introduce a ⊗-fuzzy integral that ϕ(i, j) := (∃Y ∈ P(T ) \ {∅})(∀k ∈ Y )(ϕ(i, j, k)&ψ(Y )), could be used for modeling fuzzy quantifiers like all, some, ISBN: 978-989-95079-6-8

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IFSA-EUSFLAT 2009 for nearly all, or many, etc. and then to introduce a →-fuzzy integral that could be used for modeling quantifiers like no, not all, etc. Both types of fuzzy integrals are defined over a complete residuated lattice and it could be shown than, in general, one type of fuzzy integral cannot be expressed by another one. Nevertheless, if the structure of truth values is complete MV-algebra, then it is possible to define the →-fuzzy integral from the ⊗-fuzzy integral using negation (see Theorem 4.14). Moreover, it is surprising that we are able to show that the well-known Sugeno integral [8] is, under certain conditions, a special case of our fuzzy integral (see Theorem 4.7).

2 2.1

Preliminaries

Structures of truth values

Let us suppose that the structure of truth values is a complete residuated lattice (see e.g. [9]), i.e., an algebra L =

L, ∧, ∨, →, ⊗, ⊥, " with four binary operations and two constants such that L, ∧, ∨, ⊥, " is a complete lattice, where ⊥ is the least element and " is the greatest element of L, respectively, L, ⊗, " is a commutative monoid (i.e., ⊗ is associative, commutative and the identity a ⊗ " = a holds for any a ∈ L) and the adjointness property is satisfied, i.e., a ≤ b → c iff a ⊗ b ≤ c

(4)

holds for each a, b, c ∈ L, where ≤ denotes the corresponding lattice ordering. The operations ⊗ and → are usually called the multiplication and residuum, respectively. A residuated lattice is divisible, if a ⊗ (a → b) = a ∧ b holds for arbitrary a, b ∈ L, and satisfies the law of double negation, if (a → ⊥) → ⊥ = a holds for any a ∈ L. A divisible residuated lattice satisfying the law of double negation is called an MValgebra. For other information about residuated lattices we refer to [9]. Example 2.1. It is easy to prove (see e.g. [10]) that the algebra LT = [0, 1], min, max, T, →T , 0, 1,  where T is a left continuous t-norm and a →T b = {c ∈ [0, 1] | T (a, c) ≤ b} defines the residuum, is a complete residuated lattice. Moreover, if T is the Łukasiewicz t-norm, i.e., T (a, b) = max(a + b − 1, 0) for all a, b ∈ [0, 1], then LT is a complete MV-algebra called a Łukasiewicz algebra (on [0, 1]). One checks easily that a →T b = max(1 − a + b, 0) is the residuum in the Łukasiewicz algebra. Example 2.2. One checks easily that



a→b=

b, ∞,

L-fuzzy sets

Let L be a complete residuated lattice and M be a universe of discourse. A mapping A : M → L is called an L-fuzzy set on M . A value A(m) is called a membership degree of m in the L-fuzzy set A. The set of all L-fuzzy sets on M is denoted by FL (M ). An L-fuzzy set A on M is called crisp, if there is a subset X of M such that A = 1X , where 1X denotes the characteristic function of X. Particularly, 1∅ denotes the empty L-fuzzy set on M , i.e. 1∅ (m) = ⊥ for any m ∈ M . The set of all crisp L-fuzzy sets on M is denoted by PL (M ). An L-fuzzy set A is constant, if there is c ∈ L such that A(m) = c for any m ∈ M . For simplicity, a constant L-fuzzy set is denoted by the corresponding element of L, e.g., a, b, c.2 Let us denote Supp(A) = {m | m ∈ M & A(m) > ⊥} and core(A) = {m | m ∈ M & A(m) = "} the support and core of an L-fuzzy set A, respectively. Obviously, Supp(1X ) = core(1X ) = X for any crisp L-fuzzy set. An L-fuzzy set A is called normal, if core(A) = ∅. Let {Ai | i ∈ I} be a non-empty family of L-fuzzy sets on M . Then the union of Ai is defined by   
Ai (m) = Ai (m) (6) i∈I

i∈I

for any m ∈ M and the intersection of Ai is defined by     Ai (m) = Ai (m) i∈I

(7)

i∈I

for any m ∈ M . Let A be an L-fuzzy set on M . The complement of A is an L-fuzzy set A on M defined by A(m) = ¬A(m) for any m ∈ M . Finally, an extension of the operations ⊗ and → on L to the operations on FL (M ) is given by (A ⊗ B)(m) = A(m) ⊗ B(m) (A → B)(m) = A(m) → B(m)

(8) (9)

for any A, B ∈ FL (M ) and m ∈ M , respectively. The following theorem shows the well-known relation between the operations of the union and intersection of sets which also holds for L-fuzzy sets, if we restrict ourselves to a special class of complete residuated lattices. Theorem 2.1. Let L be a complete residuated lattice satisfying the law of double negation and {Ai | i ∈ I} be a nonempty family of L-fuzzy sets on M . Then

L[0,∞] = [0, ∞], min, max, →, 0, ∞, where ⊗ = min and

2.2



if b < a, otherwise,

(5) i∈I

Ai =

 i∈I

Ai

and

 i∈I

Ai =



Ai .

(10)

i∈I

is a complete residuated lattice. Note that L[0,∞] is a special We say that an L-fuzzy set A is less than or equal to an example of more general residuated lattice called a Heyting L-fuzzy set B and denote by A ⊆ B, if, for any m ∈ M , algebra.1 we have A(m) ≤ B(m). M  be a mapping.  Let f : M → →  Let us define the following additional operations for all Then f (A)(m) = m
∈f −1 (m) A(m ) defines a mapping a, b ∈ L: f → : FL (M ) → FL (M  ). Obviously, if f is a bijective → a ↔ b = (a → b) ∧ (b → a) (biresiduum) mapping, then f (A)(f (m)) = A(m) for any m ∈ M . ¬a = a → ⊥. 1

(negation)

A Heyting algebra is a residuated lattice with ⊗ = ∧.

ISBN: 978-989-95079-6-8

2 We suppose that the meaning of this symbol will be unmistakable from the context, that is, it should be clear when an element of L is considered and when a constant L-fuzzy set is assumed.

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IFSA-EUSFLAT 2009 Example 3.4. Let (M, M) be the fuzzy measurable space of all continuous mappings from Example 3.3. It is easy to see In this section, we will introduce a notion of fuzzy measure that and complementary fuzzy measure of L-fuzzy sets. More in 1 formation about fuzzy measures could be found in [11, 12]. A(m) dm, µ(A) = For our considerations we will consider algebras of L-fuzzy 0 1 sets as a base for defining fuzzy measures of L-fuzzy sets. where 0 A(m) dm denotes the Riemann integral, defines a Definition 3.1 ([11]). Let M be a non-empty universe of dis- fuzzy measure on (M, M). course. A subset M of FL (M ) is an algebra of L-fuzzy sets Example 3.5. Let L be a complete residuated lattice with the on M , if the following conditions are satisfied support [0, 1] and N be the set of natural numbers with 0. For any non-empty countable (i.e., finite or denumerable) universe (i) 1∅ , 1M ∈ M, M , injective mapping f : M → N, n ∈ N and A ∈ FL (M ), (ii) if A ∈ M, then A ∈ M, denote  A(m), if f (m) ≤ n; (iii) if A, B ∈ M, then A ∪ B ∈ M. Af,n (m) = (11) 0, otherwise. A couple (M, M) is called a fuzzy measurable space, if M is Further, for any injective mapping f : M → N and n ∈ N, an algebra of L-fuzzy sets on M . define µf,n : FL (M ) → [0, 1] as follows Example 3.1. The sets {1∅ , 1M }, PL (M ), σ-algebras on M , m∈Supp(Af,n ) Af,n (m) or FL (M ) are algebras of L-fuzzy sets on M . (12) µf,n (A) = |Supp(1Mf,n )| Example 3.2. Let us say that an L-fuzzy set A on M is a simple L-fuzzy set on M , if there exists a family of sets {Mi | and, finally, define µf , µf : FL (M ) → [0, 1] as follows n i = 1, . . . , n} such that i=1 Mi = M , Mi = Mj for any µf = lim inf µf,n (A), (13) i = j and A(m) = A(m ) holds for each m, m ∈ Mi , where n→∞ i = 1, . . . , n. Obviously, the set of all simple L-fuzzy sets on µf = lim sup µf,n (A). (14) n→∞ M is an algebra of L-fuzzy sets on M . It is easy to see that µf,n , µf and µf are fuzzy measures on Example 3.3. Let L be the Łukasiewicz algebra on [0, 1] (see 4 Example 2.1) and M = [0, 1]. Then the set of all continuous (M, FL (M )) determined by an injective mapping f . If, for mappings A : [0, 1] → [0, 1] is an algebra of L-fuzzy sets in example, M = N and f = id, then µf (A) = µf (A) = ⊥ for any L-fuzzy set with finite universe. For the set of all even or M .3 odd numbers, both fuzzy measures give 12 and, for the set of Let us introduce the concepts of fuzzy measure and com- all prime numbers, we obtain 0. plementary fuzzy measure as follows. The first definition is a If M is finite, then µf = µg = µf = µg for any injective modification of the definition of a normed fuzzy measure with mappings f, g : M → N and respect to truth values (see e.g. [12, 13]). A(m) µf (A) = µf (A) = m∈M . (15) Definition 3.2. Let (M, M) be a fuzzy measurable space. A |M | mapping µ : M → L is called a fuzzy measure on (M, M), if Hence, it is easy to see that µf (A) = µf (h→ (A)) holds for (i) µ(1∅ ) = ⊥ and µ(1M ) = ", any non-empty finite universe M , A ∈ FL (M ), injective mapping f : M → N and bijective mapping h : M → M . (ii) if A, B ∈ M such that A ⊆ B, then µ(A) ≤ µ(B). Unfortunately, this equality fails for denumerable universes in A triplet (M, M, µ) is called a fuzzy measure space, if general. In fact, consider M = N, f = id and a bijective (M, M) is a fuzzy measurable space and µ is a fuzzy mea- mapping h : N → N such that the image of all even numbers is the set of prime numbers. Then both fuzzy measures give sure on (M, M). 1 2 for the set of all even numbers, however, 0 for the set of all Definition 3.3. Let (M, M) be a fuzzy measurable space. A prime numbers. mapping ν : M → L is called a complementary fuzzy meaExample 3.6. Let µf be one of the fuzzy measures on sure on (M, M), if (M, FL (M )) determined by f defined in (13) and (14). If (i) ν(1∅ ) = " and ν(1M ) = ⊥, h : [0, 1] → [0, 1] is a non-decreasing mapping with h(0) = 0 and h(1) = 1, then h ◦ µf is a fuzzy measure on (M, FL (M )) (ii) if A, B ∈ M such that A ≤ B, then ν(A) ≥ ν(B). determined by µf and h. If h : [0, 1] → [0, 1] is a nonA triplet (M, M, ν) is called a complementary fuzzy measure increasing mapping with h(0) = 1 and h(1) = 0, then h◦µf is space, if (M, M) is a fuzzy measurable space and ν is a com- a complementary fuzzy measure on (M, FL (M )) determined by µf and h. plementary fuzzy measure on (M, M).

3

3

Fuzzy measures

Note that the set of all continuous mappings need not be an algebra of L-fuzzy sets for other residuated lattices determined by left continuous T -norms, because the negation is not a continuous mapping in general. ISBN: 978-989-95079-6-8

4

Note that µf and µf could be understood as a generalization of lower and upper weighted densities well known in the number theory which are examples of so-called lower and upper asymptotic fuzzy measures (see [14]).

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IFSA-EUSFLAT 2009 Theorem 3.1. Let (M, M) be a fuzzy measurable space. If (iii) there exists a bijective mapping f : M → M  with µ (ν) is a fuzzy measure (a complementary fuzzy measure) on A(m) = g(A)(f (m)) for any A ∈ M and m ∈ M . (M, M), then ν  (A) = ¬µ(A) (µ (A) = ¬ν(A)) defines a (M  , M ) be fuzzy measurable complementary fuzzy measure (a fuzzy measure) on (M, M). Theorem 3.4. Let (M, M),  spaces and g : M → M be a surjective mapping. Then g is Definition 3.4. Let (M, M) be a fuzzy measurable space an isomorphism between (M, M) and (M  , M ) if and only and X ∈ FL (M ). We say that X is M-fuzzy measurable, if if there exists a bijective mapping f : M → M  such that X ∈ M. g = f →. Let (M, M) be a fuzzy measurable space and X ∈ Definition 3.6. Let (M, M) and (M  , M ) be fuzzy measurFL (M ). Denote MX the set of all M-fuzzy measurable sets able spaces. We say that a mapping g : M → M is an isowhich are contained in X, i.e., morphism between (M, M, µ) and (M  , M , µ ) (or between (M, M, ν) and (M  , M , ν  )), if (16) MX = {A | A ∈ M and A ⊆ X}. (i) g is an isomorphism between (M, M) and (M  , M ), Note that 1∅ ∈ MX for each X ∈ FL (M ) and if X is M(ii) µ(A) = µ (g(A)) (or ν(A) = ν  (g(A))) for any A ∈ fuzzy measurable set, then also X ∈ MX . If X = M , then M. we will write only M instead of MM . If g is an isomorphism between fuzzy measure spaces Theorem 3.2. Let (M, M, µ) be a fuzzy measure space. A (M, M, µ) and (M  , M , µ ) or between complementary mapping µ∗ : FL (M ) → L defined by fuzzy measure spaces (M, M, ν) and (M  , M , ν  ), then
∗ µ(A) (17) we write g(M, M, µ) = (M  , M , µ ) or g(M, M, ν) = µ (X) = (M  , M , ν  ), respectively. A∈MX Let (M, M, µ) be a fuzzy measure space. If f : M → M  is a fuzzy measure on the fuzzy measurable space is a bijective mapping, then (M  , f → (M), µf → ), where (M, FL (M )). We say that µ∗ is the inner fuzzy measure on (M, FL (M )) determined by µ. (20) µf → (f → (A)) = µ(A) Example 3.7. Let (M, PL (M ), µ) be an arbitrary fuzzy measurable space (recall that PL (M ) is the power set of M ). Then the inner fuzzy measure on (M, FL (M )) is defined by  µ(A ), if 1core(A) = A , ∗ (18) µ (A) = ⊥, otherwise.

holds for any A ∈ M, is a fuzzy measure space isomorphic to (M, M, µ). A simple consequence of Theorem 3.4 is the fact that each fuzzy measure space (M  , M , µ ) isomorphic to (M, M, µ) has the form (M  , f → (M), µf → ) for a suitable bijective mapping f : M → M  . Analogously, to every couple of isomorphic complementary fuzzy measure spaces    Thus all L-fuzzy sets that are not normal have the inner fuzzy (M, M, ν) and (M , M , ν ) there is a bijective mapping   f : M → M such that (M , f → (M), νf → ) = (M  , M , ν  ). measure equal to ⊥.

4 Fuzzy integrals Example 3.8. Let L be the Łukasiewicz algebra on [0, 1], (M, M, µ) be the fuzzy measure space of continuous LIn this section, we will introduce two types of fuzzy integrals. fuzzy sets from Example 3.4. Then, for example, we have The first one is a generalization of the formula (3) derived in µ∗ (1[a,b] ) = b − a, however, 1[a,b] ∈ M. Introduction. For more information about fuzzy integrals we Theorem 3.3. Let (M, M, ν) be a complementary fuzzy mea- refer to [11, 12]. sure space. A mapping ν ∗ : FL (M ) → L defined by 4.1 ⊗-fuzzy integral  ∗ ν(A) (19) In this part, we will introduce a type of fuzzy integral that can ν (X) = be defined on an arbitrary fuzzy measure space (M, M, µ). A∈MX The form of this integral is motivated by our need to describe is a complementary fuzzy measure on the fuzzy measurable a class of models of L-fuzzy quantifiers of the type 1. In space (M, FL (M )). We say that ν is the inner complementary [2], we show that this class of models is bounded by the modfuzzy measure on (M, FL (M )) determined by ν. els of determiners all and some. Note that models of all and In the following part we will define an isomorphism be- some are the same as the interpretations of quantifiers ∀ and tween fuzzy measure spaces and then between complementary ∃, respectively, in fuzzy logic (see e.g. [6, 7, 15, 4]). fuzzy measure spaces. 



Definition 3.5. Let (M, M) and (M , M ) be fuzzy measurable spaces. We say that a mapping g : M → M is an isomorphism between (M, M) and (M  , M ), if (i) g is a bijective mapping with g(1∅ ) = 1∅ ,

Definition 4.1. Let (M, M, µ) be a fuzzy measure space, A ∈ FL (M ) and X be a M-fuzzy measurable L-fuzzy set. The ⊗-fuzzy integral of A on X is given by

⊗ 
A dµ = (A(m) ⊗ µ(Y )). (21) X

Y ∈MX \{1∅ } m∈Supp(Y )

(ii) g(A ∪ B) = g(A) ∪ g(B) and g(A) = g(A) hold for any ⊗ A dµ. If X = 1M , then we write A, B ∈ M, ISBN: 978-989-95079-6-8

360

IFSA-EUSFLAT 2009 ⊗  Remark 4.1. It is easy to see that 1∅ A dµ = ∅ = ⊥ Theorem 4.6. Let g be an isomorphism between fuzzy mea⊗ ⊗    for any A ∈ FL (M ) and X A dµ ≤ Y A dµ, whenever sure spaces (M, M, µ) and (M , M , µ ) and X ∈ M. Then ⊗ ⊗ we have X ⊆ Y . Since 1M A dµ = " in general, µA (X) = X A dµ

⊗

⊗ does not define a fuzzy measure on (M, M) in the sense of A dµ = g(A) dµ (26) Definition 3.2. X

Remark 4.2. One can also define a ∧-fuzzy integral of A on X in such way that ⊗ is replaced by ∧ in (21). Since ⊗ and ∧ have many common properties, both types of fuzzy integral will have similar properties. Nevertheless, we prefer the ⊗fuzzy integral in this paper, because it is closely related (due to the adjointness property) to →-fuzzy integral that will be introduced in the following subsection.

g(X)

for any A ∈ FL (M ).

In the end of this part, we will show that the Sugeno integral is a special case of our proposed integral. For this purpose we will use a slight modification of the usual Sugeno integral definition with respect to the fuzzy measurable spaces over complete residuated lattices. Theorem 4.1. Let (M, M, µ) be a fuzzy measure space. Then Let L be a complete residuated lattice and (M, M) be a µ : FL (M ) → L defined by fuzzy measurable space such that A ∩ B ∈ M for any A, B ∈ M.5 Denote Aa = {m | m ∈ M & A(m) ≥ a}. We say

⊗  A dµ (22) that an L-fuzzy set A is M-Sugeno measurable, if 1Aa ∈ M µ (A) = for any a ∈ L. The Sugeno integral is given, for any fuzzy measure space (M, M, µ) with B ∩ C ∈ M for any B, C ∈ is a fuzzy measure on (M, FL (M )). M, for any M-Sugeno measurable L-fuzzy set A and for any Theorem 4.2. Let (M, M, µ) be a fuzzy measure space. Then X ∈ M, by ⊗ ⊗ ⊗

(i) X (A ∩ B) dµ ≤ X A dµ ∧ X B dµ,
A dµ = (a ∧ µ(1Aa ∩ X)). (27) ⊗ ⊗ ⊗ X (ii) X (A ∪ B) dµ ≥ X A dµ ∨ X B dµ, a∈L ⊗ ⊗ (iii) X (c ⊗ A) dµ ≥ c ⊗ X A dµ, Theorem 4.7. Let L be a complete Heyting algebra, ⊗ ⊗ (M, M, µ) be a fuzzy measure space with B ∩ C ∈ M for (iv) X (c → A) dµ ≤ c → X A dµ, any B, C ∈ M, Abe a M-Sugeno  ⊗ measurable L-fuzzy set and X ∈ M. Then X A dµ = X A dµ. hold for any X ∈ M, A, B ∈ FL (M ) and c ∈ L. Theorem 4.3. Let (M, M, µ) be a fuzzy measure space and c ∈ L. Then we have ⊗ (i) (c ⊗ 1X ) dµ = c ⊗ µ∗ (1X ) for any X ⊆ M , ⊗ (c ⊗ 1X ) dµ = c ⊗ µ(1X ) for any X ⊆ M such that (ii) 1X ∈ M, ⊗ (iii) 1X dµ = µ(1X ) for any X ⊆ M such that 1X ∈ M, ⊗ c dµ = c. (iv)

4.2

→-fuzzy integral

In this part, we will introduce another type of fuzzy integral that can be defined on an arbitrary complementary fuzzy measure space (M, M, ν). The form of this integral is motivated by our need to describe another class of models of L-fuzzy quantifiers of the type 1 which are kind of negations of the previous ones.

Definition 4.2. Let (M, M, ν) be a complementary fuzzy measure space, A ∈ FL (M ) and X be a M-fuzzy measurable Theorem 4.4. Let (M, M, µ) be a fuzzy measure space. If L-fuzzy set. The →-fuzzy integral of A on X is given by X ∈ M is such that 1Supp(Y ) ∈ MX for any Y ∈ MX ,

→
 then, for any A ∈ FL (M ), we have A dν = (A(m) → ν(Y )).

⊗ X 
Y ∈MX \{1∅ } m∈Supp(Y ) A dµ = (A(m) ⊗ µ(1Y )), (23) (28) X m∈Y 1Y ∈PX \{1∅ } → A dν. If X = 1M , then we write where PX = {1Supp(Z) | Z ∈ MX }. →  that 1∅ A dν = ∅ = " Theorem 4.5. Let L be a complete MV-algebra, (M, M, µ) Remark 4.3. It is easy tosee → → for any A ∈ FL (M ) and X A dν ≤ Y A dν, whenever be a fuzzy measure space, A ∈ FL (M ) and X ∈ M. Then → → Y ⊆ X. Since 1M A dν = ⊥ in general, νA (X) = X A dν

⊗ 


 µ(Y ) ⊗ A dµ = A(m) . (24) does not define a complementary fuzzy measure on (M, M) in the sense of Definition 3.3. X Y ∈MX \{1 } m∈Supp(Y ) ∅

Moreover,

⊗

(c ⊗ A) dµ = c ⊗

X

for any c ∈ L. ISBN: 978-989-95079-6-8

⊗

A dµ X

(25)

5 Note that, according to Theorem 2.1, p. 2, each complete residuated lattice satisfying the law of double negation has this property. Nevertheless, there are fuzzy measurable spaces which keep this property, but L does not satisfy the law of double negation. A simple example is a fuzzy measurable space (M, M) such that M ⊆ PL (M ) and L is an arbitrary complete residuated lattice (e.g. L[0,∞] from Example 2.2).

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IFSA-EUSFLAT 2009 Theorem 4.8. Let (M, M, ν) be a complementary fuzzy mea- Theorem 4.14. Let L be a complete MV-algebra and (M, M) be a fuzzy measurable space. Then sure space. Then ν  : FL (M ) → L defined by

→

→

⊗ ν  (A) = A dν (29) A dν  = ¬ A dµ, (34) X X

→

⊗ is a complementary fuzzy measure on (M, FL (M )). A dµ = ¬ A dν (35) X X Theorem 4.9. Let (M, M, ν) be a complementary fuzzy meahold for any fuzzy measure µ and complementary fuzzy measure space. Then → → → sure ν, where ν  = ¬µ and µ = ¬ν. (i) X (A ∩ B) dν ≥ X A dν ∨ X B dν, → → → 5 Conclusions (ii) X (A ∪ B) dν ≤ X A dν ∧ X B dν, → → In this contribution, new types of fuzzy integrals, that are quite (iii) X (c ⊗ A) dν ≤ c → X A dν, useful for modeling fuzzy quantifiers (of the type 1), are in→ → troduced and some of their properties are studied. The defini(iv) X (c → A) dν ≥ c ⊗ X A dν tions of fuzzy quantifiers using these types of fuzzy integrals hold for any X ∈ M, A, B ∈ FL (M ) and c ∈ L. and some of their semantical properties could be found in [2]. Theorem 4.10. Let (M, M, ν) be a complementary fuzzy Acknowledgment measure space and c ∈ L. Then we have → This paper has been supported by the Grant IAA108270901 (c ⊗ 1X ) dν = c → ν ∗ (1X ) for any X ⊆ M , (i) ˇ of the GA AV CR. → (ii) (c ⊗ 1X ) dν = c → ν(1X ) for any X ⊆ M such References that 1X ∈ M, → [1] V. Nov´ak. A formal theory of intermediate quantifiers. Fuzzy (iii) 1X dν = ν(1X ) for any X ⊆ M such that 1X ∈ M, Sets and Systems, 159(10):1229 1246, 2008. → [2] A. Dvoˇra´ k and M. Holˇcapek. Basic properties of L-fuzzy (iv) c dν = ¬c. Theorem 4.11. Let (M, M, ν) be a complementary fuzzy measure space. If X ∈ M is such that 1Supp(A) ∈ MX for any A ∈ MX , then, for any A ∈ FL (M ), we have

→ 
A dν = (A(m) → ν(1Y )), (30) X

1Y ∈PX \{1∅ } m∈Y

where PX = {1Supp(A) | A ∈ MX }. Theorem 4.12. Let L be a complete MV-algebra, (M, M, ν) be a complementary fuzzy measure space, A ∈ FL (M ) and X ∈ M. Then

→  

 ( A dν = A(m)) → ν(Y ) . X

Y ∈MX \{1∅ }

m∈Supp(Y )

(31) Moreover,



→

(c ⊗ A) dν = c →

X

quantifiers of the type 1 determined by fuzzy measures. In IFSA/EUSFLAT 2009, Lisboa, 2009.

[3] R. Mesiar. Fuzzy measures and integrals. Fuzzy Sets and Systems, 156:365–370, 2005. [4] M. Ying. Linguistic quantifiers modeled by Sugeno integrals. Artificial Intelligence, 170:581–606, 2006. [5] W. San-min and Z. Bin. Prenex normal form in linguistic quantifiers modeled by Sugeno integrals. Fuzzy Sets and Systems, 159:1719–1723, 2008. [6] I. Gl¨ockner. Fuzzy Quantifiers: A Computational Theory. Springer-Verlag, Berlin, 2006. [7] M. Holˇcapek. Monadic L-fuzzy quantifiers of the type 1n , 1. Fuzzy Sets and Systems, 159:1811–1835, 2008. [8] M. Sugeno. Theory of Fuzzy Integrals and its Applications. PhD thesis, Tokyo Institute of Technology, 1974. [9] R. Bˇelohl´avek. Fuzzy Relational Systems: Foundations and Principles. Kluwer Academic Publisher, New York, 2002.

→

A dν X

(32)

[10] P. H´ajek. Metamathematics of Fuzzy Logic. Kluwer Academic Publishers, Dordrecht, 1998. [11] Z. Wang and G.J. Klir. Fuzzy measure theory. Plenum Press, New York, 1992.

for any c ∈ L.

Theorem 4.13. Let g be an isomorphism between comple- [12] M. Grabisch, T. Murofushi, and M. Sugeno, editors. Fuzzy mentary fuzzy measure spaces (M, M, ν) and (M  , M , ν  ) Measures and Integrals. Theory and Applications. Studies in Fuzziness and Soft Computing. Physica Verlag, Heidelberg, and X ∈ M. Then we have

→

→ 2000. A dν = g(A) dν  (33) [13] E.P. Klement, R. Mesiar, and E. Pap. A universal integral based X

for any A ∈ FL (M ).

g(X)

on measures of level sets. IEEE Transactions on Fuzzy Systems, to appear. [14] L. Miˇs´ık and J. T´oth. On asymptotic behaviour of universal

The following statement shows that if we consider a comfuzzy measure. Kybernetika, 42(3):379–388, 2006. plete MV-algebra, then we can restrict ourselves, for example, [15] S. Peters and D. Westerst˚ahl. Quantifiers in Language and to ⊗-fuzzy integrals, since each →-fuzzy integral is uniquely Logic. Oxford University Press, New York, 2006. determined by the negation of a suitable ⊗-fuzzy integral. ISBN: 978-989-95079-6-8

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