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

Authors: Olivier A. Heubo-Kwegna
Journal: Advances in Fuzzy Systems
 (2016)

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## 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.


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## Body

## 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 domainR is a commutative ring with identity and no-zero divisors. Hence, its quotient ring L is a field. A group (M,+) is an R-module if there is a mapping R×M→M, (r,x)↦rx, satisfying the following conditions: 1x=x; r(x-y)=rx-ry; and (rt)x=r(tx) for all r,t∈R and x,y∈M, where 1 is the identity of R. Note that the quotient field K of an integral domain R is an R-module. An R-submoduleN of an R-module M is a subgroup of M such that rx∈N for all r∈R and x∈N. For more reading on integral domains and modules, the reader may refer to [7, 15]. Recall also that astar operation on R is a mapping A→A⋆ of F(R) into F(R) such that, for all A,B∈F(R) and for all a∈L∖{0},(i)
(a)⋆=(a) and (aA)⋆=aA⋆;(ii)
A⊆B⇒A⋆⊆B⋆;(iii)
A⊆A⋆ and A⋆⋆≔(A⋆)⋆=A⋆.For an overview of star operations, the reader may refer to [15, Sections 32 and 34].Afuzzy subset of L is a function from L into the real closed interval [0,1]. We say α⊆β if α(x)≤β(x) for all x∈L. Theintersection⋂i∈I‍αi of the fuzzy subsets αi’s is defined as ⋂i∈I‍αI(x)=⋀iα(x) and theunion⋃i∈I‍αI of the fuzzy subsets αi’s is defined as ⋃i∈I‍αi(x)=⋁iα(x) for every x∈L. Let βt={x∈R:β(x)≥t}; then, βt is called alevel subset of β. We let χA denote the characteristic function of the subset A of R. A fuzzy subset of L is a fuzzy R-submodule of L if β(x-y)≥β(x)∧β(y), β(rx)≥β(x), and β(0)=1, for every x,y∈L and every r∈R. Note that a fuzzy subset β of L is a fuzzy R-submodule of L if and only if β(0)=1 and βt is an R-submodule of L for every real number t in [0,1]. Let dt denote the fuzzy subset of L defined as follows: for each x in L,   dt(x)=t if x=d and dt(x)=0 otherwise. We call dt afuzzy singleton. A fuzzy R-submodule β of L isfinitely generated if β is generated by some finite fuzzy singletons; that is, it is the smallest fuzzy R-submodule of L containing those fuzzy singletons. Throughout this paper, Fz(R) denotes the set of all fuzzy R-submodules of L and fz(R) denotes the set of all finitely generated fuzzy R-submodules of L.Definition 1 (see [9]).
Afuzzy semistar operation on R is a mapping ⋆:F¯z(R)→F¯z(R), β↦β⋆, which satisfies the following three properties for all α,β∈F¯z(R), and 0≠d∈K: (⋆1)
(d1∘β)⋆=d1∘β⋆;(⋆2)
α⊆β⇒α⋆⊆β⋆;(⋆3)
β⊆β⋆ and β⋆⋆≔(β⋆)⋆=β⋆.Recall from [9] that a fuzzy semistar operation ⋆ on R isunion preserving if (⋃n∈Z+‍βn)⋆=⋃n∈Z+‍βn⋆. Note that the preservation of union on ⋆ is over a countable set. Now, define a mapping ⋆f from F¯z(R) into F¯z(R) as follows: (1)β⟼β⋆f≔⋃α⋆∣α∈fzR,α⊆β.Then, if ⋆ is a union preserving fuzzy semistar operation on R, then ⋆f is a fuzzy semistar operation on R [9, Theorem 3.5]. This leads to the following definition.Definition 2 (see [9]).
Let⋆ be a fuzzy semistar operation on R. (1)
If⋆f is a fuzzy semistar operation on R, then ⋆f is calledthe fuzzy semistar operation of finite character (or finite type) associated with⋆.(2)
⋆ is called afuzzy semistar operation of finite character if ⋆=⋆f.Example 3.
(1) It is clear by definition that(⋆f)f=⋆f; that is, ⋆f is of finite character whenever ⋆f is a fuzzy semistar operation on R for any fuzzy semistar operation ⋆ on R.
(2) The constant mapβ↦χL is also trivially a fuzzy semistar operation on R that is not of finite character.
(3) LetZ denote the set of all integers with quotient field Q of all rational numbers. Let L=[0,1] be the unit interval (note that the unit interval is a completely distributive lattice). Define ⋆:F¯z(Z)→F¯z(Z) by (2)β⋆x=1,ifx=0;⋁y∈Q∖0βy,ifβx≠0,x≠0;0,ifβx=0,
for anyβ∈F¯z(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).


## 3.Fuzzy Logic versus Classical Logic: An Example

Recall from [9] that a fuzzy semistar operation ⋆ on R is said to beunion preserving if (⋃n∈Z+‍βn)⋆=⋃n∈Z+‍βn⋆. 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 R. Then, ⋆f is a fuzzy semistar operation on R.LetR be an integral domain with quotient field L. Recall that Fz(R) denotes the set of all fuzzy R-submodules of L and fz(R) denotes the set of finitely generated fuzzy R-submodules of L. 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 B be a submodule of R in L and let I be a finitely generated submodule of R in L such that I⊆⋃‍{A⋆∣A∈f(R)andA⊆B}, where ⋆ is a classical semistar operation on R. Then, I is contained in some Ai⋆ with Ai∈f(R) and Ai⊆B. This classical argument is a well-known simple argument in multiplicative ideal theory. In fact, suppose we set I=x1,…,xs as a finitely generated ideal such that I⊆⋃‍{A⋆∣A∈f(R)andA⊆B}. Then, for each i=1,…,s, xi∈Ai⋆ with Ai∈f(R) and Ai⊆B. So, I=x1,…,xs⊆∑i=1s‍Ai⋆. Now, using the well-known facts that A⊆A⋆ and ∑i=1s‍Ai⋆⋆=∑i=1s‍Ai⋆, for a classical semistar operation ⋆ on R, we obtain that I⊆∑i=1s‍Ai⋆. Now, since each Ai is finitely generated, the finite sum of Ai’s is also finitely generated and this completes the proof.The Fuzzy Counterpart of the Above Classical Argument. Let β be a fuzzy R-submodule of L and let α be a finitely generated fuzzy R-submodule of L such that α⊆⋃‍{γ⋆∣γ∈fz(R)andγ⊆β}, where ⋆ is a fuzzy semistar operation on R. Then, α is contained in some γi⋆, with γi∈fz(R) and γi⊆β.


### 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):⋆ : F ¯ z ( Z ) → F ¯ z ( Z ) by (3)β⋆x=1,ifx=0;⋁y∈Q∖0βy,ifβx≠0,x≠0;0,ifβx=0,for any β∈F¯z(Z).LetQ denote the quotient field of all rational numbers. We define β:Q→[0,1] (note that the unit interval is a completely distributive lattice), and we use the known fact that β is a fuzzy Z-submonoid of Q if and only if βt is a Z-submonoid of Q for any t∈[0,1] and β(0)=1. Let f:Z→[0,1] be defined by (4)fn=121+sgnnnn+1,where sgn is the signature function and |n| denotes the absolute value of n. It is easy to see that f(n)→1 for n→∞ and f(n)→0 for n→-∞. Consider an infinite sequence of Z-submodules of Q as follows: (5)⋯2nZ⊆2n-1Z⊆⋯20Z⊆⋯12n-1Z⊆12nZ⊆⋯Q.Obviously, 2nZ is a Z-submodule of Q for any n∈Z, since r2n-s2n=(r-s)2n and r(s2n)=(rs)2n. Moreover, 2nZ⊆2mZ, whenever m<n. Indeed, r2n=(r2n-m)2m, which implies r2n∈2mZ. Then, one can define β for any x∈Q by (6)βx=⋁fn∣x∈2nZ,n∈Z.Note that if x∉2nZ for any n∈Z, then β(x)=0 (e.g., β(2)=0). On the other hand, (7)β1=β3=12,β12=β32=14,β2=β6=34.Since 0∈2nZ for any n∈Z (and 0 is the unique element having this property), a consequence of the supremum is β(0)=1. Thus, β(x)=1 if and only if x=0. It should be noted that if 0<β(x)<1, then there exists n∈Z such that β(x)=f(n). Indeed, it is easy to see from definition of f and beta that x∈⋃i∈Z‍2iZ and there exist m,m′∈Z such that 0<f(m′)<β(x)<f(m)<1 (see the remark above about the convergence of f to zero and one); therefore, x∈2m′Z and x∉2mZ. Hence, β(x)=f(n) for a suitable n∈Z for which m′<n<m, since the supremum is calculated over only a finite set of linearly ordered values.To demonstrate thatβ is a fuzzy Z-submodule of Q, let us first consider the cases t=0 and t=1. One can simply check that β0=Q and β1={0}. If t∈(0,1), then there exists exactly one n∈Z such that f(n-1)<t≤f(n) and βt=2nZ. Therefore, each t-level subset of β is a Z-submonoid of Q. Since β(0)=1, β∈Fz(Z)¯.Now, let us consider(8)β⋆=β⋆f=γ⋆∣γ∈fzR,γ⊆β,where β and ⋆ have been defined above. Put K=⋃i∈Z‍2iZ. Let us show that β⋆=χK. Since β(x)=0 for any x∈Q∖K, we have β⋆(x)=0; therefore, β⋆⊆χK. To show the opposite inequality, let x∈K and without loss of generality let x∈2n0Z and x∉2n0+1Z. Consider xn∈2nZ for n=n0+1,n0+2,…. It is easy to see that αn=〈(x)β(x),(xn)β(xn)〉⊆β and αn is finitely generated Z-submodule of Q for any n>n0. Moreover, αn(x)=β(x) and αn(xn)=β(xn), where β(xn)>β(x)>0. Then, by definition of ⋆, we obtain (9)β⋆f≥⋃n=n0+1∞αn⋆x≥⋁n=n0+1∞βx=⋁n=n0+1∞fn=1=χKx.Now, let us demonstrate the false argument. Let us considerα=〈x1〉 for some x∈K∖{0}. Obviously, α∈fz(Z) and α⊆β⋆f. According to our false argument, it should be true that “since α is finitely generated, α is contained in finitely many γi⋆ with γi∈fz(Z) and γi⊆β.” But this is impossible, because for any choice of finitely many γ1,…,γn∈fz(Z) we can find x1,…,xm∈K (it is sufficient to consider elements of K that are used for generating γ1,…,γn) such that (∑i=1n‍γi⋆)(x)≤⋁j=1mβ(xj)<1=α(x).


### 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 generatedR-submodule of L for each element of I. 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 a=⋁i∈Iai does not imply the existence of ai0, i0∈I, with a≤ai0.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,(⋃n∈Z+‍βn)⋆=⋃n∈Z+‍βn⋆, 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.


## 3.1. The Fuzzy and Classical Statements

A Classical Argument. Let B be a submodule of R in L and let I be a finitely generated submodule of R in L such that I⊆⋃‍{A⋆∣A∈f(R)andA⊆B}, where ⋆ is a classical semistar operation on R. Then, I is contained in some Ai⋆ with Ai∈f(R) and Ai⊆B. This classical argument is a well-known simple argument in multiplicative ideal theory. In fact, suppose we set I=x1,…,xs as a finitely generated ideal such that I⊆⋃‍{A⋆∣A∈f(R)andA⊆B}. Then, for each i=1,…,s, xi∈Ai⋆ with Ai∈f(R) and Ai⊆B. So, I=x1,…,xs⊆∑i=1s‍Ai⋆. Now, using the well-known facts that A⊆A⋆ and ∑i=1s‍Ai⋆⋆=∑i=1s‍Ai⋆, for a classical semistar operation ⋆ on R, we obtain that I⊆∑i=1s‍Ai⋆. Now, since each Ai is finitely generated, the finite sum of Ai’s is also finitely generated and this completes the proof.The Fuzzy Counterpart of the Above Classical Argument. Let β be a fuzzy R-submodule of L and let α be a finitely generated fuzzy R-submodule of L such that α⊆⋃‍{γ⋆∣γ∈fz(R)andγ⊆β}, where ⋆ is a fuzzy semistar operation on R. Then, α is contained in some γi⋆, with γi∈fz(R) and γi⊆β.


## 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):⋆ : F ¯ z ( Z ) → F ¯ z ( Z ) by (3)β⋆x=1,ifx=0;⋁y∈Q∖0βy,ifβx≠0,x≠0;0,ifβx=0,for any β∈F¯z(Z).LetQ denote the quotient field of all rational numbers. We define β:Q→[0,1] (note that the unit interval is a completely distributive lattice), and we use the known fact that β is a fuzzy Z-submonoid of Q if and only if βt is a Z-submonoid of Q for any t∈[0,1] and β(0)=1. Let f:Z→[0,1] be defined by (4)fn=121+sgnnnn+1,where sgn is the signature function and |n| denotes the absolute value of n. It is easy to see that f(n)→1 for n→∞ and f(n)→0 for n→-∞. Consider an infinite sequence of Z-submodules of Q as follows: (5)⋯2nZ⊆2n-1Z⊆⋯20Z⊆⋯12n-1Z⊆12nZ⊆⋯Q.Obviously, 2nZ is a Z-submodule of Q for any n∈Z, since r2n-s2n=(r-s)2n and r(s2n)=(rs)2n. Moreover, 2nZ⊆2mZ, whenever m<n. Indeed, r2n=(r2n-m)2m, which implies r2n∈2mZ. Then, one can define β for any x∈Q by (6)βx=⋁fn∣x∈2nZ,n∈Z.Note that if x∉2nZ for any n∈Z, then β(x)=0 (e.g., β(2)=0). On the other hand, (7)β1=β3=12,β12=β32=14,β2=β6=34.Since 0∈2nZ for any n∈Z (and 0 is the unique element having this property), a consequence of the supremum is β(0)=1. Thus, β(x)=1 if and only if x=0. It should be noted that if 0<β(x)<1, then there exists n∈Z such that β(x)=f(n). Indeed, it is easy to see from definition of f and beta that x∈⋃i∈Z‍2iZ and there exist m,m′∈Z such that 0<f(m′)<β(x)<f(m)<1 (see the remark above about the convergence of f to zero and one); therefore, x∈2m′Z and x∉2mZ. Hence, β(x)=f(n) for a suitable n∈Z for which m′<n<m, since the supremum is calculated over only a finite set of linearly ordered values.To demonstrate thatβ is a fuzzy Z-submodule of Q, let us first consider the cases t=0 and t=1. One can simply check that β0=Q and β1={0}. If t∈(0,1), then there exists exactly one n∈Z such that f(n-1)<t≤f(n) and βt=2nZ. Therefore, each t-level subset of β is a Z-submonoid of Q. Since β(0)=1, β∈Fz(Z)¯.Now, let us consider(8)β⋆=β⋆f=γ⋆∣γ∈fzR,γ⊆β,where β and ⋆ have been defined above. Put K=⋃i∈Z‍2iZ. Let us show that β⋆=χK. Since β(x)=0 for any x∈Q∖K, we have β⋆(x)=0; therefore, β⋆⊆χK. To show the opposite inequality, let x∈K and without loss of generality let x∈2n0Z and x∉2n0+1Z. Consider xn∈2nZ for n=n0+1,n0+2,…. It is easy to see that αn=〈(x)β(x),(xn)β(xn)〉⊆β and αn is finitely generated Z-submodule of Q for any n>n0. Moreover, αn(x)=β(x) and αn(xn)=β(xn), where β(xn)>β(x)>0. Then, by definition of ⋆, we obtain (9)β⋆f≥⋃n=n0+1∞αn⋆x≥⋁n=n0+1∞βx=⋁n=n0+1∞fn=1=χKx.Now, let us demonstrate the false argument. Let us considerα=〈x1〉 for some x∈K∖{0}. Obviously, α∈fz(Z) and α⊆β⋆f. According to our false argument, it should be true that “since α is finitely generated, α is contained in finitely many γi⋆ with γi∈fz(Z) and γi⊆β.” But this is impossible, because for any choice of finitely many γ1,…,γn∈fz(Z) we can find x1,…,xm∈K (it is sufficient to consider elements of K that are used for generating γ1,…,γn) such that (∑i=1n‍γi⋆)(x)≤⋁j=1mβ(xj)<1=α(x).


## 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 generatedR-submodule of L for each element of I. 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 a=⋁i∈Iai does not imply the existence of ai0, i0∈I, with a≤ai0.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,(⋃n∈Z+‍βn)⋆=⋃n∈Z+‍βn⋆, 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.



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