Fuzzy Logic versus Classical Logic : An Example in Multiplicative Ideal Theory

Rosenfeld in 1971 was the first classical algebraist to introduce fuzzy algebra by writing a paper on fuzzy groups [1]. The introduction of fuzzy groups then motivated several researchers to shift their interest to the extension of the seminal work of Zadeh [2] on fuzzy subsets of a set to algebraic structures such as rings and modules [3–7]. In that regard, Lee and Mordeson in [3, 4] introduced the notion of fractionary fuzzy ideal and the notion of invertible fractionary fuzzy ideal and used these notions to characterize Dedekind domains in terms of the invertibility of certain fractionary fuzzy ideals, leading to the fuzzification of one of the main results in multiplicative ideal theory. Other significant introduced notions to tackle the fuzzification of multiplicative ideal theory are the notion of fuzzy star operation [8] and the notion of fuzzy semistar operation [9, 10] on integral domains. This paper is concerned with the fuzzification of multiplicative ideal theory in commutative algebra (see, e.g., [1, 3, 5, 8–11]). In the field of commutative ring, it is customary to use star operations not only to generalize classical domains, but also to produce a common treatment anddeeper understanding of those domains. Some of the instances are the notion of Prüfer ⋆-multiplication domain which generalizes the notion of Prüfer domain [12] and the notion of ⋆-completely integrally closed domain which generalizes the notion of completely integrally closed domain [13, 14]. The importance of star operations in the classical theory has led scholars to be interested in fuzzy star operations introduced in [8] and this has been generalized to fuzzy semistar operations in [10]; this generalization has led to more fuzzification of main results in multiplicative ideal theory. In this note, we focus on some classical arguments of multiplicative ideal theory that do not hold in the fuzzy context. The example chosen is to infer that what appears to be pretty simple and even rather easy in the context of classical logic may not be true in the fuzzy context. So, one challenge of fuzzification is to detect any defect or incongruous statement that may first appear benign but is a real poison in the argument used to prove fuzzy statements. Precisely, in our example, we display the difficulty in how the natural definition in the fuzzy context may make it a little bit more challenging to work with in comparison with its equivalent classical definition. For an overview of all definitions of fuzzy submodules, fuzzy ideals, and fuzzy (semi)star operations (of finite character), the reader may refer to [8–10, 15].


Introduction
Rosenfeld in 1971 was the first classical algebraist to introduce fuzzy algebra by writing a paper on fuzzy groups [1].The introduction of fuzzy groups then motivated several researchers to shift their interest to the extension of the seminal work of Zadeh [2] on fuzzy subsets of a set to algebraic structures such as rings and modules [3][4][5][6][7].In that regard, Lee and Mordeson in [3,4] introduced the notion of fractionary fuzzy ideal and the notion of invertible fractionary fuzzy ideal and used these notions to characterize Dedekind domains in terms of the invertibility of certain fractionary fuzzy ideals, leading to the fuzzification of one of the main results in multiplicative ideal theory.Other significant introduced notions to tackle the fuzzification of multiplicative ideal theory are the notion of fuzzy star operation [8] and the notion of fuzzy semistar operation [9,10] on integral domains.This paper is concerned with the fuzzification of multiplicative ideal theory in commutative algebra (see, e.g., [1,3,5,[8][9][10][11]).
In the field of commutative ring, it is customary to use star operations not only to generalize classical domains, but also to produce a common treatment and deeper understanding of those domains.Some of the instances are the notion of Prüfer ⋆-multiplication domain which generalizes the notion of Prüfer domain [12] and the notion of ⋆-completely integrally closed domain which generalizes the notion of completely integrally closed domain [13,14].The importance of star operations in the classical theory has led scholars to be interested in fuzzy star operations introduced in [8] and this has been generalized to fuzzy semistar operations in [10]; this generalization has led to more fuzzification of main results in multiplicative ideal theory.
In this note, we focus on some classical arguments of multiplicative ideal theory that do not hold in the fuzzy context.The example chosen is to infer that what appears to be pretty simple and even rather easy in the context of classical logic may not be true in the fuzzy context.So, one challenge of fuzzification is to detect any defect or incongruous statement that may first appear benign but is a real poison in the argument used to prove fuzzy statements.Precisely, in our example, we display the difficulty in how the natural definition in the fuzzy context may make it a little bit more challenging to work with in comparison with its equivalent classical definition.For an overview of all definitions of fuzzy submodules, fuzzy ideals, and fuzzy (semi)star operations (of finite character), the reader may refer to [8][9][10]15].

Preliminaries and Notations
Recall that an integral domain  is a commutative ring with identity and no-zero divisors.Hence, its quotient ring  is a field.A group (, +) is an -module if there is a mapping × → , (, )  → , satisfying the following conditions: 2 Advances in Fuzzy Systems 1 = ; ( − ) =  − ; and () = () for all ,  ∈  and ,  ∈ , where 1 is the identity of .Note that the quotient field  of an integral domain  is an -module.An -submodule  of an -module  is a subgroup of  such that  ∈  for all  ∈  and  ∈ .For more reading on integral domains and modules, the reader may refer to [7,15].Recall also that a star operation on  is a mapping  →  ⋆ of () into () such that, for all ,  ∈ () and for all  ∈  \ {0}, For an overview of star operations, the reader may refer to [15,Sections 32 and 34].
A fuzzy subset of  is a function from  into the real closed interval [0, 1].We say  ⊆  if () ≤ () for all  ∈ .The intersection ⋂ ∈   of the fuzzy subsets   's is defined as ⋂ ∈   () = ⋀  () and the union ⋃ ∈   of the fuzzy subsets   's is defined as ⋃ ∈   () = ⋁  () for every  ∈ .Let   = { ∈  : () ≥ }; then,   is called a level subset of .We let   denote the characteristic function of the subset  of .A fuzzy subset of  is a fuzzy -submodule of  if ( − ) ≥ () ∧ (), () ≥ (), and (0) = 1, for every ,  ∈  and every  ∈ .Note that a fuzzy subset  of  is a fuzzy -submodule of  if and only if (0) = 1 and   is an -submodule of  for every real number  in [0, 1].Let   denote the fuzzy subset of  defined as follows: for each  in ,   () =  if  =  and   () = 0 otherwise.We call   a fuzzy singleton.A fuzzy -submodule  of  is finitely generated if  is generated by some finite fuzzy singletons; that is, it is the smallest fuzzy -submodule of  containing those fuzzy singletons.Throughout this paper,   () denotes the set of all fuzzy -submodules of  and   () denotes the set of all finitely generated fuzzy -submodules of .
(1) If ⋆  is a fuzzy semistar operation on , then ⋆  is called the fuzzy semistar operation of finite character (or finite type) associated with ⋆.
(1) It is clear by definition that (⋆  )  = ⋆  ; that is, ⋆  is of finite character whenever ⋆  is a fuzzy semistar operation on  for any fuzzy semistar operation ⋆ on .
(2) The constant map   →   is also trivially a fuzzy semistar operation on  that is not of finite character.
(3) Let Z denote the set of all integers with quotient field Q of all rational numbers.Let  = [0, 1] be the unit interval (note that the unit interval is a completely distributive lattice).Define ⋆ :   (Z) →   (Z) by for any  ∈   (Z).Then, ⋆ is a fuzzy semistar operation on Z of finite character (the reader may refer to [9, Example 3.8. (2)] for the proof of this fact).

Fuzzy Logic versus Classical Logic: An Example
Recall from [9] that a fuzzy semistar operation ⋆ on  is said to be union preserving if (⋃ ∈Z +   ) ⋆ = ⋃ ∈Z +  ⋆  .Note that the preservation of union on ⋆ is over a countable set.Also, recall the following result in [9].Theorem 4 (see [9,Theorem 3.5]).Let ⋆ be a union preserving fuzzy semistar operation on .Then, ⋆  is a fuzzy semistar operation on .
Let  be an integral domain with quotient field .Recall that   () denotes the set of all fuzzy -submodules of  and   () denotes the set of finitely generated fuzzy submodules of .Now, we claim that we could not get rid of the assumption in Theorem 4 because we could not use the fuzzy counterpart of the following classical argument below.
The Fuzzy Counterpart of the Above Classical Argument.Let  be a fuzzy -submodule of  and let  be a finitely generated fuzzy -submodule of  such that  ⊆ ⋃{ ⋆ |  ∈   () and  ⊆ }, where ⋆ is a fuzzy semistar operation on .Then,  is contained in some  ⋆  , with   ∈   () and   ⊆ .

A Counterexample to
Negate the Fuzzy Counterpart.We now produce an example to prove that the fuzzy counterpart statement is false.Note that the reason why the counterpart may be false is clearly the fact that the union in the fuzzy context is the supremum.So, the real challenge here is to construct a counterexample that will clearly justify the wrongness of the argument.
The Counterexample.Let ⋆ be the fuzzy semistar operation of finite character as defined in Example 3(4): for any  ∈   (Z).
Let Q denote the quotient field of all rational numbers.We define  : Q → [0, 1] (note that the unit interval is a completely distributive lattice), and we use the known fact that  is a fuzzy Z-submonoid of Q if and only if   is a Z-submonoid of Q for any  ∈ [0, 1] and (0) = 1.Let  : Z → [0, 1] be defined by where sgn is the signature function and || denotes the absolute value of .It is easy to see that () → 1 for  → ∞ and () → 0 for  → −∞.Consider an infinite sequence of Z-submodules of Q as follows: Note that if  ∉ 2  Z for any  ∈ Z, then () = 0 (e.g., ( √ 2) = 0).On the other hand, Since 0 ∈ 2  Z for any  ∈ Z (and 0 is the unique element having this property), a consequence of the supremum is (0) = 1.Thus, () = 1 if and only if  = 0.It should be noted that if 0 < () < 1, then there exists  ∈ Z such that () = ().Indeed, it is easy to see from definition of  and  that  ∈ ⋃ ∈Z 2  Z and there exist ,   ∈ Z such that 0 < (  ) < () < () < 1 (see the remark above about the convergence of  to zero and one); therefore,  ∈ 2   Z and  ∉ 2  Z.Hence, () = () for a suitable  ∈ Z for which   <  < , since the supremum is calculated over only a finite set of linearly ordered values.

Final Remark.
The proof of the classical argument holds due to the fact that the classical union is involved allowing the choice of a finitely generated -submodule of  for each element of .However, in the fuzzy counterpart statement, the fuzzy union is defined in terms of the supremum and the technique used in the proof of the classical argument cannot apply in the fuzzy context since clearly  = ⋁ ∈   does not imply the existence of   0 ,  0 ∈ , with  ≤   0 .
We must also note that the fuzzy counterpart statement is the natural one that grasps some thoughts about the context in which the crisp result can be extended.In fact, the condition of union preserving of fuzzy star operation, that is, (⋃ ∈Z +   ) ⋆ = ⋃ ∈Z +  ⋆  , which does not always hold in the fuzzy context is not needed in the crisp case to get a classical finite character semistar operation.This additional condition of union preserving of fuzzy star operation will make our fuzzy counterpart statement true.