research article fuzzy logic versus classical logic: an...

Research ArticleFuzzy Logic versus Classical Logic An Example inMultiplicative Ideal Theory

Olivier A Heubo-Kwegna

Department of Mathematical Sciences Saginaw Valley State University 7400 Bay Road University Center MI 48710-0001 USA

Correspondence should be addressed to Olivier A Heubo-Kwegna oheubokwsvsuedu

Received 6 October 2016 Accepted 24 November 2016

Academic Editor Erich Peter Klement

Copyright copy 2016 Olivier A Heubo-Kwegna This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We discuss a fuzzy result by displaying an example that shows how a classical argument fails to work when one passes from classicallogic to fuzzy logic Precisely we present an example to show that in the fuzzy context the fact that the supremum is naturally usedin 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 intro-duce fuzzy algebra by writing a paper on fuzzy groups [1]The introduction of fuzzy groups then motivated severalresearchers to shift their interest to the extension of theseminal work of Zadeh [2] on fuzzy subsets of a set toalgebraic structures such as rings and modules [3ndash7] Inthat regard Lee and Mordeson in [3 4] introduced thenotion of fractionary fuzzy ideal and the notion of invertiblefractionary fuzzy ideal and used these notions to characterizeDedekind domains in terms of the invertibility of certainfractionary fuzzy ideals leading to the fuzzification of oneof the main results in multiplicative ideal theory Othersignificant introduced notions to tackle the fuzzificationof multiplicative ideal theory are the notion of fuzzy staroperation [8] and the notion of fuzzy semistar operation[9 10] on integral domains This paper is concerned with thefuzzification of multiplicative ideal theory in commutativealgebra (see eg [1 3 5 8ndash11])

In the field of commutative ring it is customary to use staroperations not only to generalize classical domains but alsoto produce a common treatment anddeeper understanding ofthose domains Some of the instances are the notion of Prufer⋆-multiplication domain which generalizes the notion ofPrufer domain [12] and the notion of ⋆-completely integrallyclosed domain which generalizes the notion of completelyintegrally closed domain [13 14] The importance of star

operations in the classical theory has led scholars to beinterested in fuzzy star operations introduced in [8] and thishas been generalized to fuzzy semistar operations in [10] thisgeneralization has led to more fuzzification of main results inmultiplicative ideal theory

In this note we focus on some classical arguments ofmultiplicative ideal theory that do not hold in the fuzzycontext The example chosen is to infer that what appearsto be pretty simple and even rather easy in the context ofclassical logic may not be true in the fuzzy context Soone challenge of fuzzification is to detect any defect orincongruous statement that may first appear benign but is areal poison in the argument used to prove fuzzy statementsPrecisely in our example we display the difficulty in howthe natural definition in the fuzzy context may make ita little bit more challenging to work with in comparisonwith its equivalent classical definition For an overview ofall definitions of fuzzy submodules fuzzy ideals and fuzzy(semi)star operations (of finite character) the reader mayrefer to [8ndash10 15]

2 Preliminaries and Notations

Recall that an integral domain 119877 is a commutative ring withidentity and no-zero divisors Hence its quotient ring 119871 is afield A group (119872 +) is an 119877-module if there is a mapping119877times119872 rarr119872 (119903 119909) 997891rarr 119903119909 satisfying the following conditions

Hindawi Publishing CorporationAdvances in Fuzzy SystemsVolume 2016 Article ID 3839265 4 pageshttpdxdoiorg10115520163839265

2 Advances in Fuzzy Systems

1119909 = 119909 119903(119909 minus 119910) = 119903119909 minus 119903119910 and (119903119905)119909 = 119903(119905119909) for all 119903 119905 isin119877 and 119909 119910 isin 119872 where 1 is the identity of 119877 Note that thequotient field 119870 of an integral domain 119877 is an 119877-module An119877-submodule119873 of an 119877-module119872 is a subgroup of119872 suchthat 119903119909 isin 119873 for all 119903 isin 119877 and 119909 isin 119873 For more reading onintegral domains andmodules the reader may refer to [7 15]Recall also that a star operation on 119877 is a mapping 119860 rarr 119860⋆of 119865(119877) into 119865(119877) such that for all 119860 119861 isin 119865(119877) and for all119886 isin 119871 0

(i) (119886)⋆ = (119886) and (119886119860)⋆ = 119886119860⋆(ii) 119860 sube 119861 rArr 119860⋆ sube 119861⋆(iii) 119860 sube 119860⋆ and 119860⋆⋆ fl (119860⋆)⋆ = 119860⋆

For an overview of star operations the readermay refer to [15Sections 32 and 34]

A fuzzy subset of 119871 is a function from119871 into the real closedinterval [0 1] We say 120572 sube 120573 if 120572(119909) le 120573(119909) for all 119909 isin 119871The intersection ⋂119894isin119868 120572119894 of the fuzzy subsets 120572119894rsquos is definedas ⋂119894isin119868 120572119868(119909) = ⋀119894120572(119909) and the union ⋃119894isin119868 120572119868 of the fuzzysubsets 120572119894rsquos is defined as⋃119894isin119868 120572119894(119909) = ⋁119894120572(119909) for every 119909 isin 119871Let 120573119905 = 119909 isin 119877 120573(119909) ge 119905 then 120573119905 is called a level subset of120573 We let 120594119860 denote the characteristic function of the subset119860 of 119877 A fuzzy subset of 119871 is a fuzzy 119877-submodule of 119871 if120573(119909 minus 119910) ge 120573(119909) and 120573(119910) 120573(119903119909) ge 120573(119909) and 120573(0) = 1 forevery 119909 119910 isin 119871 and every 119903 isin 119877 Note that a fuzzy subset 120573 of119871 is a fuzzy 119877-submodule of 119871 if and only if 120573(0) = 1 and 120573119905is an 119877-submodule of 119871 for every real number 119905 in [0 1] Let119889119905 denote the fuzzy subset of 119871 defined as follows for each 119909in 119871 119889119905(119909) = 119905 if 119909 = 119889 and 119889119905(119909) = 0 otherwise We call119889119905 a fuzzy singleton A fuzzy 119877-submodule 120573 of 119871 is finitelygenerated if120573 is generated by somefinite fuzzy singletons thatis it is the smallest fuzzy 119877-submodule of 119871 containing thosefuzzy singletonsThroughout this paper119865119911(119877) denotes the setof all fuzzy 119877-submodules of 119871 and 119891119911(119877) denotes the set ofall finitely generated fuzzy 119877-submodules of 119871Definition 1 (see [9]) A fuzzy semistar operation on 119877 is amapping ⋆ 119865119911(119877) rarr 119865119911(119877) 120573 997891rarr 120573⋆ which satisfies thefollowing three properties for all 120572 120573 isin 119865119911(119877) and 0 = 119889 isin119870

(⋆1) (1198891 ∘ 120573)⋆ = 1198891 ∘ 120573⋆(⋆2) 120572 sube 120573 rArr 120572⋆ sube 120573⋆(⋆3) 120573 sube 120573⋆ and 120573⋆⋆ fl (120573⋆)⋆ = 120573⋆Recall from [9] that a fuzzy semistar operation ⋆ on 119877

is union preserving if (⋃119899isinZ+ 120573119899)⋆ = ⋃119899isinZ+ 120573⋆119899 Note that thepreservation of union on⋆ is over a countable set Now definea mapping ⋆119891 from 119865119911(119877) into 119865119911(119877) as follows

120573 997891997888rarr 120573⋆119891 fl⋃120572⋆ | 120572 isin 119891119911 (119877) 120572 sube 120573 (1)

Then if ⋆ is a union preserving fuzzy semistar operation on119877 then ⋆119891 is a fuzzy semistar operation on 119877 [9 Theorem35] This leads to the following definition

Definition 2 (see [9]) Let ⋆ be a fuzzy semistar operation on119877

(1) If ⋆119891 is a fuzzy semistar operation on 119877 then ⋆119891 iscalled the fuzzy semistar operation of finite character(or finite type) associated with ⋆

(2) ⋆ is called a fuzzy semistar operation of finite characterif ⋆ = ⋆119891

Example 3 (1) It is clear by definition that (⋆119891)119891 = ⋆119891 thatis ⋆119891 is of finite character whenever ⋆119891 is a fuzzy semistaroperation on 119877 for any fuzzy semistar operation ⋆ on 119877

(2) The constant map 120573 997891rarr 120594119871 is also trivially a fuzzysemistar operation on 119877 that is not of finite character

(3) Let Z denote the set of all integers with quotient fieldQ of all rational numbers Let 119871 = [0 1] be the unit interval(note that the unit interval is a completely distributive lattice)Define ⋆ 119865119911(Z) rarr 119865119911(Z) by

120573⋆ (119909) =

1 if 119909 = 0⋁119910isinQ0

120573 (119910) if 120573 (119909) = 0 119909 = 00 if 120573 (119909) = 0

(2)

for any 120573 isin 119865119911(Z) Then ⋆ is a fuzzy semistar operation onZ of finite character (the reader may refer to [9 Example 38(2)] for the proof of this fact)

3 Fuzzy Logic versus ClassicalLogic An Example

Recall from [9] that a fuzzy semistar operation ⋆ on 119877 is saidto be union preserving if (⋃119899isinZ+ 120573119899)⋆ = ⋃119899isinZ+ 120573⋆119899 Note thatthe preservation of union on ⋆ is over a countable set Alsorecall the following result in [9]

Theorem4 (see [9Theorem 35]) Let⋆ be a union preservingfuzzy semistar operation on 119877 Then ⋆119891 is a fuzzy semistaroperation on 119877

Let 119877 be an integral domain with quotient field 119871 Recallthat 119865119911(119877) denotes the set of all fuzzy 119877-submodules of 119871and 119891119911(119877) denotes the set of finitely generated fuzzy 119877-submodules of 119871 Now we claim that we could not get rid ofthe assumption in Theorem 4 because we could not use thefuzzy counterpart of the following classical argument below

31 The Fuzzy and Classical Statements

A Classical Argument Let 119861 be a submodule of 119877 in 119871 andlet 119868 be a finitely generated submodule of 119877 in 119871 such that119868 sube ⋃119860⋆ | 119860 isin 119891(119877) and 119860 sube 119861 where ⋆ is a classicalsemistar operation on 119877 Then 119868 is contained in some 119860⋆119894with 119860 119894 isin 119891(119877) and 119860 119894 sube 119861 This classical argument is awell-known simple argument in multiplicative ideal theoryIn fact suppose we set 119868 = ⟨1199091 119909119904⟩ as a finitely generatedideal such that 119868 sube ⋃119860⋆ | 119860 isin 119891(119877) and 119860 sube 119861 Thenfor each 119894 = 1 119904 119909119894 isin 119860⋆119894 with 119860 119894 isin 119891(119877) and 119860 119894 sube 119861So 119868 = ⟨1199091 119909119904⟩ sube sum119904119894=1 119860⋆119894 Now using the well-knownfacts that 119860 sube 119860⋆ and (sum119904119894=1 119860⋆119894 )⋆ = (sum119904119894=1 119860 119894)⋆ for a classical

Advances in Fuzzy Systems 3

semistar operation ⋆ on 119877 we obtain that 119868 sube (sum119904119894=1 119860 119894)⋆Now since each119860 119894 is finitely generated the finite sum of 119860 119894rsquosis also finitely generated and this completes the proof

The Fuzzy Counterpart of the Above Classical Argument Let 120573be a fuzzy 119877-submodule of 119871 and let 120572 be a finitely generatedfuzzy 119877-submodule of 119871 such that 120572 sube ⋃120574⋆ | 120574 isin119891119911(119877) and 120574 sube 120573 where⋆ is a fuzzy semistar operation on119877Then 120572 is contained in some 120574⋆119894 with 120574119894 isin 119891119911(119877) and 120574119894 sube 12057332 A Counterexample to Negate the Fuzzy Counterpart Wenow produce an example to prove that the fuzzy counterpartstatement is false Note that the reason why the counterpartmay be false is clearly the fact that the union in the fuzzycontext is the supremum So the real challenge here isto construct a counterexample that will clearly justify thewrongness of the argument

TheCounterexample Let ⋆ be the fuzzy semistar operation offinite character as defined in Example 3(4)⋆ 119865119911(Z) rarr 119865119911(Z) by

120573⋆ (119909) =

1 if 119909 = 0⋁119910isinQ0

120573 (119910) if 120573 (119909) = 0 119909 = 00 if 120573 (119909) = 0

(3)

for any 120573 isin 119865119911(Z)Let Q denote the quotient field of all rational numbers

We define 120573 Q rarr [0 1] (note that the unit interval is acompletely distributive lattice) and we use the known factthat 120573 is a fuzzy Z-submonoid of Q if and only if 120573119905 is aZ-submonoid of Q for any 119905 isin [0 1] and 120573(0) = 1 Let119891 Zrarr [0 1] be defined by

119891 (119899) = 12 (1 + sgn (119899) |119899||119899| + 1) (4)

where sgn is the signature function and |119899| denotes theabsolute value of 119899 It is easy to see that 119891(119899) rarr 1 for 119899 rarr infinand 119891(119899) rarr 0 for 119899 rarr minusinfin Consider an infinite sequence ofZ-submodules ofQ as follows

sdot sdot sdot 2119899Z sube 2119899minus1Z sube sdot sdot sdot 20Z sube sdot sdot sdot 12119899minus1Z sube 12119899Zsube sdot sdot sdotQ

(5)

Obviously 2119899Z is a Z-submodule of Q for any 119899 isin Z since1199032119899 minus 1199042119899 = (119903 minus 119904)2119899 and 119903(1199042119899) = (119903119904)2119899 Moreover 2119899Z sube2119898Z whenever 119898 lt 119899 Indeed 1199032119899 = (1199032119899minus119898)2119898 whichimplies 1199032119899 isin 2119898Z Then one can define 120573 for any 119909 isin Q

by

120573 (119909) = ⋁119891 (119899) | 119909 isin 2119899Z 119899 isin Z (6)

Note that if 119909 notin 2119899Z for any 119899 isin Z then 120573(119909) = 0 (eg120573(radic2) = 0) On the other hand

120573 (1) = 120573 (3) = 12 120573 (12) = 120573(32) = 14 120573 (2) = 120573 (6) = 34

(7)

Since 0 isin 2119899Z for any 119899 isin Z (and 0 is the unique elementhaving this property) a consequence of the supremum is120573(0) = 1 Thus 120573(119909) = 1 if and only if 119909 = 0 It should benoted that if 0 lt 120573(119909) lt 1 then there exists 119899 isin Z such that120573(119909) = 119891(119899) Indeed it is easy to see from definition of 119891 and119887119890119905119886 that 119909 isin ⋃119894isinZ 2119894Z and there exist 1198981198981015840 isin Z such that0 lt 119891(1198981015840) lt 120573(119909) lt 119891(119898) lt 1 (see the remark above aboutthe convergence of 119891 to zero and one) therefore 119909 isin 21198981015840Zand 119909 notin 2119898Z Hence 120573(119909) = 119891(119899) for a suitable 119899 isin Z forwhich 1198981015840 lt 119899 lt 119898 since the supremum is calculated overonly a finite set of linearly ordered values

To demonstrate that 120573 is a fuzzy Z-submodule of Q letus first consider the cases 119905 = 0 and 119905 = 1 One can simplycheck that 1205730 = Q and 1205731 = 0 If 119905 isin (0 1) then there existsexactly one 119899 isin Z such that119891(119899minus1) lt 119905 le 119891(119899) and 120573119905 = 2119899ZTherefore each 119905-level subset of 120573 is a Z-submonoid of QSince 120573(0) = 1 120573 isin 119865119911(Z)

Now let us consider

120573⋆ = 120573⋆119891 = 120574⋆ | 120574 isin 119891119911 (119877) 120574 sube 120573 (8)

where 120573 and ⋆ have been defined above Put K = ⋃119894isinZ 2119894ZLet us show that 120573⋆ = 120594K Since 120573(119909) = 0 for any 119909 isin Q Kwe have 120573⋆(119909) = 0 therefore 120573⋆ sube 120594K To show the oppositeinequality let 119909 isin K and without loss of generality let 119909 isin21198990Z and 119909 notin 21198990+1Z Consider 119909119899 isin 2119899Z for 119899 = 1198990 + 1 1198990 +2 It is easy to see that 120572119899 = ⟨(119909)120573(119909) (119909119899)120573(119909119899)⟩ sube 120573 and120572119899 is finitely generated Z-submodule of Q for any 119899 gt 1198990Moreover 120572119899(119909) = 120573(119909) and 120572119899(119909119899) = 120573(119909119899) where 120573(119909119899) gt120573(119909) gt 0 Then by definition of ⋆ we obtain120573⋆119891 ge ( infin⋃

119899=1198990+1

120572⋆119899) (119909) geinfin⋁119899=1198990+1

120573 (119909) = infin⋁119899=1198990+1

119891 (119899) = 1= 120594K (119909)

(9)

Now let us demonstrate the false argument Let usconsider 120572 = ⟨1199091⟩ for some 119909 isin K 0 Obviously120572 isin 119891119911(Z) and 120572 sube 120573⋆119891 According to our false argumentit should be true that ldquosince 120572 is finitely generated 120572 iscontained in finitely many 120574⋆119894 with 120574119894 isin 119891119911(Z) and 120574119894 sube120573rdquo But this is impossible because for any choice of finitelymany 1205741 120574119899 isin 119891119911(Z) we can find 1199091 119909119898 isin K

(it is sufficient to consider elements of K that are used forgenerating 1205741 120574119899) such that (sum119899119894=1 120574⋆119894 )(119909) le ⋁119898119895=1120573(119909119895) lt1 = 120572(119909)


33 Final Remark The proof of the classical argument holdsdue to the fact that the classical union is involved allowingthe choice of a finitely generated 119877-submodule of 119871 for eachelement of 119868 However in the fuzzy counterpart statementthe fuzzy union is defined in terms of the supremum and thetechnique used in the proof of the classical argument cannotapply in the fuzzy context since clearly 119886 = ⋁119894isin119868119886119894 does notimply the existence of 1198861198940 1198940 isin 119868 with 119886 le 1198861198940

We must also note that the fuzzy counterpart statementis the natural one that grasps some thoughts about thecontext in which the crisp result can be extended In fact thecondition of union preserving of fuzzy star operation that is(⋃119899isinZ+ 120573119899)⋆ = ⋃119899isinZ+ 120573⋆119899 which does not always hold in thefuzzy context is not needed in the crisp case to get a classicalfinite character semistar operationThis additional conditionof union preserving of fuzzy star operation will make ourfuzzy counterpart statement true

Competing Interests

The author declares that there are no competing interestsregarding the publication of this paper

References

[1] A Rosenfeld ldquoFuzzy groupsrdquo Journal of Mathematical Analysisand Applications vol 35 pp 512ndash517 1971

[2] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[3] K H Lee and J N Mordeson ldquoFractionary fuzzy ideals andfuzzy invertible fractionary idealsrdquo Fuzzy Sets and Systems vol5 pp 875ndash883 1997

[4] K H Lee and J N Mordeson ldquoFractionary fuzzy ideals andDedekind domainsrdquo Fuzzy Sets and Systems An InternationalJournal in Information Science and Engineering vol 99 no 1pp 105ndash110 1998

[5] W J Liu ldquoOperations on fuzzy idealsrdquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 11 no 1 pp 31ndash41 1983

[6] P Lubczonok ldquoFuzzy vector spacesrdquo Fuzzy Sets and SystemsAn International Journal in Information Science andEngineeringvol 38 no 3 pp 329ndash343 1990

[7] H Matsumura Commutative Ring Theory Cambridge Univer-sity Press Cambridge UK 1986

[8] H Kim M O Kim S-M Park and Y S Park ldquoFuzzy star-operations on an integral domainrdquo Fuzzy Sets and Systems vol136 no 1 pp 105ndash114 2003

[9] O A Heubo-Kwegna ldquoFuzzy semistar operations of finitecharacter on integral domainsrdquo Information Sciences vol 269pp 366ndash377 2014

[10] O A Heubo-Kwegna ldquoFuzzy semistar operations on integraldomainsrdquo Fuzzy Sets and Systems vol 210 pp 117ndash126 2013

[11] J N Mordeson and D S Malik Fuzzy commutative algebraWorld Scientific Publishing Singapore Asia 1998

[12] G W Chang ldquoPrufer lowast-multiplication domains Nagata ringsand Kronecker function ringsrdquo Journal of Algebra vol 319 no1 pp 309ndash319 2008

[13] D D Anderson D F Anderson M Fontana and M Zahfrul-lah ldquoOn v-domains and star operationsrdquo Communications inAlgebra vol 2 pp 141ndash145 2008

[14] E G Houston S B Malik and J L Mott ldquoCharacterizationof lowast-multiplication domainsrdquo Canadian Mathematical Bulletinvol 27 pp 48ndash52 1984

[15] R Gilmer Multiplicative Ideal Theory Corrected Reprint of the1972 Edition vol 90 of Queenrsquos Papers in Pure and AppliedMathematics Queenrsquos University Kingston Canada 1992

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1119909 = 119909 119903(119909 minus 119910) = 119903119909 minus 119903119910 and (119903119905)119909 = 119903(119905119909) for all 119903 119905 isin119877 and 119909 119910 isin 119872 where 1 is the identity of 119877 Note that thequotient field 119870 of an integral domain 119877 is an 119877-module An119877-submodule119873 of an 119877-module119872 is a subgroup of119872 suchthat 119903119909 isin 119873 for all 119903 isin 119877 and 119909 isin 119873 For more reading onintegral domains andmodules the reader may refer to [7 15]Recall also that a star operation on 119877 is a mapping 119860 rarr 119860⋆of 119865(119877) into 119865(119877) such that for all 119860 119861 isin 119865(119877) and for all119886 isin 119871 0

(i) (119886)⋆ = (119886) and (119886119860)⋆ = 119886119860⋆(ii) 119860 sube 119861 rArr 119860⋆ sube 119861⋆(iii) 119860 sube 119860⋆ and 119860⋆⋆ fl (119860⋆)⋆ = 119860⋆

For an overview of star operations the readermay refer to [15Sections 32 and 34]

A fuzzy subset of 119871 is a function from119871 into the real closedinterval [0 1] We say 120572 sube 120573 if 120572(119909) le 120573(119909) for all 119909 isin 119871The intersection ⋂119894isin119868 120572119894 of the fuzzy subsets 120572119894rsquos is definedas ⋂119894isin119868 120572119868(119909) = ⋀119894120572(119909) and the union ⋃119894isin119868 120572119868 of the fuzzysubsets 120572119894rsquos is defined as⋃119894isin119868 120572119894(119909) = ⋁119894120572(119909) for every 119909 isin 119871Let 120573119905 = 119909 isin 119877 120573(119909) ge 119905 then 120573119905 is called a level subset of120573 We let 120594119860 denote the characteristic function of the subset119860 of 119877 A fuzzy subset of 119871 is a fuzzy 119877-submodule of 119871 if120573(119909 minus 119910) ge 120573(119909) and 120573(119910) 120573(119903119909) ge 120573(119909) and 120573(0) = 1 forevery 119909 119910 isin 119871 and every 119903 isin 119877 Note that a fuzzy subset 120573 of119871 is a fuzzy 119877-submodule of 119871 if and only if 120573(0) = 1 and 120573119905is an 119877-submodule of 119871 for every real number 119905 in [0 1] Let119889119905 denote the fuzzy subset of 119871 defined as follows for each 119909in 119871 119889119905(119909) = 119905 if 119909 = 119889 and 119889119905(119909) = 0 otherwise We call119889119905 a fuzzy singleton A fuzzy 119877-submodule 120573 of 119871 is finitelygenerated if120573 is generated by somefinite fuzzy singletons thatis it is the smallest fuzzy 119877-submodule of 119871 containing thosefuzzy singletonsThroughout this paper119865119911(119877) denotes the setof all fuzzy 119877-submodules of 119871 and 119891119911(119877) denotes the set ofall finitely generated fuzzy 119877-submodules of 119871Definition 1 (see [9]) A fuzzy semistar operation on 119877 is amapping ⋆ 119865119911(119877) rarr 119865119911(119877) 120573 997891rarr 120573⋆ which satisfies thefollowing three properties for all 120572 120573 isin 119865119911(119877) and 0 = 119889 isin119870

(⋆1) (1198891 ∘ 120573)⋆ = 1198891 ∘ 120573⋆(⋆2) 120572 sube 120573 rArr 120572⋆ sube 120573⋆(⋆3) 120573 sube 120573⋆ and 120573⋆⋆ fl (120573⋆)⋆ = 120573⋆Recall from [9] that a fuzzy semistar operation ⋆ on 119877

is union preserving if (⋃119899isinZ+ 120573119899)⋆ = ⋃119899isinZ+ 120573⋆119899 Note that thepreservation of union on⋆ is over a countable set Now definea mapping ⋆119891 from 119865119911(119877) into 119865119911(119877) as follows

120573 997891997888rarr 120573⋆119891 fl⋃120572⋆ | 120572 isin 119891119911 (119877) 120572 sube 120573 (1)

Then if ⋆ is a union preserving fuzzy semistar operation on119877 then ⋆119891 is a fuzzy semistar operation on 119877 [9 Theorem35] This leads to the following definition

Definition 2 (see [9]) Let ⋆ be a fuzzy semistar operation on119877

(1) If ⋆119891 is a fuzzy semistar operation on 119877 then ⋆119891 iscalled the fuzzy semistar operation of finite character(or finite type) associated with ⋆

(2) ⋆ is called a fuzzy semistar operation of finite characterif ⋆ = ⋆119891

Example 3 (1) It is clear by definition that (⋆119891)119891 = ⋆119891 thatis ⋆119891 is of finite character whenever ⋆119891 is a fuzzy semistaroperation on 119877 for any fuzzy semistar operation ⋆ on 119877

(2) The constant map 120573 997891rarr 120594119871 is also trivially a fuzzysemistar operation on 119877 that is not of finite character

(3) Let Z denote the set of all integers with quotient fieldQ of all rational numbers Let 119871 = [0 1] be the unit interval(note that the unit interval is a completely distributive lattice)Define ⋆ 119865119911(Z) rarr 119865119911(Z) by

120573⋆ (119909) =

1 if 119909 = 0⋁119910isinQ0

120573 (119910) if 120573 (119909) = 0 119909 = 00 if 120573 (119909) = 0

(2)

for any 120573 isin 119865119911(Z) Then ⋆ is a fuzzy semistar operation onZ of finite character (the reader may refer to [9 Example 38(2)] for the proof of this fact)

3 Fuzzy Logic versus ClassicalLogic An Example

Recall from [9] that a fuzzy semistar operation ⋆ on 119877 is saidto be union preserving if (⋃119899isinZ+ 120573119899)⋆ = ⋃119899isinZ+ 120573⋆119899 Note thatthe preservation of union on ⋆ is over a countable set Alsorecall the following result in [9]

Theorem4 (see [9Theorem 35]) Let⋆ be a union preservingfuzzy semistar operation on 119877 Then ⋆119891 is a fuzzy semistaroperation on 119877

Let 119877 be an integral domain with quotient field 119871 Recallthat 119865119911(119877) denotes the set of all fuzzy 119877-submodules of 119871and 119891119911(119877) denotes the set of finitely generated fuzzy 119877-submodules of 119871 Now we claim that we could not get rid ofthe assumption in Theorem 4 because we could not use thefuzzy counterpart of the following classical argument below

31 The Fuzzy and Classical Statements

A Classical Argument Let 119861 be a submodule of 119877 in 119871 andlet 119868 be a finitely generated submodule of 119877 in 119871 such that119868 sube ⋃119860⋆ | 119860 isin 119891(119877) and 119860 sube 119861 where ⋆ is a classicalsemistar operation on 119877 Then 119868 is contained in some 119860⋆119894with 119860 119894 isin 119891(119877) and 119860 119894 sube 119861 This classical argument is awell-known simple argument in multiplicative ideal theoryIn fact suppose we set 119868 = ⟨1199091 119909119904⟩ as a finitely generatedideal such that 119868 sube ⋃119860⋆ | 119860 isin 119891(119877) and 119860 sube 119861 Thenfor each 119894 = 1 119904 119909119894 isin 119860⋆119894 with 119860 119894 isin 119891(119877) and 119860 119894 sube 119861So 119868 = ⟨1199091 119909119904⟩ sube sum119904119894=1 119860⋆119894 Now using the well-knownfacts that 119860 sube 119860⋆ and (sum119904119894=1 119860⋆119894 )⋆ = (sum119904119894=1 119860 119894)⋆ for a classical





120573⋆ (119909) =

1 if 119909 = 0⋁119910isinQ0

120573 (119910) if 120573 (119909) = 0 119909 = 00 if 120573 (119909) = 0

(3)



119891 (119899) = 12 (1 + sgn (119899) |119899||119899| + 1) (4)



(5)


by

120573 (119909) = ⋁119891 (119899) | 119909 isin 2119899Z 119899 isin Z (6)


120573 (1) = 120573 (3) = 12 120573 (12) = 120573(32) = 14 120573 (2) = 120573 (6) = 34

(7)



Now let us consider

120573⋆ = 120573⋆119891 = 120574⋆ | 120574 isin 119891119911 (119877) 120574 sube 120573 (8)


119899=1198990+1

120572⋆119899) (119909) geinfin⋁119899=1198990+1

120573 (119909) = infin⋁119899=1198990+1

119891 (119899) = 1= 120594K (119909)

(9)






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120573⋆ (119909) =

1 if 119909 = 0⋁119910isinQ0

120573 (119910) if 120573 (119909) = 0 119909 = 00 if 120573 (119909) = 0

(3)



119891 (119899) = 12 (1 + sgn (119899) |119899||119899| + 1) (4)



(5)


by

120573 (119909) = ⋁119891 (119899) | 119909 isin 2119899Z 119899 isin Z (6)


120573 (1) = 120573 (3) = 12 120573 (12) = 120573(32) = 14 120573 (2) = 120573 (6) = 34

(7)



Now let us consider

120573⋆ = 120573⋆119891 = 120574⋆ | 120574 isin 119891119911 (119877) 120574 sube 120573 (8)


119899=1198990+1

120572⋆119899) (119909) geinfin⋁119899=1198990+1

120573 (119909) = infin⋁119899=1198990+1

119891 (119899) = 1= 120594K (119909)

(9)






Competing Interests


References























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in






Competing Interests


References























Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in










Advances in

FuzzySystems


Volume 2014












Journal of

Journal of





Advances in

Multimedia


Biomedical Imaging



Advances in


RoboticsJournal of










Advances in



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