Convexity Invariance of Fuzzy Sets under the Extension Principles

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.


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

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 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 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 where f is a membership function for fuzzy grade μ A x and is defined as

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 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 * M y min x, y , minimum 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 x 1 , . . ., x n ∈ 0, 1 n n ≥ 3 the value x 1 * • • • * x n defined recurrently by 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 X 1 × • • • × X r , and let f be a mapping from X to a universe Y such that y f x 1 , . . ., x r .Then a mapping f from F X 1 × • • • × F X r to F Y can be induced by f as follows: for all v ∈ Y and all n-tuple of fuzzy sets A 1 , . . ., A r which are fuzzy sets in X 1 , . . ., 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 X 1 ×•••×X r , and let f be a mapping from X to a universe Y such that y f x 1 , . . ., x r .Then a mapping for all v ∈ Y and all n-tuple of fuzzy sets A 1 , . . ., A r which are fuzzy sets in X 1 , . . ., 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"

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 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 Then the continuity and the monotonicity of * and the arbitrariness of u in f −1 x imply that

3.3
Similarly from the continuity and the monotonicity of * and the arbitrariness of v in f −1 z , it follows that 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 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 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 , By the arbitrariness of ε, we obtain that for every α ∈ 0, 1 , A αx 1 − α y ≥ min A x , A y .

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: holds for all λ ∈ 0, 1 .

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 * Mconvex 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 12 because f| uv is monotonous.Thus, from the * -convexity of B, it follows that

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 t 0 ∈ 0, 1 which satisfies

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 satisfies min g 0 , g 1 ≤ g t ≤ max g 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 E 1 × • • • × 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 A 1 , . . ., A r be * -convex fuzzy sets in E 1 , . . ., 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 Thus, without loss of generality, suppose that f −1 x and f −1 z are nonempty sets.For any two points u u

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 t 0 ∈ 0, 1 such that g t 0 y which implies that f −1 y is nonempty.Then by the commutativity and the associativity of * and the * -convexities of A 1 , . . ., A r , we have that

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 f A 1 , . . ., A r y

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 f A 1 , . . ., A r y which implies that f A 1 , . . ., A r is an * -convex fuzzy set of Y .We complete the whole proof here.

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.

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

4.1
Then the hyperalgebra operations for type-2 fuzzy sets are expressed as follows by using the extension principles, generalized union: and generalized intersection:

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 .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.

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 ÷ : R 2 → 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 ∈ R 2 , respectively.It is easy to see that the first three mappings are continuous on R 2 , 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.

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.
-1 a * 1 a, boundary condition 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 .

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.

Corollary 4 . 2 Corollary 4 . 3 .Corollary 4 . 4 .Corollary 4 . 5 .
see 4, 8 .If A and B are two convex type-2 fuzzy sets, then A∪ * M B, A∩ * M B are convex type-2 fuzzy sets.If A and B are two * P -convex type-2 fuzzy sets, then A∪ * P B, A∩ * P B are * P -convex type-2 fuzzy sets.If A and B are two * L -convex type-2 fuzzy sets, then A∪ * L B, A∩ * L B are * L -convex type-2 fuzzy sets.If A and B are two * D -convex type-2 fuzzy sets, then A∪ * D B, A∩ * D B are * D -convex type-2 fuzzy sets.

Theorem 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.
Definition 2.12.Let E be a Cartesian product of real linear topological spaces E E 1 × • • • × 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 A 1 , . . ., A r of E into an * -convex fuzzy set of Y , where A 1 , . . ., A r are * -convex fuzzy sets in X 1 , . . ., X r , respectively.
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, μ y ≥ 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.