## Function Spaces, Hyperspaces, and Asymmetric and Fuzzy Structures

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Dong Qiu, Weiquan Zhang, "Convexity Invariance of Fuzzy Sets under the Extension Principles", *Journal of Function Spaces*, vol. 2012, Article ID 849104, 13 pages, 2012. https://doi.org/10.1155/2012/849104

# Convexity Invariance of Fuzzy Sets under the Extension Principles

**Academic Editor:**Manuel Sanchis

#### Abstract

We discuss the convexity invariance of fuzzy sets under the extension principles. Particularly, we give a necessary and sufficient condition for a mapping to be an inverse *-convex transformation, and also obtain some sufficient conditions for a mapping to be an *-convex transformation. Two applications are given to illustrate the obtained results. Finally, we give some applications of the main results to the hyperstructure convexity invariance of type 2 fuzzy sets under hyperalgebra operations, and to the convexity invariance of fuzzy numbers under basic arithmetic operations.

#### 1. Introduction

As a suitable mathematical model to handle vagueness and uncertainty, fuzzy set theory is emerging as a powerful theory and has attracted the attention of many researchers and practitioners who contributed to its development and applications [1–12]. Convexity plays the most useful role in the theory and applications of fuzzy sets; and the research on convexity and generalized convexity is one of the most important aspects of fuzzy set theory [9, 10, 13–16].

Moreover, the extension principle for fuzzy sets is in essence a basic identity which allows the domain of the definition of a mapping or a relation to be extended from points in a set to fuzzy subsets of . This, in fact, is the underlying basis for many operations of the most basic concepts of fuzzy set theory such as arithmetic operations of fuzzy numbers, hyperalgebra operations of type 2 fuzzy sets, and synthetic operations of fuzzy relations.

Hence it is important and interesting to study the convexity invariance of concepts of fuzzy set theory under the extension principles, which has been explored by many researchers [4, 5, 8, 16–22]. In this paper, we will discuss the convexity invariance of fuzzy sets under the extension principles in the general case. We hope that our results of the convexity invariance may lead to significant, new, and innovative results in those related fields [1, 3, 6, 8].

#### 2. Preliminaries

In this section, some basic definitions of -norms, extension principles, convex fuzzy sets, and type 2 fuzzy sets are reviewed. Throughout this paper, the letters and will denote the set of all positive integers and real numbers, respectively. A fuzzy set in a universe of discourse is characterized by a membership function which associates with each point in a real number in the interval , with the value of at representing the “grade of membership" of in [10]. The symbol denotes the family of all fuzzy subsets of a set .

*Definition 2.1 (see [1, 12]). * For a fuzzy set in a universe and each , the strong -level set of , denoted by , is defined as
Specially, the set is called the support set of fuzzy set .

According to Zadeh’s definition, a type 2 fuzzy set is a fuzzy set with a fuzzy membership function.

*Definition 2.2 (see [21]). *A fuzzy set of type-2, , in a universe of discourse is characterized by a fuzzy membership function as
where the value is a fuzzy grade and is a fuzzy set in the unit interval . A fuzzy grade is represented by
where is a membership function for fuzzy grade and is defined as

*Definition 2.3 (see [2]). * A -norm is a binary operation on the unit interval , that is, a function , such that for all , the following four axioms are satisfied: (T-1) (T-2) , whenever and (T-3) (T-4) .

A -norm is said to be continuous if it is a continuous function on .

*Example 2.4 (see [2]). *The following are the four basic -norms , , , and given by, respectively,
The -norms , , and are continuous, but is not.

A -norm can be extended (by associativity) in a unique way to a -ary operation taking for the value defined recurrently by

*Definition 2.5 2.5 (see [14]). *Let be a real linear space. A fuzzy set of is said to be a convex fuzzy set if for all and ,
Yuan and Lee [9] introduced the notion of -norm convexity for fuzzy sets as follows.

*Definition 2.6 2.6 (see [9]). *Let be a real linear space, and let be a -norm. A fuzzy set of is said to be an -convex fuzzy set if for all and ,
A convex fuzzy set is indeed an -convex fuzzy set according to Definition 2.6.

*Definition 2.7 (see [1, 21, 23) (Zadeh’s extension principle). * Let be a mapping from a universe to another universe . Two mappings from to and from to can be induced by as follows, respectively:
for all , , , and , where is the inverse image set of .

*Definition 2.8 (see [1, 21, 23) (Zadeh’s multivariable extension principle). *Let be a Cartesian product of universes, , and let be a mapping from to a universe such that . Then a mapping from to can be induced by as follows:
for all and all -tuple of fuzzy sets which are fuzzy sets in , respectively.

The multivariable extension principle as stated in Definition 2.8 can and has been generalized by using sup-(*t*-norm) convolution rather than sup-min convolution in [17].

*Definition 2.9 (see [1, 17) (generalized multivariable extension principle). *Let be a -norm, let be a Cartesian product of universes, , and let be a mapping from to a universe such that . Then a mapping from to can be induced by as follows:
for all and all -tuple of fuzzy sets which are fuzzy sets in , respectively.

In set theory, a total order is a binary relation (here denoted by infix ) on some set X. The relation is transitive, antisymmetric, and total. A set paired with a total order is called a totally ordered set. If is a totally ordered set, the order topology on is generated by the subbase of “open rays" for all in [24].

*Remark 2.10. *It should be noted that a fuzzy set in a totally ordered set is an -convex fuzzy set if and only if for all with ,

*Definition 2.11. *Let be a real linear space, be a -norm, and let be a mapping from to , a totally ordered set equipped with the order topology. The mapping is said to be an -convex transformation from to if the induced mapping (by Zadeh’s extension principle) transforms every -convex fuzzy set of into an -convex fuzzy set of ; the mapping is said to be an inverse -convex transformation from to if the induced mapping (by Zadeh’s extension principle) transforms every -convex fuzzy set of into an -convex fuzzy set of .

*Definition 2.12. *Let be a Cartesian product of real linear topological spaces , let be a *t*-norm, and let be a mapping from to , a totally ordered set equipped with the order topology. The mapping is said to be a multivariable -convex transformation from to if the induced mapping (by the generalized multivariable extension principle) transforms every n-tuple of -convex fuzzy sets of into an -convex fuzzy set of , where are -convex fuzzy sets in , respectively.

#### 3. Main Results

Let be a real linear topological space. For arbitrary two points , the line segment joining and is the set of all points of the form , .

Theorem 3.1. *Let be a real linear topological space, let be a continuous -norm, and let be a mapping from to , a totally ordered set equipped with the order topology. If the restriction of to every line segment , , is continuous, then is an -convex transformation. *

* Proof. *Let be an arbitrary -convex fuzzy set of . For any given three points with , if either or is an empty set, then it is obvious that
Thus, without loss of generality, suppose that and are nonempty sets. For any two points and , the restriction is continuous. In addition, we have that and . From the generalized intermediate value theorem [24], it follows that there is a point with which implies that is nonempty. Then by the -convexity of , we have
Then the continuity and the monotonicity of and the arbitrariness of in imply that
Similarly from the continuity and the monotonicity of and the arbitrariness of in , it follows that
which implies that is an -convex fuzzy set of . This completes the whole proof.

Corollary 3.2. *Let be a continuous t-norm. If is a continuous mapping from to , a totally ordered set equipped with the order topology, then is an -convex transformation. *

* Proof. *It is obvious that satisfies the conditions of Theorem 3.1. Then the desired result follows quickly from Theorem 3.1.

In Corollary 3.2, the continuity of is not a necessary condition for a mapping to be an -convex transformation. To give such a counterexample, we need some lemmas.

Lemma 3.3. *A fuzzy set in a real linear space is an -convex fuzzy set if and only if all of its strong -level sets, , are convex sets. *

* Proof. *Suppose that is an -convex fuzzy set. For every and any and , the inequalities
imply that the point belongs to . Thus, is convex.

Conversely, suppose that all of the strong -level sets, , are convex sets. For any and , if either or is , then it is obvious that
Thus, without loss of generality, suppose that and are not . Taking , where , we have that and belong to the convex set , which implies that for every ,
By the arbitrariness of , we obtain that for every ,
Thus, is an -convex fuzzy set. We complete the whole proof here.

Lemma 3.4 (see [23]). *If is a mapping from a universe to another universe . Then for any , the following equation:
**
holds for all . *

*Example 3.5. *Define the function by
This function is not continuous at because the limit of as tends to 0 does not exist. However, it is an -convex transformation.

In order to show this, let be an arbitrary -convex fuzzy set in . Thus, Lemma 3.3 implies that every strong -level set is a convex set in . In addition, the convex sets in are intervals. If the interval does not contain , then is an interval because is a continuous function on . If the interval contains , then because any neighborhood of can always include an interval or an interval for sufficiently large .

Thus, by Lemma 3.4, we have proved that all of the strong -level sets, , are convex sets, which shows that is an -convex fuzzy set in . Then is an -convex transformation.

*Definition 3.6. *Let be a real linear topological space, let be a -norm, and let be a mapping from to , a totally ordered set equipped with the order topology. For a line segment , define the mapping by . is said to be monotonous on a line segment , or is said to be monotonous, if is monotonous on with respect to .

Theorem 3.7. *Let be a real linear topological space, be a -norm, and be a mapping from to , a totally ordered set equipped with the order topology. is an inverse -convex transformation if and only if the restriction of to every line segment is monotonous. *

* Proof. *Suppose that the restriction of to every line segment is monotonous. Let be an arbitrary -convex fuzzy set in . For any and , we have that
or
because is monotonous. Thus, from the -convexity of , it follows that
which implies that is an -convex fuzzy set. Therefore, is an inverse -convex transformation.

Conversely, suppose that is an inverse -convex transformation. If there is a line segment on which is not monotonous. Then there is a which satisfies but (or but ).

Now define the fuzzy set in by
It is easy to check that is an -convex fuzzy set. But is not an -convex fuzzy set in because
This is a contradiction. Thus, we can complete the whole proof here.

Corollary 3.8. *Let be a real linear topological space, and let be a t-norm. If is a real linear functional, then is an inverse -convex transformation. *

* Proof. *For any line segment , the function defined as
satisfies
for all . From the arbitrariness of the line segment and the above inequalities, one can easily deduce that is monotonous on . Then by Theorem 3.7, we get the desired result.

Theorem 3.9. *Let be a Cartesian product of real linear topological spaces , let be a continuous t-norm, and let be a mapping from to , a totally ordered set equipped with the order topology. If the restriction of to every line segment , , is continuous, then is a multivariable -convex transformation. *

* Proof. *Let be -convex fuzzy sets in , respectively. For any given three points with , if either or is an empty set, then it is obvious that
Thus, without loss of generality, suppose that and are nonempty sets. For any two points and , define the mapping by
The mapping is continuous on with respect to because the restriction is continuous. Since and , from the generalized intermediate value theorem [24], it follows that there is a such that which implies that is nonempty. Then by the commutativity and the associativity of and the -convexities of , we have that
Combining with the continuity and the monotonicity of and the arbitrariness of in , the inequality (3.20) implies that
Similarly from the continuity and the monotonicity of and the arbitrariness of in , it follows from the above inequality that
which implies that is an -convex fuzzy set of . We complete the whole proof here.

#### 4. Applications and Examples

Now we give some applications of the main results to the hyperstructure convexity invariance of type-2 fuzzy sets under hyperalgebra operations, and to the convexity invariance of fuzzy numbers under basic arithmetic operations.

##### 4.1. Convexity Invariance of Type-2 Fuzzy Sets under Set Operations

Let be a -norm, and let and be fuzzy grades for type-2 fuzzy sets, and , represented as Then the hyperalgebra operations for type-2 fuzzy sets are expressed as follows by using the extension principles, generalized union: and generalized intersection: It should be noted that a type-2 fuzzy set is -convex if all of its fuzzy grades are -convex fuzzy sets. The following theorem is a generalization of the results in [4, 8].

Theorem 4.1. *Let be a continuous t-norm. If and are two -convex type-2 fuzzy sets, then , are -convex type-2 fuzzy sets. *

* Proof. *Define two mappings as and for , respectively. It is easy to see that the two mappings are continuous on . Thus, by Theorem 3.9, we can get the desired results.

Corollary 4.2 (see [4, 8]). *If and are two convex type-2 fuzzy sets, then , are convex type-2 fuzzy sets. *

Corollary 4.3. *If and are two -convex type-2 fuzzy sets, then , are -convex type-2 fuzzy sets. *

Corollary 4.4. *If and are two -convex type-2 fuzzy sets, then , are -convex type-2 fuzzy sets. *

Corollary 4.5. *If and are two -convex type-2 fuzzy sets, then , are -convex type-2 fuzzy sets. *

##### 4.2. Convexity Invariance of Fuzzy Numbers under Basic Arithmetic Operations

In the early literature, fuzzy number is defined as a convex fuzzy set in a real line [1, 5, 11, 12, 22]. Although this definition is very often modified nowadays, the convexity is always one of the conditions for a fuzzy set to be a fuzzy number. The arithmetic operations of for fuzzy numbers can be defined by using the Zadeh’s multivariable extension principle.

Define four mappings , and by , , and , for , respectively. It is easy to see that the first three mappings are continuous on , and is continuous on (or ). Thus, let the -norm be , and let the mapping be one of the above four mappings in Theorem 3.9 in turn, one can get the following theorem.

Theorem 4.6 4.6 (see [1, 5, 12]). *If and are fuzzy numbers in a real line , then , , and are fuzzy numbers. In addition, if does not belong to the support set of , then is a fuzzy number. *

#### 5. Conclusions

In this work, we have discussed the convexity invariance of fuzzy sets under the extension principles. Particularly, we have given a necessary and sufficient condition for a mapping to be an inverse -convex transformation and have also obtained some sufficient conditions for a mapping to be an -convex transformation. Finally, two applications are given to illustrate the obtained results. The properties of the introduced concept, -convex transformation, certainly deserve further investigation.

#### Acknowledgments

The authors thank the anonymous reviewers for their valuable comments. This work was supported by the National Natural Science Foundation of China (Grant no. 11201512) and the Natural Science Foundation Project of CQ CSTC (cstc2012jjA00001).

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#### Copyright

Copyright © 2012 Dong Qiu and Weiquan Zhang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.