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

Abstract

We discuss a fuzzy result by displaying an example that shows how a classical argument fails to work when one passes from classical logic to fuzzy logic. Precisely, we present an example to show that, in the fuzzy context, the fact that the supremum is naturally used in lieu of the union can alter an argument that may work in the classical context.

1. 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 [3–7]. 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 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 [8–10, 15].

2. 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: ; ; 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 ,(i) and ;(ii);(iii) and .For an overview of star operations, the reader may refer to [15, Sections and ].

A fuzzy subset of is a function from into the real closed interval . 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 , for every and every . Note that a fuzzy subset of is a fuzzy -submodule of if and only if and is an -submodule of for every real number in . Let denote the fuzzy subset of defined as follows: for each in ,    if and 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 .

Definition 1 (see [9]). A fuzzy semistar operation on is a mapping , , which satisfies the following three properties for all , and : ();();() and .

Recall from [9] that a fuzzy semistar operation on is union preserving if . Note that the preservation of union on is over a countable set. Now, define a mapping from into as follows: Then, if is a union preserving fuzzy semistar operation on , then is a fuzzy semistar operation on [9, Theorem ]. This leads to the following definition.

Definition 2 (see [9]). Let be a fuzzy semistar operation on . (1)If is a fuzzy semistar operation on , then is called the fuzzy semistar operation of finite character (or finite type) associated with .(2) is called a fuzzy semistar operation of finite character if .

Example 3. (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 denote the set of all integers with quotient field of all rational numbers. Let be the unit interval (note that the unit interval is a completely distributive lattice). Define by for any . Then, is a fuzzy semistar operation on of finite character (the reader may refer to [9, Example . (2)] for the proof of this fact).

3. Fuzzy Logic versus Classical Logic: An Example

Recall from [9] that a fuzzy semistar operation on is said to be union preserving if . Note that the preservation of union on is over a countable set. Also, recall the following result in [9].

Theorem 4 (see [9, Theorem ]). 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.

3.1. The Fuzzy and Classical Statements

A Classical Argument. Let be a submodule of in and let be a finitely generated submodule of in such that , where is a classical semistar operation on . Then, is contained in some with and . This classical argument is a well-known simple argument in multiplicative ideal theory. In fact, suppose we set as a finitely generated ideal such that . Then, for each , with and . So, . Now, using the well-known facts that and , for a classical semistar operation on , we obtain that . Now, since each is finitely generated, the finite sum of ’s is also finitely generated and this completes the proof.

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 , where is a fuzzy semistar operation on . Then, is contained in some , with and .

3.2. 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):

by for any .

Let denote the quotient field of all rational numbers. We define (note that the unit interval is a completely distributive lattice), and we use the known fact that is a fuzzy -submonoid of if and only if is a -submonoid of for any and . Let be defined by where sgn is the signature function and denotes the absolute value of . It is easy to see that for and for . Consider an infinite sequence of -submodules of as follows: Obviously, is a -submodule of for any , since and . Moreover, , whenever . Indeed, , which implies . Then, one can define for any by Note that if for any , then (e.g., ). On the other hand, Since for any (and 0 is the unique element having this property), a consequence of the supremum is . Thus, if and only if . It should be noted that if , then there exists such that . Indeed, it is easy to see from definition of and that and there exist such that (see the remark above about the convergence of to zero and one); therefore, and . Hence, for a suitable for which , since the supremum is calculated over only a finite set of linearly ordered values.

To demonstrate that is a fuzzy -submodule of , let us first consider the cases and . One can simply check that and . If , then there exists exactly one such that and . Therefore, each -level subset of is a -submonoid of . Since , .

Now, let us consider where and have been defined above. Put . Let us show that . Since for any , we have ; therefore, . To show the opposite inequality, let and without loss of generality let and . Consider for . It is easy to see that and is finitely generated -submodule of for any . Moreover, and , where . Then, by definition of , we obtain

Now, let us demonstrate the false argument. Let us consider for some . Obviously, and . According to our false argument, it should be true that “since is finitely generated, is contained in finitely many with and .” But this is impossible, because for any choice of finitely many we can find (it is sufficient to consider elements of that are used for generating such that .

3.3. 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 , , with .

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, , 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.

Competing Interests

The author declares that there are no competing interests regarding the publication of this paper.