On Integral Transforms for Residuated Lattice-Valued Functions

Abstract

The article aims to introduce four types of integral transforms for functions whose function values belong to a complete residuated lattice. The integral transforms are defined using so-called qualitative residuum based fuzzy integrals and integral kernels in the form of binary fuzzy relations. We present some of the basic properties of proposed integral transforms including a linearity property that is satisfied under specific conditions for comonotonic functions.

Introduction

Mathematical operators known as integral transforms produce a new function g(y) by integrating the product of an existing function f(x) and an integral kernel function K(x, y) between suitable limits. The Fourier and Laplace transforms belong among the most popular integral transforms and are applied for real or complex-valued functions. The importance of the integral transforms is mainly in solving (partial) differential equations, algebraic equations, signal and image processing, spectral analysis of stochastic processes (see, e.g., [2, 21, 23]).

In fuzzy set theory we often deal with functions whose function values belong to an appropriate algebra of truth values as a residuated lattice and its special variants as the BL-algebra, MV-algebra, IMTL-algebra (see, e.g. [1, 5, 18]). In [20], Perfilieva introduced lattice-valued upper and lower fuzzy transforms that are, among others, used for an approximation of functions. A deeper investigation of fuzzy transforms properties can be found in [13–17, 19, 22]. In a recent article [9], we demonstrated that the lower and upper fuzzy transforms can be introduced as two type of integral transforms, where the multiplication based fuzzy integral is applied [3, 4]. Namely, for a fuzzy measure space , an integral kernel and a function , where L is a complete residuated lattice, we proposed the integral transforms given by the following formulas:

where becomes the upper fuzzy transform if for any such that and becomes the lower fuzzy transform if for any such that .1 Moreover, to get the exact definitions of lower and upper fuzzy transforms, the family of fuzzy sets has to form a fuzzy partition of X (see [20]).

The aim of this article is to introduce further integral transforms for residuated lattice-valued functions and analyze their basic properties for which we consider the residuum based fuzzy integrals that were proposed by Dvořák and Holčapek in [4] and Dubois, Prade and Rico in [3]. Together with the integral transform with the multiplication based fuzzy integral introduced in [9] we get a class of nonstandard integral transforms for the residuated lattice-valued functions based on fuzzy (or also qualitative) integrals that are often used in data processing. Note that the fuzzy integrals aggregate data and, in this way, provide summary information that is not directly visible from data. Obviously, the proposed integral transforms also provide an aggregation of function values, mainly, if the set Y has a significantly smaller size than the set X. This can be used, for example, in hierarchical decision making, classification problem or signal and image processing, where kernels can express relationships between different levels of criteria, object attributes and classes or introduce windows for some kind of filtering, respectively.

The article is structured as follows. In the next section, we recall the definition of complete residuated lattices and the basic concepts of fuzzy set theory and the theory of fuzzy measure spaces. The third section introduces two types of the residuum based fuzzy integrals and shows their basic properties. The integral transforms for residuated lattice-valued functions are established in the fourth section. We present their elementary properties and demonstrate the linearity property under the restriction to comonotonic functions. The last section is a conclusion.

Because of the space limitation almost all proofs are omitted in this article.

Preliminary

Truth Value Structures

We assume that the structure of truth values is a complete residuated lattice, i.e., an algebra with four binary operations and two constants such that is a complete lattice, where is the least element and is the greatest element of L, is a commutative monoid (i.e., is associative, commutative and the identity holds for any ) and the adjointness property is satisfied, i.e.,

1

holds for each , where denotes the corresponding lattice ordering, i.e., if for . A residuated lattice L is said to be divisible if holds for arbitrary . The operation of negation on L is defined as for . A residuated lattice L satisfies the law of double negation if holds for any . A divisible residuated lattice satisfying the law of double negation is called an MV-algebra. A residuated lattice is said to be linearly ordered if the corresponding lattice ordering is linear, i.e., or holds for any .

Theorem 1

Let be a non-empty set of elements from L, and let . Then

1. ,

2. ,

3. ,

4. ,

5. ,

6. .

If L is a complete MV-algebra the above inequalities may be replaced by equalities.

For more information about residuated lattices, we refer to [1, 18]. In what follows, we present two examples of linearly ordered lattice.

Example 1

It is easy to prove that the algebra

where T is a left continuous t-norm (see, e.g., [11]) and , defines the residuum, is a complete residuated lattice. In this article, we will refer to complete residuated lattices determined by the Łukasiewicz t-norm and nilpotent minimum, i.e.,

respectively. Their residua are as follows:

In the first case, the complete residuated lattice will be denoted by . Note that is a complete MV-algebra called the Łukasiewicz algebra (on [0, 1]), where, for example, the distributivity of over is satisfied.2 The residuated lattice determined by the nilpotent minimum is an example of a residuated lattice in which the above-mentioned distributivity fails.

Example 2

Let be such that . One checks easily that , where

2

is a complete residuated lattice. Note that is a special example of a more general residuated lattice called a Heyting algebra.3

In the end of this section, we introduce two families of subsets of L from which important algebras of sets are later generated. Let be defined as

3

for any . Obviously, . A set , for which holds, is called the upper set or upset. We use to denote the family of all upsets in L, i.e., .4 Similarly, let be defined as

4

for any . A set for which holds is called the lower set or loset. The family of all losets in L is denoted .

Fuzzy Sets

Let L be a complete residuated lattice, and let X be a non-empty universe of discourse. A function is called a fuzzy set (-fuzzy set) on X. A value A(x) is called a membership degree of x in the fuzzy set A. The set of all fuzzy sets on X is denoted by . A fuzzy set A on X is called crisp if for any . Obviously, a crisp fuzzy set can be uniquely identified with a subset of X. The symbol denotes the empty fuzzy set on X, i.e., for any . The set of all crisp fuzzy sets on X (i.e., the power set of X) is denoted by . A constant fuzzy set A on X (denoted as ) satisfies for any , where . The sets and are called the support and the core of a fuzzy set A, respectively. A fuzzy set A is called normal if .

Let A, B be fuzzy sets on X. The extension of the operations , , and on L to the operations on is given by

5

for any . Obviously, and are the standard definitions of the intersection and union of fuzzy sets A and B, respectively, but we prefer here the symbols of infimum () and supremum () over the classical and .

Let X, Y be non-empty universes. A fuzzy set is called a (binary) fuzzy relation. A fuzzy relation K is said to be normal, whenever , and normal in the first coordinate, whenever for any . Similarly, a fuzzy relation is normal in the second component. A fuzzy relation K is said to be complete normal whenever K is normal in the first and the second coordinates. A relaxation of the normality of fuzzy relation is a semi–normal fuzzy relation defined as , i.e., for certain . Similarly one can define semi-normal in the the first (second) coordinate and complete semi-normal fuzzy relation.

Fuzzy Measure Spaces

Measurable spaces and functions. Let us consider algebras of sets as follows.

Definition 1

Let X be a non-empty set. A subset of is an algebra of sets on X provided that.

1. ,

2. if , then ,

3. if , then .

Definition 2

An algebra of sets on X is a -algebra of sets if

• (A4)if , , then .

It is easy to see that if is an algebra (-algebra) of sets, then the intersection of finite (countable) number of sets belongs to . A pair is called a measurable space (on X) if is an algebra (-algebra) of sets on X. Let be a measurable space and . We say that A is -measurable if . Obviously, the sets and are -algebras of fuzzy sets on X.

A beneficial tool how to introduce an algebra or a -algebra of sets on X is an algebra (-algebra) generated by a non-empty family of sets.

Definition 3

Let be a non-empty family of sets. The smallest algebra (-algebra) on X containing is denoted by () and is called the generated algebra (-algebra) by the family .

Note that the intersection of algebras (-algebras) is again an algebra (-algebra), hence, the smallest algebra (-algebra) on X containing always exists and its unique. Moreover, the generated algebra , in contrast to , can be simply constructed from the elements as the set which consists of all finite unions applied on the set of all finite intersections over the elements of and their complements. Note that the construction of a generated -algebra needs a transfinite approach. In this article, we will consider the algebras generated from the families of upsets and losets .

Let and be measurable spaces, and let be a function. We say that f is --measurable if for any . The following theorem shows that the verification of --measurability of functions can be simplified if is a generated algebra (-algebra).

Theorem 2

Let be a subset such that , and let be a measurable space. A function is --measurable if and only if for any .

Proof

() The implication is a simple consequence of .

() Let . Note that is called the preimage algebra on Y. From the definition of the generated algebra by the family , we find that . Hence, we obtain that for any , which means that f is --measurable.

Note that the previous theorem remains true if is replaced by . In the following three statements we provide sufficient conditions under which the functions obtained applying the operations to measurable functions remain measurable. For the purpose of this article, we restrict to fuzzy sets and algebras determined by upsets and losets.

Theorem 3

Let be linearly ordered, let be an algebra, and let be a set of all --measurable fuzzy sets. Then for any .

Theorem 4

Let be an algebra, and let be a set of all --measurable fuzzy sets. If is closed over arbitrary unions, then

Theorem 5

Let L be linearly ordered and dense. Let be an algebra, and let be a set of all --measurable fuzzy sets. If is closed over arbitrary unions, then

The previous theorems become true if the algebra is replaced by and the --measurability is considered.

Fuzzy Measures. The concept of a fuzzy measure on a measurable space is a slight extension of the standard definition of the normed measure where the unit interval (or the real line) is replaced by a complete residuated lattice L (e.g., [6, 12]).

Definition 4

A map is called a fuzzy measure on a measurable space if

• (i) and ,
• (ii)if such that , then .

A triplet is called a fuzzy measure space whenever is a measurable space and is a fuzzy measure on .

Example 3

Let be an algebra from Example 1, where T is a continuous t-norm. Let be a finite non-empty set, and let be an arbitrary algebra. A relative fuzzy measure on can be given as

for all , where |A| and |X| denote the cardinality of A and X, respectively. Let be a monotonically non-decreasing map with and . The relative measure can be generalized as a fuzzy measure on given by for any .

Residuum Based Fuzzy Integrals

In the following part, we introduce two types of fuzzy (qualitative) integrals based on the operation of residuum. The first type of this fuzzy integral was proposed by Dvořák and Holčapek in [4] for fuzzy quantifiers modelling, the second type was proposed by Dubois, Prade and Rico in [3], known also under the name desintegral, for the reasoning with a decreasing evaluation scale. A comparison of both fuzzy integrals can be found in [10].

–Fuzzy Integral

The integrated functions are fuzzy sets on X. We consider a modified version of the original definition of the residuum based fuzzy integral presented in [4].

Definition 5

Let be a complementary fuzzy measure space, and let . The -fuzzy integral of f on X is given by

6

Note that the original definition in [4] and the previous definition of residuum based integrals coincide on MV-algebras. The following statement presents basic properties of -fuzzy integral.

Theorem 6

For any and , we have

• (i) if ;
• (ii);
• (iii);
• (iv).

Note that the inequality (iii) of the previous theorem becomes the equality in a complete MV-algebra. An equivalent, and useful from the practical point of view, definition of –fuzzy integrals can be obtained for --measurable functions.

Theorem 7

If be --measurable, then

7

We say that are comonotonic if and only if there is no pair such that and simultaneously . Note that the Sugeno integral preserves the infimum and supremum for the commonotonic functions, i.e., it is comonotonically minitive and comonotonically maxitive, (see, [7, Theorem 4.44]). A similar result for the residuum based fuzzy integral can be simply derived using the following lemma whose proof can be found in [9].

Lemma 1

Let L be linearly ordered, and let . Denote , where . Then is a chain with respect to , and if f and g are comonotonic, then or for any , where .

Theorem 8

Let be linearly ordered, and let be comonotonic --measurable functions. Then

8

Note that a dual formula to (8), where the infimum is replaced by the supremum and vice versa, is not true in general even if we restrict ourselves to linearly ordered Heyting algebra (cf. Theorem 3.4 in [9] for the multiplication based fuzzy integral).

–Fuzzy Integrals

The integrated functions are again fuzzy sets on X.

Definition 6

Let be a fuzzy measure space, and let . The -fuzzy integral of f on X is given by

9

Note that if , then , hence, the empty set has no influence on the value of the –fuzzy integral. The following statement presents basic properties of -fuzzy integral.

Theorem 9

For any and , we have

• (i) if ;
• (ii);
• (iii);
• (iv).

Note that the inequality (iv) of the previous theorem becomes the equality in a complete MV-algebra. An equivalent formula to (9) under the assumption of --measurability of functions is as follows.

Theorem 10

Let be a fuzzy measure space, and let be --measurable. Then

10

Lemma 2

Let be linearly ordered, and let . Denote , where . Then is a chain with respect to , and if f and g are comonotonic, then or for any , where .

Theorem 11

Let be linearly ordered, and let be comonotonic --measurable functions. Then

Integral Transforms for Lattice-Valued Functions

In this section, we propose four types of integral transforms for functions whose function values are evaluated in a complete residuated lattice. For their definitions, we use the residuum based fuzzy integral introduced in Sect. 3. The integral transforms transform fuzzy sets from to fuzzy sets from .

–Integral Transforms

In this part, we propose two types of integral transform based on –fuzzy integral. For their definitions we are inspired by a straightforward generalization of the lower and upper fuzzy transforms in terms of the multiplication based fuzzy integral presented in [9]. We start with the definition of –integral transforms merging an integral kernel and a transformed function by the multiplication operation.

Definition 7

Let be a complementary fuzzy measure space, and let be a semi-normal in the second component fuzzy relation. A map defined by

11

is called a –integral transform.

It is easy to see that a complementary measure and a semi-normal in the second component fuzzy relation K are parameters of –integral transform. The fuzzy relation K will be called the integral kernel, which corresponds to the standard notation in the theory of integral transforms. Note that the semi-normality in the second component of integral kernels is considered as a natural assumption avoiding the trivial case, namely, if for any and some , we trivially obtain as a consequence of (see Theorem 6(b)).

Remark 1

If an integral kernel K is normal in the second component for any and, moreover, the family of sets forms a partition of X, the family of fuzzy sets is called a fuzzy partition of X, which is a crucial concept in the definition of lower and upper fuzzy transforms [20]. In this article, we significantly relax the concept of fuzzy partition because we require only that for any . Nevertheless, the fulfillment of certain integral transforms properties usually forces to introduce specific conditions for integral kernels (see Theorems 4.4 and 4.7 in [9]).

The following theorem shows basic properties of –integral transforms.

Theorem 12

For any and , we have

• (i) if ;
• (ii);
• (iii);
• (iv);
• (v).

Moreover, if L is a complete MV-algebra, the equality in (iv) holds.

Proof

The first three statements are trivial consequences of the monotonicity of the operation (i.e., monotonically non-decreasing) and the –fuzzy integral (Theorem 6(i)). Using Theorem 6(iii) and the commutativity of , for any , we have

Moreover, if L is a complete MV-algebra, the previous inequality becomes the equality and hence, the equality in (iv) holds. Since , using (i) and (iv) of Theorem 6, one can simply prove (v).    

Let us continue with another type of –integral transforms, where the integral kernels are combined with the transformed functions using the residuum operation.

Definition 8

Let be a complementary fuzzy measure space, and let be a semi-normal in the second component fuzzy relation. A map defined by

12

is called a -integral transform.

Note that if K is not semi-normal in the second component, i.e., for some and any , we trivially obtain as a consequence of (see Theorem 6(b)). In what follows, we present some basic properties of the –integral transform.

Theorem 13

For any and , we have

• (i) if ;
• (ii);
• (iii);
• (iv);
• (v).

Proof

The first three statements are trivial consequences of the monotonicity of the operation (i.e., monotonically non-decreasing in the second component) and the –fuzzy integral (Theorem 6(i)). Since , then using (i) and (iii) of Theorem 6, for any , we obtain

Since , using Theorem 6(iv), one can simply prove (v).    

One could see that, although, the –integral transforms are defined by different operations, i.e., and , their basic properties coincide.

We showed in Theorem 8 that under the assumption of the linearity of complete residuated lattices, the –fuzzy integral is a linear operator in the sense that the –fuzzy integral of the supremum of comonotonic functions is the infimum of –fuzzy integrals of these functions. The linearity property of –fuzzy integral can be used to prove the analogous property for –integral transforms.

Theorem 14

Let L be a linearly ordered and assume that the algebra is closed over arbitrary unions. Let be --measurable for any . If and are comonotonic for , then

13

–Integral Transform

Similarly to the previous subsection we propose two types of integral transforms based now on the –fuzzy integral. Again we start with the definition of integral transform, where integral kernels and transformed functions are merged by the multiplication operation.

Definition 9

Let be a fuzzy measure space, and let be a semi-normal in the second component fuzzy relation. A map defined by

14

is called a –integral transform.

The following theorem shows several basic properties of –integral transforms.

Theorem 15

For any and , we have

• (i) if ;
• (ii);
• (iii);
• (iv);
• (v);

Proof

Similarly to the proof of Theorem 12 one can simply prove all the statements using the properties of –fuzzy integral presented in Theorem 9.    

Definition 10

Let be a fuzzy measure space, and let be a semi-normal in the second component fuzzy relation. A map defined by

15

is called a –integral transform.

Some of basic properties of -integral transform are presented in the following theorem.

Theorem 16

For any and , we have

• (i) if ;
• (ii);
• (iii);
• (iv);
• (v).

Moreover, if L is a complete MV-algebra, the equality in (v) holds.

Again one could notice that although, the –integral transforms are defined by different operations, their basic properties are identical. The following linear property ensured for comonotonic functions is a straightforward consequence of Theorem 11.

Theorem 17

Let L be a linearly ordered and assume that the algebra is closed over arbitrary unions. Let be --measurable for any . If and are comonotonic for , then

16

Conclusion

In this article, we introduced four types of integral transforms, where we used the residuum based fuzzy (qualitative) integrals, namely, the –fuzzy integral proposed by Dvořák and Holčapek in [4] and the –fuzzy integral proposed by Dubois, Prade and Rico in [3]. We presented some of the basic properties of the residuum based fuzzy integrals including a linearity property for the comonotonic functions which holds in the linearly ordered complete residuated lattices. Using these properties we provided an initial analysis of elementary properties of proposed integral transforms. The further development of the theory of integral transforms for residuated lattice-valued functions is a subject of our future research, where, among others, we want to focus on the seeking of inverse integral kernels to be able to approximate the original functions from the transformed functions. Our motivation comes from the relationship between the lower and upper fuzzy transforms and their related inverse fuzzy transforms.