Convexity Invariance of Fuzzy Sets under the Extension Principles

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 𝑈 to fuzzy subsets of 𝑈. 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 𝑡-norms, extension principles, convex fuzzy sets, and type 2 fuzzy sets are reviewed. Throughout this paper, the letters ℕ and ℝ will denote the set of all positive integers and real numbers, respectively. A fuzzy set 𝐴 in a universe of discourse 𝑋 is characterized by a membership function 𝐴(𝑥) which associates with each point in 𝑋 a real number in the interval [0,1], with the value of 𝐴(𝑥) at 𝑥 representing the “grade of membership" of 𝑥 in 𝐴 [10]. The symbol ℱ(𝑋) denotes the family of all fuzzy subsets of a set 𝑋.

Definition 2.1 (see [1, 12]). For a fuzzy set 𝐴 in a universe 𝑋 and each 𝜆∈[0,1), the strong 𝜆-level set of 𝐴, denoted by [𝐴]_̇𝜆, is defined as [𝐴]_̇𝜆={𝑥∈𝑋∣𝐴(𝑥)>𝜆}. (2.1) Specially, the set [𝐴]_̇0 is called the support set of fuzzy set 𝐴.

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, 𝐴, in a universe of discourse 𝑋 is characterized by a fuzzy membership function 𝜇_𝐴 as 𝜇_𝐴∶𝑋→[0,1]^[0,1], (2.2) where the value 𝜇_𝐴(𝑥) is a fuzzy grade and is a fuzzy set in the unit interval [0,1]. A fuzzy grade 𝜇_𝐴(𝑥) is represented by 𝜇_𝐴(𝑥)=𝑓(𝑢)𝑢,𝑢∈[0,1], (2.3) where 𝑓 is a membership function for fuzzy grade 𝜇_𝐴(𝑥) and is defined as 𝑓∶[0,1]→[0,1]. (2.4)

Definition 2.3 (see [2]). A 𝑡-norm is a binary operation on the unit interval [0,1], that is, a function ∗∶[0,1]²→[0,1], such that for all 𝑎,𝑏,𝑐,𝑑∈[0,1], the following four axioms are satisfied: (T-1) 𝑎∗1=𝑎,(boundarycondition) (T-2) 𝑎∗𝑏≤𝑐∗𝑑, whenever 𝑎≤𝑐 and 𝑏≤𝑑,(monotonicity) (T-3) 𝑎∗𝑏=𝑏∗𝑎,(commutativity) (T-4) 𝑎∗(𝑏∗𝑐)=(𝑎∗𝑏)∗𝑐(associativity).
A 𝑡-norm ∗ is said to be continuous if it is a continuous function on [0,1]².

Example 2.4 (see [2]). The following are the four basic 𝑡-norms ∗_𝑀, ∗_𝑃, ∗_𝐿, and ∗_𝐷 given by, respectively, 𝑥∗_𝑀𝑦=min𝑥,𝑦,(minimum)𝑥∗_𝑃𝑦=𝑥⋅𝑦,product𝑥∗_𝐿𝑦=max𝑥+𝑦−1,0,(Lukasiewiczt𝑡-norm)𝑥∗_𝐷∈[)𝑦=0if𝑥,𝑦0,1²,min𝑥,𝑦otherwise.drasticproduct (2.5) The 𝑡-norms ∗_𝑀, ∗_𝑃, and ∗_𝐿 are continuous, but ∗_𝐷 is not.
A 𝑡-norm ∗ can be extended (by associativity) in a unique way to a 𝑛-ary operation taking for (𝑥₁,…,𝑥_𝑛)∈[0,1]^𝑛(𝑛≥3) the value 𝑥₁∗⋯∗𝑥_𝑛 defined recurrently by 𝑥₁∗⋯∗𝑥_𝑛=𝑥₁∗⋯∗𝑥_𝑛−1∗𝑥_𝑛,∀𝑛≥3. (2.6)

Definition 2.5 2.5 (see [14]). Let 𝐸 be a real linear space. A fuzzy set 𝐴 of 𝐸 is said to be a convex fuzzy set if for all 𝑥,𝑦∈𝐸 and 𝜆∈[0,1], 𝐴𝜆𝑥+(1−𝜆)𝑦≥min𝐴(𝑥),𝐴𝑦. (2.7) Yuan and Lee [9] introduced the notion of 𝑡-norm convexity for fuzzy sets as follows.

Definition 2.6 2.6 (see [9]). Let 𝐸 be a real linear space, and let ∗ be a 𝑡-norm. A fuzzy set 𝐴 of 𝐸 is said to be an ∗-convex fuzzy set if for all 𝑥,𝑦∈𝐸 and 𝜆∈[0,1], 𝐴𝜆𝑥+(1−𝜆)𝑦≥𝐴(𝑥)∗𝐴𝑦. (2.8) A convex fuzzy set 𝐴 is indeed an ∗_𝑀-convex fuzzy set according to Definition 2.6.

Definition 2.7 (see [1, 21, 23) (Zadeh’s extension principle). Let 𝑓 be a mapping from a universe 𝑋 to another universe 𝑌. Two mappings 𝑓 from ℱ(𝑋) to ℱ(𝑌) and 𝑓⁻¹ from ℱ(𝑌) to ℱ(𝑋) can be induced by 𝑓 as follows, respectively: 𝑓(𝐴)(𝑣)=sup_{𝑓(𝑢)=𝑣}𝐴(𝑢)if𝑓⁻¹(𝑣)𝑓≠∅,0otherwise,₋₁(𝐵)(𝑢)=𝐵𝑓(𝑢), (2.9) for all 𝑣∈𝑌,   𝑢∈𝑋, 𝐴∈ℱ(𝑋), and 𝐵∈ℱ(𝑌), where 𝑓⁻¹(𝑣) is the inverse image set of 𝑣.

Definition 2.8 (see [1, 21, 23) (Zadeh’s multivariable extension principle). Let 𝑋 be a Cartesian product of universes, 𝑋=𝑋₁×⋅⋅⋅×𝑋_𝑟, and let 𝑓 be a mapping from 𝑋 to a universe 𝑌 such that 𝑦=𝑓(𝑥₁,…,𝑥_𝑟). Then a mapping 𝑓 from ℱ(𝑋₁)×⋅⋅⋅×ℱ(𝑋_𝑟) to ℱ(𝑌) can be induced by 𝑓 as follows: 𝑓𝐴₁,…,𝐴_𝑟(𝑣)=0if𝑓⁻¹(𝑣)=∅,sup_{𝑓𝑢1,…,𝑢𝑟=𝑣}min𝐴₁𝑢₁,…,𝐴_𝑟𝑢_𝑟, (2.10) for all 𝑣∈𝑌 and all 𝑛-tuple of fuzzy sets (𝐴₁,…,𝐴_𝑟) which are fuzzy sets in 𝑋₁,…,𝑋_𝑟, 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 𝑡-norm, let 𝑋 be a Cartesian product of universes, 𝑋=𝑋₁×⋅⋅⋅×𝑋_𝑟, and let 𝑓 be a mapping from 𝑋 to a universe 𝑌 such that 𝑦=𝑓(𝑥₁,…,𝑥_𝑟). Then a mapping 𝑓 from ℱ(𝑋₁)×⋅⋅⋅×ℱ(𝑋_𝑟) to ℱ(𝑌) can be induced by 𝑓 as follows: 𝑓𝐴₁,…,𝐴_𝑟(𝑣)=0if𝑓⁻¹(𝑣)=∅,sup_{𝑓𝑢1,…,𝑢𝑟=𝑣}𝐴₁𝑢₁∗⋅⋅⋅∗𝐴_𝑟𝑢_𝑟, (2.11) for all 𝑣∈𝑌 and all 𝑛-tuple of fuzzy sets (𝐴₁,…,𝐴_𝑟) which are fuzzy sets in 𝑋₁,…,𝑋_𝑟, 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 𝑋 is a totally ordered set, the order topology on 𝑋 is generated by the subbase of “open rays" ()={},()={},𝑎,∞𝑥∣𝑎<𝑥−∞,𝑏𝑥∣𝑥<𝑏 (2.12) for all 𝑎,𝑏 in 𝑋 [24].

Remark 2.10. It should be noted that a fuzzy set 𝐴 in a totally ordered set 𝑌 is an ∗-convex fuzzy set if and only if for all 𝑥,𝑦,𝑧∈𝑌 with 𝑥≤𝑦≤𝑧, 𝜇𝑦≥𝜇(𝑥)∗𝜇(𝑧). (2.13)

Definition 2.11. Let 𝐸 be a real linear space, ∗ be a 𝑡-norm, and let 𝑓 be a mapping from 𝐸 to (𝑌,≥), a totally ordered set equipped with the order topology. The mapping 𝑓 is said to be an ∗-convex transformation from 𝐸 to 𝑌 if the induced mapping 𝑓 (by Zadeh’s extension principle) transforms every ∗-convex fuzzy set of 𝐸 into an ∗-convex fuzzy set of 𝑌; the mapping 𝑓 is said to be an inverse ∗-convex transformation from 𝐸 to 𝑌 if the induced mapping 𝑓⁻¹ (by Zadeh’s extension principle) transforms every ∗-convex fuzzy set of 𝑌 into an ∗-convex fuzzy set of 𝐸.

Definition 2.12. Let 𝐸 be a Cartesian product of real linear topological spaces 𝐸=𝐸₁×⋅⋅⋅×𝐸_𝑟, let ∗ be a t-norm, and let 𝑓 be a mapping from 𝐸 to (𝑌,≥), a totally ordered set equipped with the order topology. The mapping 𝑓 is said to be a multivariable ∗-convex transformation from 𝐸 to 𝑌 if the induced mapping 𝑓 (by the generalized multivariable extension principle) transforms every n-tuple of ∗-convex fuzzy sets (𝐴₁,…,𝐴_𝑟) of 𝐸 into an ∗-convex fuzzy set of 𝑌, where (𝐴₁,…,𝐴_𝑟) are ∗-convex fuzzy sets in 𝑋₁,…,𝑋_𝑟, respectively.

3. Main Results

Let 𝐸 be a real linear topological space. For arbitrary two points 𝑥,𝑦∈𝐸, the line segment 𝑥𝑦 joining 𝑥 and 𝑦 is the set of all points of the form 𝑡𝑥+(1−𝑡)𝑦, 𝑡∈[0,1].

Theorem 3.1. Let 𝐸 be a real linear topological space, let ∗ be a continuous 𝑡-norm, and let 𝑓 be a mapping from 𝐸 to (𝑌,≥), a totally ordered set equipped with the order topology. If the restriction of 𝑓 to every line segment 𝑥𝑦, 𝑓|_𝑥𝑦, is continuous, then 𝑓 is an ∗-convex transformation.

Proof. Let 𝐴 be an arbitrary ∗-convex fuzzy set of 𝐸. For any given three points 𝑥,𝑦,𝑧∈𝑌 with 𝑥≤𝑦≤𝑧, if either 𝑓⁻¹(𝑥) or 𝑓⁻¹(𝑧) is an empty set, then it is obvious that 𝑓(𝐴)𝑦≥𝑓(𝐴)(𝑥)∗𝑓(𝐴)(𝑧)=0. (3.1) Thus, without loss of generality, suppose that 𝑓⁻¹(𝑥) and 𝑓⁻¹(𝑧) are nonempty sets. For any two points 𝑢∈𝑓⁻¹(𝑥) and 𝑣∈𝑓⁻¹(𝑧), the restriction 𝑓|_𝑢𝑣 is continuous. In addition, we have that 𝑓(𝑢)=𝑥 and 𝑓(𝑣)=𝑧. From the generalized intermediate value theorem [24], it follows that there is a point 𝑤∈𝑢𝑣 with 𝑓(𝑤)=𝑦 which implies that 𝑓⁻¹(𝑦) is nonempty. Then by the ∗-convexity of 𝐴, we have 𝑓(𝐴)𝑦=sup_{𝑚∈𝑓−1𝑦}𝐴(𝑚)≥𝐴(𝑤)≥𝐴(𝑢)∗𝐴(𝑣). (3.2) Then the continuity and the monotonicity of ∗ and the arbitrariness of 𝑢 in 𝑓⁻¹(𝑥) imply that 𝑓(𝐴)𝑦≥sup_{𝑢∈𝑓−1(𝑥)}(𝐴(𝑢)(𝑣=∗𝐴))sup_{𝑢∈𝑓−1(𝑥)}𝐴(𝑢)(𝑣)=𝑓(𝐴𝑥(𝑣).∗𝐴)()∗𝐴 (3.3) Similarly from the continuity and the monotonicity of ∗ and the arbitrariness of 𝑣 in 𝑓⁻¹(𝑧), it follows that 𝑓(𝐴)𝑦≥sup_{𝑣∈𝑓−1(𝑧)}𝑓(𝐴𝑥)(𝑣=𝑓(𝐴𝑥)∗)(∗𝐴))(sup_{𝑣∈𝑓−1(𝑧)}𝐴(𝑣)=𝑓(𝐴)(𝑥)∗𝑓(𝐴)(𝑧), (3.4) which implies that 𝑓(𝐴) is an ∗-convex fuzzy set of 𝑌. This completes the whole proof.

Corollary 3.2. Let ∗ be a continuous t-norm. If 𝑓 is a continuous mapping from ℝ to (𝑌,≥), a totally ordered set equipped with the order topology, then 𝑓 is an ∗-convex transformation.

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

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

Lemma 3.3. A fuzzy set 𝐴 in a real linear space 𝐸 is an ∗_𝑀-convex fuzzy set if and only if all of its strong 𝜆-level sets, [𝐴]_̇𝜆, are convex sets.

Proof. Suppose that 𝐴 is an ∗_𝑀-convex fuzzy set. For every 𝜆∈[0,1) and any 𝑥,𝑦∈[𝐴]_̇𝜆 and 𝛼∈[0,1], the inequalities 𝐴𝛼𝑥+(1−𝛼)𝑦≥min𝐴(𝑥),𝐴𝑦>𝜆 (3.5) imply that the point 𝛼𝑥+(1−𝛼)𝑦 belongs to [𝐴]_̇𝜆. Thus, [𝐴]_̇𝜆 is convex.
Conversely, suppose that all of the strong 𝜆-level sets, [𝐴]_̇𝜆, are convex sets. For any 𝑥,𝑦∈𝐸 and 𝛼∈[0,1], if either 𝐴(𝑥) or 𝐴(𝑦) is 0, then it is obvious that 𝐴𝛼𝑥+(1−𝛼)𝑦≥min𝐴(𝑥),𝐴𝑦=0. (3.6) Thus, without loss of generality, suppose that 𝐴(𝑥) and 𝐴(𝑦) are not 0. Taking 𝜆=min(𝐴(𝑥),𝐴(𝑦))−𝜀, where 𝜀∈(0,min(𝐴(𝑥),𝐴(𝑦))], we have that 𝑥 and 𝑦 belong to the convex set [𝐴]_̇𝜆, which implies that for every 𝛼∈[0,1], 𝐴𝛼𝑥+(1−𝛼)𝑦>𝜆=min𝐴(𝑥),𝐴𝑦−𝜀. (3.7) By the arbitrariness of 𝜀, we obtain that for every 𝛼∈[0,1], 𝐴𝛼𝑥+(1−𝛼)𝑦≥min𝐴(𝑥),𝐴𝑦. (3.8) Thus, 𝐴 is an ∗_𝑀-convex fuzzy set. We complete the whole proof here.

Lemma 3.4 (see [23]). If 𝑓 is a mapping from a universe 𝑋 to another universe 𝑌. Then for any 𝐴∈ℱ(𝑋), the following equation: 𝑓(𝐴)_̇𝜆=𝑓[𝐴]_̇𝜆 (3.9) holds for all 𝜆∈[0,1).

Example 3.5. Define the function 𝑓∶ℝ→[−1,1] by 𝑓(𝑥)=1sin𝑥if𝑥≠0,0𝑥=0. (3.10) This function is not continuous at 𝑥=0 because the limit of 𝑓(𝑥) as 𝑥 tends to 0 does not exist. However, it is an ∗_𝑀-convex transformation.
In order to show this, let 𝐴 be an arbitrary ∗_𝑀-convex fuzzy set in ℝ. Thus, Lemma 3.3 implies that every strong 𝜆-level set [𝐴]_̇𝜆 is a convex set in ℝ. In addition, the convex sets in ℝ are intervals. If the interval [𝐴]_̇𝜆 does not contain 0, then 𝑓([𝐴]_̇𝜆) is an interval because 𝑓 is a continuous function on [𝐴]_̇𝜆. If the interval [𝐴]_̇𝜆 contains 0, then 𝑓([𝐴]_̇𝜆)=[−1,1] because any neighborhood of 0 can always include an interval [1/2(𝑛+1)𝜋,1/2𝑛𝜋] or an interval [−1/2𝑛𝜋,−1/2(𝑛+1)𝜋] for sufficiently large 𝑛∈ℕ.
Thus, by Lemma 3.4, we have proved that all of the strong 𝜆-level sets, [𝑓(𝐴)]_̇𝜆, are convex sets, which shows that 𝑓(𝐴) is an ∗_𝑀-convex fuzzy set in [−1,1]. Then 𝑓 is an ∗_𝑀-convex transformation.

Definition 3.6. Let 𝐸 be a real linear topological space, let ∗ be a 𝑡-norm, and let 𝑓 be a mapping from 𝐸 to (𝑌,≥), a totally ordered set equipped with the order topology. For a line segment 𝑥𝑦, define the mapping 𝑔∶[0,1]→𝑌 by 𝑔(𝑡)=𝑓(𝑡𝑥+(1−𝑡)𝑦). 𝑓 is said to be monotonous on a line segment 𝑥𝑦, or 𝑓|_𝑥𝑦 is said to be monotonous, if 𝑔(𝑡) is monotonous on [0,1] with respect to 𝑡.

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

Proof. Suppose that the restriction of 𝑓 to every line segment is monotonous. Let 𝐵 be an arbitrary ∗-convex fuzzy set in 𝑌. For any 𝑢,𝑣∈𝑋 and 𝑡∈[0,1], we have that 𝑓(𝑢)≤𝑓(𝑡𝑢+(1−𝑡)𝑣)≤𝑓(𝑣) (3.11) or 𝑓(𝑣)≤𝑓(𝑡𝑢+(1−𝑡)𝑣)≤𝑓(𝑢), (3.12) because 𝑓|_𝑢𝑣 is monotonous. Thus, from the ∗-convexity of 𝐵, it follows that 𝑓⁻¹(𝐵()𝑣)𝑓(()𝑣)𝑓(𝑢)𝑓(𝑣)=𝑓)(𝑡𝑢+1−𝑡=𝐵𝑡𝑢+1−𝑡≥𝐵∗𝐵₋₁(𝐵)(𝑢)∗𝑓⁻¹(𝐵)(𝑣), (3.13) which implies that 𝑓⁻¹(𝐵) is an ∗-convex fuzzy set. Therefore, 𝑓 is an inverse ∗-convex transformation.
Conversely, suppose that 𝑓 is an inverse ∗-convex transformation. If there is a line segment 𝑢𝑣 on which 𝑓 is not monotonous. Then there is a 𝑡₀∈(0,1) which satisfies 𝑓(𝑢)<𝑓(𝑡₀𝑢+(1−𝑡₀)𝑣) but 𝑓(𝑡₀𝑢+(1−𝑡₀)𝑣)>𝑓(𝑣) (or 𝑓(𝑢)>𝑓(𝑡₀𝑢+(1−𝑡₀)𝑣) but 𝑓(𝑡₀𝑢+(1−𝑡₀)𝑣)<𝑓(𝑣)).
Now define the fuzzy set 𝐵 in 𝑌 by 𝐵(𝑥)=1if𝑥∈min𝑓(𝑢),𝑓(𝑣),max𝑓(𝑢),𝑓(𝑣),0otherwise. (3.14) It is easy to check that 𝐵 is an ∗-convex fuzzy set. But 𝑓⁻¹(𝐵) is not an ∗-convex fuzzy set in 𝐸 because 𝑓⁻¹(𝐵)𝑡₀𝑢+1−𝑡₀𝑣=𝐵𝑓𝑡₀𝑢+1−𝑡₀𝑣𝑓(𝑢)𝑓(𝑣)=𝑓=0<1=𝐵∗𝐵₋₁(𝐵)(𝑢)∗𝑓⁻¹(𝐵)(𝑣). (3.15) This is a contradiction. Thus, we can complete the whole proof here.

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

Proof. For any line segment 𝑥𝑦, the function 𝑔∶[0,1]→ℝ defined as 𝑔(𝑡)=𝑓𝑡𝑥+(1−𝑡)𝑦=𝑡𝑓(𝑥)+(1−𝑡)𝑓𝑦 (3.16) satisfies min𝑔(0),𝑔(1)≤𝑔(𝑡)≤max𝑔(0),𝑔(1), (3.17) for all 𝑡∈[0,1]. From the arbitrariness of the line segment 𝑥𝑦 and the above inequalities, one can easily deduce that 𝑔(𝑡) is monotonous on [0,1]. Then by Theorem 3.7, we get the desired result.

Theorem 3.9. Let 𝐸 be a Cartesian product of real linear topological spaces 𝐸=𝐸₁×⋯×𝐸_𝑟, let ∗ be a continuous t-norm, and let 𝑓 be a mapping from 𝐸 to (𝑌,≥), a totally ordered set equipped with the order topology. If the restriction of 𝑓 to every line segment 𝑥𝑦, 𝑓|_𝑥𝑦, is continuous, then 𝑓 is a multivariable ∗-convex transformation.

Proof. Let 𝐴₁,…,𝐴_𝑟 be ∗-convex fuzzy sets in 𝐸₁,…,𝐸_𝑟, respectively. For any given three points 𝑥,𝑦,𝑧∈𝑌 with 𝑥≤𝑦≤𝑧, if either 𝑓⁻¹(𝑥) or 𝑓⁻¹(𝑧) is an empty set, then it is obvious that 𝑓𝐴₁,…,𝐴_𝑟𝑦≥0=𝑓𝐴₁,…,𝐴_𝑟(𝑥)∗𝑓𝐴₁,…,𝐴_𝑟(𝑧). (3.18) Thus, without loss of generality, suppose that 𝑓⁻¹(𝑥) and 𝑓⁻¹(𝑧) are nonempty sets. For any two points 𝑢=(𝑢₁,…,𝑢_𝑟)∈𝑓⁻¹(𝑥) and 𝑣=(𝑣₁,…,𝑣_𝑟)∈𝑓⁻¹(𝑧), define the mapping 𝑔∶[0,1]→𝑌 by 𝑔(𝑡)(()𝑣)𝑡𝑢=𝑓𝑡𝑢+1−𝑡=𝑓₁,…,𝑢_𝑟+(1−𝑡)𝑣₁,…,𝑣_𝑟=𝑓𝑡𝑢₁+(1−𝑡)𝑣₁,…,𝑡𝑢_𝑟+(1−𝑡)𝑣_𝑟. (3.19) The mapping 𝑔(𝑡) is continuous on [0,1] with respect to 𝑡 because the restriction 𝑓|_𝑢𝑣 is continuous. Since 𝑔(1)=𝑥 and 𝑔(0)=𝑧, from the generalized intermediate value theorem [24], it follows that there is a 𝑡₀∈[0,1] such that 𝑔(𝑡₀)=𝑦 which implies that 𝑓⁻¹(𝑦) is nonempty. Then by the commutativity and the associativity of ∗ and the ∗-convexities of 𝐴₁,…,𝐴_𝑟, we have that 𝑓𝐴₁,…,𝐴_𝑟𝑦=sup_{𝑤1,…,𝑤𝑟∈𝑓−1(𝑦)}𝐴₁𝑤₁∗⋅⋅⋅∗𝐴_𝑟𝑤_𝑟≥𝐴₁𝑡₀𝑢₁+1−𝑡₀𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑡₀𝑢₁+1−𝑡₀𝑣₁≥𝐴₁𝑢₁∗𝐴₁𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑢_𝑟∗𝐴_𝑟𝑣_𝑟=𝐴₁𝑢₁∗⋅⋅⋅∗𝐴_𝑟𝑢_𝑟∗𝐴₁𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑣_𝑟. (3.20) Combining with the continuity and the monotonicity of ∗ and the arbitrariness of 𝑢 in 𝑓⁻¹(𝑥), the inequality (3.20) implies that 𝑓𝐴₁,…,𝐴_𝑟𝑦≥sup_{𝑢∈𝑓−1(𝑥)}𝐴₁𝑢₁∗⋅⋅⋅∗𝐴_𝑟𝑢_𝑟∗𝐴₁𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑣_𝑟=sup_{𝑢∈𝑓−1(𝑥)}𝐴₁𝑢₁∗⋅⋅⋅∗𝐴_𝑟𝑢_𝑟∗𝐴₁𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑣_𝑟=𝑓𝐴₁,…,𝐴_𝑟(𝑥)∗𝐴₁𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑣_𝑟. (3.21) Similarly from the continuity and the monotonicity of ∗ and the arbitrariness of 𝑣 in 𝑓⁻¹(𝑧), it follows from the above inequality that 𝑓𝐴₁,…,𝐴_𝑟𝑦≥sup_{𝑣∈𝑓−1(𝑧)}𝑓𝐴₁,…,𝐴_𝑟(𝑥)∗𝐴₁𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑣_𝑟=𝑓𝐴₁,…,𝐴_𝑟(𝑥)∗sup_{𝑣∈𝑓−1(𝑧)}𝐴₁𝑣₁∗⋅⋅⋅∗𝐴_𝑟𝑣_𝑟=𝑓𝐴₁,…,𝐴_𝑟(𝑥)∗𝑓𝐴₁,…,𝐴_𝑟(𝑧), (3.22) which implies that 𝑓(𝐴₁,…,𝐴_𝑟) is an ∗-convex fuzzy set of 𝑌. 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 𝑡-norm, and let 𝜇_𝐴(𝑥) and 𝜇_𝐵 be fuzzy grades for type-2 fuzzy sets, 𝐴 and 𝐵, represented as 𝜇_𝐴(𝑥)=𝑔(𝑢)𝑢[],𝜇,𝑢∈0,1_𝐵(𝑥)ℎ(𝑣)=𝑣,𝑣∈[0,1]. (4.1) Then the hyperalgebra operations for type-2 fuzzy sets are expressed as follows by using the extension principles, generalized union: 𝐴∪_∗𝐵⟺𝜇_{𝐴∪∗𝐵}(𝑥)=𝜇_𝐴(𝑥)∪_∗𝜇_𝐵(𝑥)=𝑔(𝑢)𝑢∪_∗ℎ(𝑣)𝑣=sup_{𝑤=max(𝑢,𝑣)}𝑔(𝑢)∗ℎ(𝑣)𝑤 (4.2) and generalized intersection: 𝐴∩_∗𝐵⟺𝜇_{𝐴∩∗𝐵}(𝑥)=𝜇_𝐴(𝑥)∩_∗𝜇_𝐵(𝑥)=𝑔(𝑢)𝑢∩_∗ℎ(𝑣)𝑣=sup_{𝑤=min(𝑢,𝑣)}𝑔(𝑢)∗ℎ(𝑣)𝑤. (4.3) 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 𝐴 and 𝐵 are two ∗-convex type-2 fuzzy sets, then 𝐴∪_∗𝐵, 𝐴∩_∗𝐵 are ∗-convex type-2 fuzzy sets.

Proof. Define two mappings 𝑓_∪∗,𝑓_∩∗∶[0,1]²→[0,1] as 𝑓(𝑥,𝑦)=max(𝑥,𝑦) and 𝑓_{∩∗(𝑥,𝑦)}=min(𝑥,𝑦) for (𝑥,𝑦)∈[0,1]², respectively. It is easy to see that the two mappings are continuous on [0,1]². Thus, by Theorem 3.9, we can get the desired results.

Corollary 4.2 (see [4, 8]). If 𝐴 and 𝐵 are two convex type-2 fuzzy sets, then 𝐴∪_∗𝑀𝐵, 𝐴∩_∗𝑀𝐵 are convex type-2 fuzzy sets.

Corollary 4.3. If 𝐴 and 𝐵 are two ∗_𝑃-convex type-2 fuzzy sets, then 𝐴∪_∗𝑃𝐵, 𝐴∩_∗𝑃𝐵 are ∗_𝑃-convex type-2 fuzzy sets.

Corollary 4.4. If 𝐴 and 𝐵 are two ∗_𝐿-convex type-2 fuzzy sets, then 𝐴∪_∗𝐿𝐵, 𝐴∩_∗𝐿𝐵 are ∗_𝐿-convex type-2 fuzzy sets.

Corollary 4.5. If 𝐴 and 𝐵 are two ∗_𝐷-convex type-2 fuzzy sets, then 𝐴∪_∗𝐷𝐵, 𝐴∩_∗𝐷𝐵 are ∗_𝐷-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 ℝ [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 𝑓₊,𝑓₋,𝑓_×, and 𝑓_÷∶ℝ²→ℝ by 𝑓₊(𝑥,𝑦)=𝑥+𝑦, 𝑓₋(𝑥,𝑦)=𝑥−𝑦,𝑓_×(𝑥,𝑦)=𝑥×𝑦, and 𝑓_÷(𝑥,𝑦)=𝑥÷𝑦, for (𝑥,𝑦)∈ℝ², respectively. It is easy to see that the first three mappings are continuous on ℝ², and 𝑓_÷ is continuous on ℝ×(0,+∞) (or ℝ×(−∞,0)). Thus, let the 𝑡-norm be ∗_𝑀, 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 𝐴 and 𝐵 are fuzzy numbers in a real line ℝ, then 𝐴+𝐵, 𝐴−𝐵, and 𝐴×𝐵 are fuzzy numbers. In addition, if 0 does not belong to the support set [𝐵]_̇0 of 𝐵, then 𝐴÷𝐵 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).