New Arithmetic Operations of Non-Normal Fuzzy Sets Using Compatibility

: The new arithmetic operations of non-normal fuzzy sets are studied in this paper by using the extension principle and considering the general aggregation function. Usually, the aggregation functions are taken to be the minimum function or t-norms. In this paper, we considered a general aggregation function to set up the arithmetic operations of non-normal fuzzy sets. In applications, the arithmetic operations of fuzzy sets are always transferred to the arithmetic operations of their corresponding α -level sets. When the aggregation function is taken to be the minimum function, this transformation is clearly realized. Since the general aggregation function was adopted in this paper, the concept of compatibility with α -level sets is needed and is proposed, which can cover the conventional case using minimum functions as the special case.


Introduction
In order to simplify the notations, the membership function ξ F of a fuzzy set F is identified with F by simply writing ξ F (x) = F(x).Let F and G be two fuzzy sets in R, and let denote any one of the arithmetic operations ⊕, , ⊗, between F and G.According to the extension principle, the membership function of F G is defined by F G(u) = sup {(x,y):u=x•y} min{ F(x), G(y)} (1) for all u ∈ R, where the arithmetic operations ∈ {⊕, , ⊗, } correspond to the arithmetic operations • ∈ {+, −, * , ÷}.The case of • = ÷ should avoid the division of x/y for y = 0.
In general, we can consider the t-norm instead of the minimum function by referring to Dubois and Prade [1] and Weber [2].For more detailed properties, we can refer to the monographs by Dubois and Prade [3] and Klir and Yuan [4].In this paper, we used the general function to propose the arithmetic operations of fuzzy sets, and we present the compatibility with the conventional definition using the minimum functions.We can also refer to Gebhardt [5], Fullér and Keresztfalvi [6], Mesiar [7], Ralescu [8], and Yager [9] and Wu [10] for the arithmetic operations of fuzzy sets based on the extension principle.
The generalization of Zadeh's extension principle in (1) can also be used to set up the arithmetic operations without using the minimum function.Coroianua and Fuller [11,12] used the so-called joint probability distribution to generalize the extension principle (1), which is given by F C G(u) = sup where D does not need to satisfy some extra conditions.The main difference between (2) and ( 4) is that the domains of the joint probabilitydistribution C : R 2 → [0, 1] and function D : [0, 1] 2 → [0, 1] are different.We can also refer to Coroianua and Fuller [11] for the comparison between ( 2) and ( 4).Although D in ( 4) is a general function, some sufficient conditions regarding D are still needed to obtain some desired properties.Therefore, the second motivation of this paper was to propose the concept of compatibility.We shall say that the function D is compatible with the arithmetic operations of α-level sets when the following equality: The sufficient conditions imposed upon the function D will be studied to guarantee the compatibility.Under the general function D, the associativity of the arithmetic operations is also an important issue.Therefore, many rules regarding the associativity were also studied.
There is some other interesting arithmetic of fuzzy numbers, which will be shown below.Holčapek, Škorupová, and Štěpnička [13,14] proposed the arithmetic of extensional fuzzy numbers based on a similarity relation S : R 2 → [0, 1] such that S satisfies some required conditions.On the other hand, based on the concept of the extensional hull, given a fixed real number x ∈ R, the so-called extensional fuzzy number generated by x and a similarity relation S is a fuzzy set xS in R with membership degree xS (y) = S(x, y) for all y ∈ R.
Given any two extensional fuzzy numbers xS and ỹS , the addition ⊕ S and multiplication ⊗ S are defined by xS ⊕ S ỹS = (x + y) S and xS ⊗ S ỹS = (xy) S , where S is assumed to be the so-called separated similarity relation for the purpose of well-defined arithmetic.In general, based on a system S of so-called nested similarity relations, the addition ⊕ S and multiplication ⊗ S are defined by xS ⊕ S ỹT = (x + y) max(S,T) and xS ⊗ S ỹT = (xy) max(S,T) for S, T ∈ S.
Esmi et al. [15] and Pedro et al. [16] used the extension principle in (3) to study the fuzzy differential equations.They considered the interactivity between fuzzy numbers.Let P be a fuzzy set in R. Given any fuzzy numbers F and G, we say that P is a joint probability distribution of F and G when sup x∈R P(x, y) = G(y) and sup We say that F and G are non-interactive when P(x, y) = min F(x), G(x) .
Otherwise, they are called interactive.The disadvantage is that the non-interactivity depends on their joint probability distributions.We cannot just say that F and G are non-interactive without considering the role of the joint probability distribution.Let denote any one of the arithmetic operations ⊕ P, P, ⊗ P, P between fuzzy numbers F and G along with a joint probability distribution P. The membership function of F P G is defined by for all u ∈ R, where the case of • P = ÷ should avoid the division of x/y for y = 0.The arithmetic of fuzzy intervals is an important issue.Wu [17] considered the form of expression in the decomposition theorem to study the arithmetic of fuzzy intervals.Wu [18] also used the form of expression in the decomposition theorem to study the different types of binary operations of fuzzy sets, which were also applied to study the difference of fuzzy intervals and covered the so-called generalized differences proposed by Bede and Stefanini [19] and Gomes and Barros [20] as the special cases.The fuzzy axiom of choice, the fuzzy Zorn's lemma, and the fuzzy Hausdorff maximal principle studied by Zulqarnian et al. [21] were also based on normal fuzzy sets.It is also possible to extend those results based on the non-normal fuzzy sets.
The fuzzy sets considered in Wu [17,18] were implicitly assumed to be normal.Without using the form of expression in the decomposition theorem, in this paper, we shall use the extension principle based on a general function rather than the t-norm to study the arithmetic of non-normal fuzzy intervals.In this case, the concept of compatibility with α-level sets can be proposed and the equivalence with conventional arithmetic operations using the minimum function can also be established.
In Section 2, the concept and basic properties of non-normal fuzzy sets will be presented, and the arithmetic operations of non-normal fuzzy sets will be studied using the extension principle based on the general functions.In Section 3, we shall propose the concept of compatibility with the α-level sets, which can cover the conventional case using the minimum functions as the special case.

Arithmetic Operations of Fuzzy Sets
Let F be a fuzzy set in R. Recall that a fuzzy set F in a universal set U is called normal when there exists x ∈ U satisfying F(x) = 1.For α ∈ (0, 1], the α-level set of F is denoted and defined by Fα = x ∈ R : F(x) ≥ α .
The support of a fuzzy set F is the crisp set defined by The 0-level set F0 is defined to be the topological closure of the support of F, i.e., F0 = cl( F0+ ).We write R F to denote the range of the membership function of F. In general, we have R F = [0, 1].The following result is very useful.Proposition 1.Let F be a fuzzy set in R with membership function F. Define α * = sup R F and Then, Fα = ∅ for all α ∈ I F and Fα = ∅ for all α ∈ I F.Moreover, we have R F ⊆ I F and Fα .
The interval I F is called an interval range of F.
We considered three arithmetic operations , and between any two fuzzy sets F and G in R. The extension principle says that the membership functions are given by for all u ∈ R, where the arithmetic operations ∈ { , , } correspond to the arithmetic operations • ∈ {+, −, * }.The case of division was not considered in this paper, since it can be similarly obtained.
Instead of the minimum function, we can consider a general function In this case, the membership functions are defined by In general, the arithmetic operations are defined below.

Definition 1. Given any fuzzy sets
When the function D n is taken to be the minimum function given by 7), (8), and ( 9).We can also insert the parentheses into the expression The following example shows the way of inserting parentheses.
Example 2. We present an example from mathematical finance.The well-known Black-Scholes formula (see Black and Scholes [22]) for the European call option on a stock is described as follows.
Let the function f be given by the formula: where s denotes the stock price, t denotes the time, K denotes the strike price, r denotes the interest rate, σ denotes the volatility, and N stands for the cumulative distribution function of a standard normal random variable N(0, 1).The quantities d 1 and d 2 are given by Let T be the expiry date, and let C t denote the price of a European call option at time t ∈ [0, T].Then, we have where S t denotes the stock price at time t.On the other hand, the price P t of a European put option at time t with the same expiry date T and strike price K can be obtained by the following put-call parity relationship (see Musiela and Rutkowski [23]): Under the considerations of the fuzzy interest rate r, fuzzy volatility σ, and fuzzy stock price S, we can obtain the fuzzy price Ht of a European call option at time t according to (12) and the extension principle.Therefore, the membership function of Ht is given by According to the put-call parity relationship in (13), we can also study the fuzzy price Pt of a European put option at time t.Let Then, we can obtain the fuzzy price Pt of a European put option at time t in which the membership function of Pt is given by Let F(1) , • • • , F(n) be fuzzy sets in R, and let α * i = sup R F(i) .From Proposition 1, we see that F(i) α = ∅ for all α ∈ I F(i) and F(i) α = ∅ for all α ∈ I F(i) , where the interval range I F(i) is given by Let Then, I * is also an interval of the form [0, α] or [0, α) for some α ∈ (0, 1].For α ∈ I * , we see that , and let I F be the interval range of F. We also write Therefore, the definition of interval range says satisfies the following condition: We also assumed that the supremum where Then, we have Proof.Since the supremum sup R F(i) is obtained for i = 1, • • • , n, we have for some x * i ∈ R and for all i = 1, • • • , n.It is also clear that From (15), we have On the other hand, since (15), again, we also have Therefore, we obtain F(u * ) = α * .From (15), we conclude that the supremum sup R F is obtained at u * .From (16), it follows that I F = [0, α * ].This completes the proof.

Compatibility
Let S 1 , • • • , S n be subsets of R. We write Given any fuzzy sets F (1) it is clear that the α-level sets Fα and F(i) α are nonempty for i = 1, • • • , n.Therefore, we propose the following definition.Definition 2. Given any fuzzy sets F(1) , • • • , F(n) in R, we considered the arithmetic operations i ∈ {⊕, , ⊗}, which correspond to the arithmetic operations is said to be compatible with the arithmetic operations of α-level sets when the following equality is satisfied: is said to be strongly compatible with the arithmetic operations of α-level sets when the following equality is satisfied: The purpose of this paper was to present some sufficient conditions such that the compatibility with the arithmetic operations of α-level sets can be satisfied.
Recall that the real-valued function f : R → R is upper semi-continuous on R if and only if the set {x ∈ R : f (x) ≥ α} is a closed set in R for each α ∈ R. Especially, if F is a fuzzy set in R such that its membership function F is upper semi-continuous on R, then each α-level set Fα is a closed subset of R for α ∈ I F.
Lemma 1 (Royden ([24] p. 161)).Let K be a closed and bounded subset of R, and let f be a real-valued function defined on R. Suppose that f is upper semi-continuous on R.Then, f assumes its maximum on K; that is, the supremum is obtained in the following sense: Theorem 1.Given any fuzzy sets F(1) , • • • , F(n) in R, we considered the arithmetic operations i ∈ {⊕, , ⊗}, which correspond to the arithmetic operations Then, we have the following properties: (i) For any α ∈ I * ∩ I F with α > 0, we assumed that the function D n satisfies the following condition: Then, the following inclusion: holds true for all α ∈ I * ∩ I F. (ii) Suppose that the membership functions of F(i) are upper semi-continuous for all i = 1, • • • , n.
We also assumed that the function D n satisfies the following conditions: • Given any α ∈ I * with α ∈ (0, 1], for any α j ∈ [0, 1] with j = i.Then, the following equality: holds true for all α ∈ I * ∩ I F with α > 0. We further assumed that the supports F(i) 0+ are bounded for all i = 1, • • • , n.Then, the following equality: regarding the 0-level sets holds true.
Proof.To prove Part (i), given any α ∈ I * ∩ I F with α > 0, we have Fα = ∅ and there exist We see that Using the assumption (20) of D n , we also have Therefore, we have This shows Therefore, we obtain the following inclusion: Next, we considered the 0-level sets.For α = 0, given any For each fixed i, since the concept of closure says that there exists a sequence We considered a function η : R n → R defined by The above cases conclude that the function f (x 1 , • • • , x n ) is indeed upper semicontinuous.Lemma 1 says that the function f assumes the maximum on the set F. Therefore, using (29), we have max Therefore, there exists Using the assumption (21), we obtain which shows the following inclusion: Using Part (i), we obtain the desired equality (23).
Considering the 0-level sets, for α = 0, we further assumed that the supports Since I * ∩ I F is an interval beginning from 0, using the denseness of R, there exists α ∈ I * ∩ I F with α > 0 satisfying Using the assumption (21), we have α i ≥ α > 0 for all i = 1, • • • , n, which says that the following statement holds true: Now, considering the 0-level set, we have Therefore, there exists a sequence {u m } ∞ m=1 in the following set: Using the above arguments by referring to (30), we can obtain Therefore, there exist x 1m , • • • , x nm satisfying Using (31), we have m=1 is also a bounded sequence.Therefore, there exists a convergent subsequence {x im k } ∞ k=1 of {x im } ∞ m=1 .In other words, we have lim Then, we see that {u m k } ∞ k=1 is a subsequence of {u m } ∞ n=1 , i.e., lim k→∞ u m k = u 0 . Since Therefore, we obtain the following inclusion: Using Part (i), we obtain the desired equality (24), and the proof is complete.
Theorem 2. Given any fuzzy sets F(1) , • • • , F(n) in R, we considered the arithmetic operations i ∈ {⊕, , ⊗}, which correspond to the arithmetic operations Suppose that the function D n satisfies the following conditions: • Given any α ∈ I * with α ∈ (0, 1], If ã is a fuzzy interval, then its 0-level set ã0 is a closed and bounded subset of R.
More precisely, we have We further assumed that the suprema: is a closed interval.
Proof.Given any α ∈ I ã ∩ I b ∩ I ã b, it is clear that the α-level sets ( ã b) α , ãα , and bα are nonempty.Therefore, the desired results follow immediately from Corollary 1 and Proposition 2. This completes the proof.

Conclusions
The arithmetic operations of non-normal fuzzy sets using the extension principle based on general functions were investigated in this paper.The membership function of arithmetic operation F where the way of calculation F This kind of arithmetic operation generalizes the conventional one given by The main issue of arithmetic operations is studying their α-level sets.Therefore, the concept of compatibility with α-level sets is proposed by saying that the function D n : [0, 1] n → [0, 1] is (strongly) compatible with the arithmetic operations of α-level sets when α for all α ∈ I * ∩ I F with α > 0.
It is clear that the minimum function: considered in the conventional case is compatible with arithmetic operations of α-level sets.Theorems 1 and 2 present the sufficient conditions to guarantee the compatibility with the arithmetic operations of α-level sets.This means that Theorems 1 and 2 are the general situation.Therefore, Corollary 1 and Proposition 3, which are the conventional cases, are the special cases of Theorems 1 and 2. This was the main purpose of this paper: to generalize the conventional cases.In other words, from some other functions D n that can satisfy the sufficient conditions, the desired results can be obtained as the conventional cases.The main focus was on the functions D n and the non-normal fuzzy sets, rather than the t-norm and the normal fuzzy sets.As we can see in Part (i) of Theorem 2, the equality (32) holds true for non-normal fuzzy sets.The case of normal fuzzy sets is just the special case of (32).Therefore, Theorems 1 and 2 indeed generalize the conventional cases.The limitation of Theorems 1 and 2 is checking the assumptions of general function D n .Since those assumptions are satisfied for the conventional cases, as shown in Corollary 1 and Proposition 3, this also means that those assumptions are not too strong to be used in real applications.
The interval ranges of non-normal fuzzy sets comprise an important tool to handle the arithmetic of non-normal fuzzy sets.The future research will focus on the applications by using non-normal fuzzy sets and will solve the difficulty caused by the different forms of the interval ranges of non-normal fuzzy sets.

Funding:
The APC was funded by NSTC Taiwan.

Conflicts of Interest:
The author declares no conflict of interest.

Proposition 3 .
Given any fuzzy intervals ã and b with interval ranges I ã and I b, respectively, let I ã b denote the interval range of ã b for ∈ { , , }.Then, ã b is also a fuzzy interval, and its α-level set is given by The conditions in Definition 3 says that each α-level set ãα is a bounded closed interval for α ∈ [0, 1].It is also clear that α ] with degree α, which explains the terminology of the fuzzy interval.