CLASSES OF FUZZY FILTERS OF RESIDUATED LATTICE ORDERED MONOIDS

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1 135(2010) MATHEMATICA BOHEMICA No. 1, CLASSES OF FUZZY FILTERS OF RESIDUATED LATTICE ORDERED MONOIDS Jiří Rachůnek, Olomouc, Dana Šalounová, Ostrava (Received February 3, 2009) Abstract. The logical foundations of processes handling uncertainty in information use some classes of algebras as algebraic semantics. Bounded residuated lattice ordered monoids (Rl-monoids) are common generalizations of BL-algebras, i.e., algebras of the propositional basic fuzzy logic, and Heyting algebras, i.e., algebras of the propositional intuitionistic logic. From the point of view of uncertain information, sets of provable formulas in inference systems could be described by fuzzy filters of the corresponding algebras. In the paper we investigate implicative, positive implicative, Boolean and fantastic fuzzy filters of bounded Rl-monoids. Keywords: residuated l-monoid, non-classical logics, basic fuzzy logic, intuitionistic logic, filter, fuzzy filter, BL-algebra, MV-algebra, Heyting algebra MSC 2010: 03B47, 03B52, 03G25, 06D35, 06F05 1. Introduction As is well known, while information processing dealing with certain information is based on the classical two-valued logic, non-classical logics including logics behind fuzzy reasoning handle information with various facets of uncertainty such as fuzziness, randomness, vagueness, etc. The classical two-valued logic has Boolean algebras as an algebraic semantics. Similarly, for important non-classical logics there are algebraic semantics in the form of classes of algebras. Using these classes, one can obtain an algebraization of inference systems that handle various kinds of uncertainty. The sets of provable formulas in inference systems are described by filters, and from the point of view of uncertain information, by fuzzy filters of the corresponding algebras. The first author has been supported by the Council of Czech Government, MSM

2 BL-algebras were introduced by P. Hájek as an algebraic counterpart of the basic fuzzy logic BL[7]. Omitting the requirement of pre-linearity in the definition of a BL-algebra, one obtains the definition of a bounded commutative residuated lattice ordered monoid(rl-monoid). Nevertheless, bounded commutative Rl-monoids are a generalization not only of BL-algebras but also of Heyting algebras which are an algebraic counterpart of the intuitionistic propositional logic. Therefore, bounded commutative Rl-monoids could be taken as an algebraic semantics of a more general logic than Hájek s fuzzy logic. It is known that every BL-algebra(and consequently every MV-algebra[3]) is a subdirect product of linearly ordered BL-algebras. Moreover, a bounded commutative Rl-monoid is a subdirect product of linearly ordered Rl-monoidsifandonlyifitisaBL-algebra[17]. Ontheotherhand, bounded commutative Rl-monoids which need not be BL-algebras can be constructed from BL-algebras by means of other natural operations, e.g. by means of pasting, i.e. ordinal sums. In both the BL-algebras and bounded commutative Rl-monoids, filters coincide with deductive systems of those algebras and are exactly the kernels of their congruences. Various types of filters of BL-algebras(Boolean deductive systems, implicative filters, positive implicative filters, fantastic filters) were studied in[23],[9] and[15]. Generalizations of these kinds of filters were introduced and investigated in[18] and[20]. Fuzzy ideals(or, in the dual form, fuzzy filters) of MV-algebras were introduced and developed in[11],[12], and their generalizations for pseudo MV-algebras in[14] and[5]. Moreover, fuzzy filters of bounded Rl-monoids were recently introduced and studiedin[21].somerelatedresultsonecanalsofindin[25]. Inthepaperwefurtherdevelopthetheoryoffuzzyfiltersofboundedcommutative Rl-monoids. We introduce and investigate implicative fuzzy filters, positive implicative fuzzy filters, Boolean fuzzy filters and fantastic fuzzy filters of bounded commutative Rl-monoids and describe their mutual connection, as well as their relations to the corresponding filters. For concepts and results concerning MV-algebras, BL-algebras and Heyting algebrasseeforinstance[3],[7],[1]. 2. Preliminaries Aboundedcommutative Rl-monoidisanalgebra M = (M;,,,, 0, 1)oftype 2, 2, 2, 2, 0, 0 satisfying the following conditions: 82 (Rl1) (M;, 1)isacommutativemonoid. (Rl2) (M;,, 0, 1)isaboundedlattice.

3 (Rl3) x y zifandonlyif x y zforany x, y, z M. (Rl4) x (x y) = x yforany x, y M. In the sequel, by an Rl-monoid we will mean a bounded commutative Rl-monoid. Onany Rl-monoid Mletusdefineaunaryoperationnegation by x := x 0 forany x M. Remark2.1.Infact,boundedcommutative Rl-monoidscanbealsorecognized as commutative residuated lattices [24], [6] satisfying the divisibility condition or as divisible integral residuated commutative l-monoids [10] or as bounded integral commutative generalized BL-algebras[2],[13],[4]. The above mentioned algebras can be characterized in the class of all Rl-monoids asfollows:an Rl-monoid Mis a) BL-algebraifandonlyif M satisfiestheidentityofpre-linearity (x y) (y x) = 1; b) an MV-algebraifandonlyif Mfulfilsthedoublenegationlaw x = x; c) aheytingalgebraifandonlyiftheoperations isidempotent. When doing calculations, we will use the following list of basic rules for bounded Rl-monoids. Lemma2.2[19],[22]. Inanyboundedcommutative Rl-monoid Mwehavefor any x, y, z M: (1) 1 x = x, (2) x y x y = 1, (3) x y x y, (4) x y x, (5) (x y) z = x (y z) = y (x z), (6) (x y) z = (x z) (y z), (7) x (y z) = (x y) (x z), (8) x x, x = x, (9) x y = y x, (10) (x y) = y x = y x = x y = x y, (11) x y = z x z y, y z x z, (12) x y y x, (13) x y ((x y) y) ((y x) x), (14) x y (y z) (x z), (15) x y (z x) (z y). Anon-emptysubset Fofan Rl-monoid Miscalledafilterof Mif (F1) x, y Fimply x y F; (F2) x F, y M, x yimply y F. 83

4 Asubset Dofan Rl-monoid Miscalledadeductivesystemof Mif (i) 1 D; (ii) x D, x y Dimply y D. Proposition2.3[4]. Let Hbeanon-emptysubsetofan Rl-monoid M. Then Hisafilterof Mifandonlyif Hisadeductivesystemof M. Filters of commutative Rl-monoids are exactly the kernels of their congruences. If Fisafilterof M,then Fisthekerneloftheuniquecongruence Θ(F)suchthat x, y Θ(F)ifandonlyif (x y) (y x) Fforany x, y M.Hencewewill consider quotient Rl-monoids M/F of Rl-monoids M by their filters F. 3. Fuzzy filters of Rl-monoids Let [0, 1]betheclosedunitintervalofrealsandlet M beaset. Recallthat afuzzysetin Misanyfunction ν : M [0, 1]. Afuzzyset νinan Rl-monoid M iscalledafuzzyfilter of M ifany x, y M satisfy (f1) ν(x y) ν(x) ν(y), (f2) x y = ν(x) ν(y). By(f2), it follows immediately that (f3) ν(1) ν(x)forevery x M. Lemma3.1. Let νbeafuzzyfilterofan Rl-monoid M.Thenforany x, y M we have (i) ν(x y) ν(x) ν(y), (ii) ν(x y) = ν(x) ν(y), (iii) ν(x y) = ν(x) ν(y). Proof. Forany x, y Mwehave x y x y x y.then(f2)and(f1) imply ν(x y) ν(x y) ν(x) ν(y).since x y x y x, y,itfollowsby(f1) and(f2)that ν(x) ν(y) ν(x y) ν(x y) ν(x) ν(y). Theorem3.2.Afuzzyset νinan Rl-monoid Misafuzzyfilterof Mifandonly ifitsatisfies(f1)and (f4) ν(x y) ν(x)forany x, y M. Proof. If ν isafuzzyfilterofan Rl-monoid M then x x y implies ν(x) ν(x y). Conversely,if νsatisfies(f1)and(f4)and x y,then ν(y) = ν(x y) ν(x). Hence νisafuzzyfilterof M. 84

5 Theorem3.3. Let νbeafuzzysetinan Rl-monoid M. Thenthefollowing conditions are equivalent. (1) νisafuzzyfilterof M. (2) νsatisfies (f3)andforall x, y M, ( ) ν(y) ν(x) ν(x y). Proof. (1) (2): Let νbeafuzzyfilterof M andlet x, y M. Then,by Lemma3.1(iii), ν(y) ν(x y) = ν((x y) x) = ν(x y) ν(x). Hence ν satisfies the condition(2). (2) (1):Let νbeafuzzysetin Msatisfying(f3)and( ).Let x, y M, x y. Then x y = 1.Thus ν(y) ν(x) ν(1) = ν(x),hence(f2)holds. Further,since x y (x y),by( )and(f2)weget ν(x y) ν(y) ν(y (x y)) ν(y) ν(x). Therefore(f1)isalsosatisfiedandhence νisafuzzyfilter of M. Let F beasubsetof Mandlet α, β [0, 1]besuchthat α > β. Defineafuzzy subset ν F (α, β)in Mby ν F (α, β)(x) := { α, if x F, β, otherwise. Inparticular, ν F (1, 0)isthecharacteristicfunction χ F of F.Wewillusethenotation ν F insteadof ν F (α, β)forevery α, β [0, 1], α > β. Let νbeafuzzysetin Mandlet α [0, 1].Theset U(ν; α) := {x M : ν(x) α} iscalledthelevelsubsetof νdeterminedby α. Kondo and Dudek in[16] formulated and proved the so-called Transfer Principle (TP)whichcanbeusedtoany(general)algebraofanytype: Transfer Principle. Afuzzyset λdefinedina(general)algebra Ahasa property Pifandonlyifallnon-emptylevelsubsets U(λ; α)havetheproperty P. (Aproperty Pisdefinedinastandardwaybymeansoftermsofalgebras. For more information concerning(tp) see[16].) Someoftheassertionsofourpaperwillbeimmediateconsequencesof(TP)and ofitscorollariesin[16],hencetheproofsofthemwillbeomited.thefirstofthemis 85

6 Theorem3.4.Let Fbeanon-emptysubsetofan Rl-monoid M.Thenthefuzzy set ν F isafuzzyfilterof Mifandonlyif Fisafilterof M. Let νbeafuzzysetinan Rl-monoid M.Denoteby M ν theset M ν := {x M : ν(x) = ν(1)}. Notethat M ν = U(ν; ν(1)),hence M ν isaspecialcaseofalevelsubsetof M. Theorem3.5.If νisafuzzyfilterofan Rl-monoid M,then M ν isafilterof M. Proof. Let νbeafuzzyfilterof M. Let x, y M ν,i.e. ν(x) = ν(1) = ν(y). Then ν(x y) ν(x) ν(y) = ν(1),hence ν(x y) = ν(1),thus x y M ν. Further, let x M ν, y M and x y. Then ν(1) = ν(x) ν(y), hence ν(y) = ν(1),andtherefore y M ν. Thatmeans M ν isafilterof M. TheconverseimplicationtothatfromTheorem3.5isnottrueingeneral,noteven for pseudo MV-algebras, as was shown in[5, Example 3.9]. Theorem3.6. Let νbeafuzzysetinan Rl-monoid M.Then νisafuzzyfilter of Mifandonlyifitslevelsubset U(ν; α)isafilterof Mor U(ν; α) = foreach α [0, 1]. Proof. Itfollowsfrom(TP). Theorem3.7. Let νbeafuzzysubsetinan Rl-monoid M.Thenthefollowing conditions are equivalent. (1) νisafuzzyfilterof M. (2) x, y, z M; x (y z) = 1 = ν(z) ν(x) ν(y). Proof. (1) (2): Let ν beafuzzyfilterof M. Let x, y, z M and x (y z) = 1. ThenbyTheorem3.3, ν(y z) ν(x) ν(x (y z)) = ν(x) ν(1) = ν(x). Moreover,alsobyTheorem3.3, ν(z) ν(y) ν(y z),henceweobtain ν(z) ν(y) ν(x). (2) (1): Letafuzzyset νin Msatisfythecondition(2). Let x, y M. Since x (x 1) = 1,wehave ν(1) ν(x) ν(x) = ν(x),hence(f3)issatisfied. Further,since (x y) (x y) = 1weget ν(y) ν(x y) ν(x),thus ν satisfies( ),whichmeans,bytheorem3.3,that νisafuzzyfilterof M. 86

7 Corollary3.8. Afuzzyset νinan Rl-monoid M isafuzzyfilterof M ifand onlyifforall x, y, z M, x y zimplies ν(z) ν(x) ν(y). 4. Implicative fuzzy filters Let Mbean Rl-monoidand F asubsetof M. Then F iscalledanimplicative filterof Mif (I) 1 F; (II) x (y z) F, x y Fimply x z Fforany x, y, z M. By[20],everyimplicativefilterisafilterof M. Afuzzyset νinan Rl-monoid Miscalledanimplicativefuzzyfilterof Miffor any x, y, z M (1) ν(1) ν(x); (2) ν(x (y z)) ν(x y) ν(x z). Proposition 4.1. Every implicative fuzzy filter of an Rl-monoid M is a fuzzy filterof M. Proof. Let νbeanimplicativefuzzyfilterof M. Let α [0, 1]besuchthat U(ν; α).thenforany x U(ν; α)wehave ν(1) ν(x),thus 1 U(ν; α). Let x, x y U(ν; α), i.e. ν(x), ν(x y) α. Then ν(1 x), ν(1 (x y)) α,hence ν(1 (x y)) ν(1 x) α,thusby(2), ν(1 y) α. Thatmeans ν(y) α,andtherefore y U(ν; α). HencebyTheorem3.6, νisa fuzzyfilterof M. Theorem4.2. Afilter F ofan Rl-monoid Misimplicativeifandonlyif ν F is animplicativefuzzyfilterof M. Proof. Itfollowsfrom(TP). Theorem4.3([20,Theorem3.3]). Let F beafilterofan Rl-monoid M. Then the following conditions are equivalent. (a) Fisanimplicativefilterof M. (b) y (y x) Fimplies y x Fforany x, y M. (c) z (y x) Fimplies (z y) (z x) Fforany x, y, z M. (d) z (y (y x)) Fand z Fimply y x Fforany x, y, z M. (e) x (x x) Fforany x M. 87

8 Theorem4.4.Let Fbeafilterofan Rl-monoid M.Thenthefollowingconditions are equivalent. (a) ν F isanimplicativefuzzyfilterof M. (b) ν F (y (y x)) ν F (y x)forany x, y M. (c) ν F (z (y x)) ν F ((z y) (z x))forany x, y, z M. (d) ν F (z (y (y x))) ν F (z) ν F (y x)forany x, y, z M. (e) ν F (x (x x)) = ν F (1). Proof. (a) (b): Let ν F beanimplicativefuzzyfilterof M. ThenbyTheorem4.2, Fisanimplicativefilterof M,andhencebyTheorem4.3, y (y x) F implies y x F forany x, y M. Let x, y Mandlet ν F (y (y x)) = α. Then ν F (y x) = α,andthus ν F satisfiesthecondition(b). Conversely,let ν F satisfy(b). Let x, y M and y (y x) F. Then ν F (y (y x)) = α,hencealso ν F (y x) = α,thatmeans y x F.Therefore by4.3, Fisanimplicativefuzzyfilterof M. The proofs of the equivalences(a) (c),(a) (d) and(a) (e) are analogous, and hence they are omitted. Theorem4.5.Let νbeafuzzyfilterofan Rl-monoid M.Then νisanimplicative fuzzyfilterof Mifandonlyif U(ν; α)isanimplicativefilterforany α [0, 1]such that U(ν; α). Proof. Itfollowsfrom(TP). As a consequence we obtain the following theorem. Theorem4.6. If νisafuzzyfilterofan Rl-monoid M,then νisanimplicativefuzzyfilterof M ifandonlyif U(ν; α)satisfiesanyofconditions(b) (e)of Theorem 4.3foreach α [0, 1]suchthat U(ν; α). Theorem4.7([20,Theorem3.4]). If Fisafilterofan Rl-monoid M,then Fis animplicativefilterifandonlyifthequotient Rl-monoid M/FisaHeytingalgebra. The following assertion follows from Theorems 4.6 and 4.7. Theorem4.8. If νisafuzzyfilterofan Rl-monoid M,then νisanimplicative fuzzyfilterof Mifandonlyifthequotient Rl-monoid M/U(ν; α)isaheytingalgebra forany α [0, 1]suchthat U(ν; α). 88

9 Theorem 4.9([20, Theorem 3.10]). Let M be an Rl-monoid. Then the following conditions are equivalent. (a) MisaHeytingalgebra. (b) Everyfilterof Misimplicative. (c) {1}isanimplicativefilterof M. Theorem Let M be an Rl-monoid. Then the following conditions are equivalent. (a) MisaHeytingalgebra. (b) Everyfuzzyfilterof Misimplicative. (c) Everyfuzzyfilter νof Msuchthat ν(1) = 1isimplicative. (d) χ {1} isanimplicativefuzzyfilterof M. Proof. (a) (b): Let M beaheytingalgebraand νafuzzyfilterof M. If α [0, 1]and U(ν; α),then U(ν; α)is,bytheorem4.9,animplicativefilterof M.HencebyTheorem4.5, νisanimplicativefuzzyfilterof M. (b) (c),(c) (d): Obvious. (d) (a):ifthefuzzyfilter χ {1} = ν {1} (1, 0)isimplicative,thenbyTheorem4.2, {1}isanimplicativefilterof M,andhencebyTheorem4.9, MisaHeytingalgebra. 5. Positive implicative and Boolean fuzzy filters Let M bean Rl-monoidand F asubsetof M. Then F iscalledapositiveimplicativefilterof Mif (I) 1 F; (III) x ((y z) y) Fand x Fimply y Fforany x, y, z M. By[20],everypositiveimplicativefilterof Misafilterof M. Afuzzyset νinan Rl-monoid Miscalledapositiveimplicativefuzzyfilterof M ifforany x, y, z M, (1) ν(1) ν(x); (3) ν(x ((y z) y)) ν(x) ν(y). Proposition 5.1. Every positive implicative fuzzy filter of an Rl-monoid M is a fuzzyfilterof M. Proof. Let νbeapositiveimplicativefuzzyfilterof M, α [0, 1]and U(ν; α).then 1 U(ν; α). 89

10 Further,let x, x y U(ν; α),i.e. ν(x), ν(x y) α.then ν(x ((y 1) y)) = ν(x (1 y)) = ν(x y),hence ν(x ((y 1) y) ν(x) αandthus by(3), ν(y) α.therefore y U(ν; α). Thatmeans,byTheorem3.6, νisafuzzyfilterof M. Theorem5.2. Afilter Fofan Rl-monoid Mispositiveimplicativeifandonly if ν F isapositiveimplicativefuzzyfilterof M. Proof. Let Fbeafilterof M.Letussupposethat Fispositiveimplicative.Let ν F (x ((y z) y)) ν F (x) = α. Then ν F (x ((y z) y)) = α = ν F (x), thus x ((y z) y), x F,andhence y F,thatmeans ν F (y) = α.therefore wegetthat ν F isapositiveimplicativefuzzyfilterof M. Conversely,let ν F beapositiveimplicativefuzzyfilterof M.Let x ((y z) y) F and x F. Then ν F (x ((y z) y)) = α = ν F (x),hence ν F (y) = α andso y F.Thatmeans Fisapositiveimplicativefilterof M. Theorem5.3([20,Theorem3.8]). Let F beafilterofan Rl-monoid M. Then the following conditions are equivalent. (a) Fisapositiveimplicativefilterof M. (b) (x y) x Fimplies x Fforany x, y M. (c) (x x) x Fforany x M. Theorem5.4.Let Fbeafilterofan Rl-monoid M.Thenthefollowingconditions are equivalent. (a) ν F isapositiveimplicativefuzzyfilterof M. (b) ν F ((x y) x) ν F (x)forany x, y M. (c) ν F ((x x) x) = ν F (1). Proof. AnalogoustothatforTheorem4.4. Theorem5.5. Let νbeafuzzyfilterofan Rl-monoid M. Then νisapositive implicativefuzzyfilterof Mifandonlyif U(ν; α)isapositiveimplicativefilterof Mforevery α [0, 1]suchthat U(ν; α). Proof. Letussupposethat νisafuzzyfilterof M.Let νbepositiveimplicative, α [0, 1], U(ν; α), x, y, z Mand x ((y z) y) U(ν; α), x U(ν; α). Then ν(x ((y z) y)), ν(x) α,hence ν(x ((y z) y)) ν(x) α. Since ν(y) ν(x ((y z) y)) ν(x),weget y U(ν; α).thereforethefilter U(ν; α) is positive implicative. Conversely,let νbesuchthat U(ν; α)isapositiveimplicativefilterforany α [0, 1]suchthat U(ν; α).if x, y, z M,then x ((y z) y), x U(ν; (x 90

11 ((y z) y)) x),thusalso y U(ν; (x ((y z) y)) x),hence ν(y) ν((x ((y z) y)) x) = ν(x ((y z) y))) ν(x). Thatmeans νisa positive implicative fuzzy filter. Theorem 5.6. Every positive implicative fuzzy filter of an Rl-monoid M is implicative. Proof. Let νbeapositiveimplicativefuzzyfilterof M.ThenbyTheorem5.5, if α [0, 1]issuchthat U(ν; α) then U(ν; α)isapositiveimplicativefilterof M. Henceby[20,Proposition3.7], U(ν; α)isalsoanimplicativefilterof M.Therefore bytheorem4.5, νisanimplicativefuzzyfilterof M. Theorem5.7([20,Proposition3.11]). Let F beanimplicativefilterofan Rlmonoid M. Then the following conditions are equivalent. (a) Fisapositiveimplicativefilterof M. (b) (x y) y Fimplies (y x) x Fforany x, y M. Theorem 5.8. Let F beanimplicativefilterofan Rl-monoid M. Thenthe following conditions are equivalent. (a) ν F isapositiveimplicativefuzzyfilterof M. (b) ν F ((x y) y) = ν F ((y x) x)forany x, y M. Proof. (a) (b): Let ν F beapositiveimplicativefuzzyfilterof M. Then bytheorem5.2, F isapositiveimplicativefilterof M hence (x y) y F implies (y x) x F forany x, y M. Let ν F ((x y) y) = α. Thenalso ν F ((y x) x) = α,andthus ν F satisfies(b). (b) (a): Let ν F satisfy(b)andlet ν F (x ((y z) y)) = α = ν F (x). Then x ((y z) y), x F,andhencealso (y z) y F. Further, (y z) y (y z) ((y z) z),therefore (y z) ((y z) z) F. Since Fisanimplicativefilterof M,byTheorem4.3weget (y z) z F,and consequentlybytheorem5.7, F isapositiveimplicativefilterof M. Thereforeby Theorem5.2, ν F isapositiveimplicativefuzzyfilterof M. Theorem 5.9([20, Theorem 3.12]). Let M be an Rl-monoid. Then the following conditions are equivalent. (a) {1} is a positive implicative filter. (b) Every filter of M is positive implicative. (c) MisaBooleanalgebra. 91

12 Theorem Let M be an Rl-monoid. Then the following conditions are equivalent. (a) MisaBooleanalgebra. (b) Every fuzzy filter of M is positive implicative. (c) Everyfuzzyfilter νof Msuchthat ν(1) = 1ispositiveimplicative. (d) χ {1} isapositiveimplicativefuzzyfilterof M. Proof. (a) (b):let MbeaBooleanalgebraand νafuzzyfilterof M.Let α [0, 1]besuchthat U(ν; α).thenbytheorem5.9, U(ν; α)isapositiveimplicative filterof M. HencebyTheorem5.5weobtainthat νisapositiveimplicativefuzzy filterof M. (b) (c),(c) (d): Obvious. (d) (a): Let χ {1} = ν {1} (1; 0)beapositiveimplicativefuzzyfilterof M. Then bytheorem5.2wegetthat {1}isapositiveimplicativefilterof M,andtherefore bytheorem5.9, MisaBooleanalgebra. Afilter F ofan Rl-monoid MiscalledaBooleanfilterof M,ifforany x M, x x F. Afuzzyfilter νofan Rl-monoid MiscalledaBooleanfuzzyfilterof M,ifforany x M, ν(x x ) = ν(1). Theorem5.11. Afilter F ofan Rl-monoid MisBooleanifandonlyif ν F isa Booleanfuzzyfilterof M. Proof. Let FbeaBooleanfilterof Mandlet x M.Then ν F (x x ) = α = ν F (1),hence ν F isabooleanfuzzyfilterof M. Conversely,let ν F beabooleanfuzzyfilterof Mandlet x F.Then ν F (x x ) = ν F (1) = α,then x x F,thatmeans FisaBooleanfilterof M. Theorem5.12. Let νbeafuzzyfilterofan Rl-monoid M. Thenthefollowing conditions are equivalent. (a) νisabooleanfuzzyfilterof M. (b) If α [0, 1]issuchthat U(ν; α),then U(ν; α)isabooleanfilterof M. (c) M ν = U(ν; ν(1))isabooleanfilterof M. Proof. Itfollowsfrom(TP). 92

13 Theorem5.13.Let νbeafuzzyfilterofan Rl-monoid M.Then νisbooleanif andonlyifthequotient Rl-monoid M/U(ν; α)isabooleanalgebraforany α [0, 1] suchthat U(ν; α). Proof. In[18]itisprovedthatafilter Fofan Rl-monoid MisBooleanifand onlyif M/FisaBooleanalgebra.Hencetheassertionisacorollaryofthepreceding theorem. Theorem5.14([21]). Afilter Fofan Rl-monoid Mispositiveimplicativeifand onlyif FisaBooleanfilter. AsaconsequenceofTheorems5.2,5.11and5.14weget Theorem5.15. If F isafilterofan Rl-monoid Mthenforthefuzzyfilter ν F the following conditions are equivalent. (a) ν F isapositiveimplicativefuzzyfilterof M. (b) ν F isabooleanfuzzyfilterof M. Analogously, from Theorems 5.5, 5.12 and 5.14 we obtain Theorem5.16. Let νbeafuzzyfilterofan Rl-monoid M. Thenthefollowing conditions are equivalent. (a) If α [0, 1]issuchthat U(ν; α),then U(ν; α)isapositiveimplicativefilter of M. (b) If α [0, 1]issuchthat U(ν; α),then U(ν; α)isabooleanfilterof M. Remark5.17.Theorems4.8and5.16giveanalternativeproofofTheorem Fantastic fuzzy filters Let Mbean Rl-monoidand Fasubsetof M.Then Fiscalledafantasticfilter of Mif (I) 1 F; (IV) z (y x) Fand z Fimply ((x y) y) x Fforany x, y, z M. By[20],everyfantasticfilterisafilterof M. Afuzzysubset νinan Rl-monoid Miscalledafantasticfuzzyfilterof Miffor any x, y, z M, (1) ν(1) ν(x); (4) ν(z (y x)) ν(z) ν(((x y) y) x). 93

14 Proposition 6.1. Every fantastic fuzzy filter of an Rl-monoid M is a fuzzy filter of M. Proof. Let νbeafantasticfuzzyfilterof M. Let α [0, 1]and U(ν; α). Then 1 U(ν; α). Let x, x y U(ν; α), i.e., ν(x), ν(x y) α. Then ν(x (1 y)) = ν(x y) α,hence ν(x (1 y)) ν(x) α,thusby(4), ν(y) = ν(1 y) = ν(((y 1) 1) y) ν(x (1 y)) ν(x) α,andso y U(ν; α).thereforebytheorem3.6, νisafuzzyfilterof M. Theorem6.2. Afilter F ofan Rl-monoid Misfantasticifandonlyif ν F isa fantasticfuzzyfilterof M. Proof. Let F beafilterof M. Letussupposethat F isfantastic. Let ν F (z (y x)) ν F (z) = α. Then ν F (z (y x)) = α = ν F (z), thus z (y x) F, z F, andhence ((x y) y) x F, thatmeans ν(((x y) y) x) = α.thereforewegetthat ν F isafantasticfuzzyfilterof M. Conversely,let ν F beafantasticfuzzyfilterof M. Let z (y x) F and z F. Then ν F (z (y x)) = α = ν F (z),hence ν F (((x y) y) x) = α, andtherefore ((x y) y) x F.Thatmeans, Fisafantasticfilterof M. Theorem6.3([20,Theorems4.2,4.4]). Let F beafilterofan Rl-monoid M. Then the following conditions are equivalent: (a) Fisafantasticfilterof M. (b) y x Fimplies ((x y) y) x Fforevery x, y M. (c) x x Fforevery x M. (d) x z Fand y z Fimply ((x y) y) z Fforevery x, y, z M. Theorem6.4.Let Fbeafilterofan Rl-monoid M.Thenthefollowingconditions are equivalent. (a) ν F isafantasticfuzzyfilterof M. (b) ν F (y x) ν F (((x y) y) x)forany x, y M. (c) ν F (x x) = ν F (1)forany x M. (c) ν F (x z) ν F (y z) ν F (((x y) y) z)forany x, y, z M. Proof. AnalogoustothatforTheorem4.4. Theorem6.5. Let νbeafuzzyfilterofan Rl-monoid M.Then νisafantastic fuzzyfilterof Mifandonlyif U(ν; α)isafantasticfilterof Mforevery α [0, 1] suchthat U(ν; α). Proof. Itfollowsfrom(TP). As a consequence we obtain the following theorem. 94

15 Theorem 6.6. Let ν beafuzzyfilterofan Rl-monoid M. Then ν isafantasticfuzzyfilterof Mifandonlyif U(ν; α)satisfieseachofconditions(b) (d)of Theorem 6.3forevery α [0, 1]suchthat U(ν; α). Theorem 6.7. Every positive implicative fuzzy filter of an Rl-monoid M is fantastic. Proof. If νisapositiveimplicativefuzzyfilterof M,thenbyTheorem5.5, U(ν; α)isapositiveimplicativefilterof M forevery U(ν; α). Henceby[20, Theorem4.3], U(ν; α)isalsoafantasticfilterof M,andhence,bythepreceding theorem, νisafantasticfilterof M. Theorem6.8([20,Theorem4.6]). Afilter F ofan Rl-monoid Misfantasticif andonlyif M/Fisan MV-algebra. Theorem6.9.If νisafuzzyfilterofan Rl-monoid M,then νisafantasticfuzzy filterof M ifandonlyifthequotient Rl-monoid M/U(ν; α)isan MV-algebrafor every α [0, 1]suchthat U(ν; α). Proof. FollowsfromTheorems6.5and6.8. Theorem 6.10 ([20, Proposition 4.10]). Let M be an Rl-monoid. Then the following conditions are equivalent: (1) Misan MV-algebra. (2) Everyfilterof Misfantastic. (3) {1}isafantasticfilterof M. Theorem Let M be an Rl-monoid. Then the following conditions are equivalent: (a) Misan MV-algebra. (b) Everyfuzzyfilterof Misfantastic. (c) Everyfuzzyfilter νof Msuchthat ν(1) = 1isfantastic. (d) χ {1} isafantasticfuzzyfilterof M. Proof. (a) (b):let Mbean MV-algebraandlet νbeafuzzyfilterof M.Let α [0, 1]besuchthat U(ν; α). ThenbyTheorem6.10, U(ν; α)isafantastic filterof M,hencebyTheorem6.5wegetthat νisafantasticfuzzyfilterof M. (b) (c),(c) (d): Obvious. (d) (a): Let χ {1} = ν {1} (1; 0)beafantasticfuzzyfilterof M. ThenbyTheorem6.2wehavethat {1}isafantasticfilterof M,andthereforebyTheorem6.8, M is an MV-algebra. 95

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17 [24] M. Ward, R. P. Dilworth: Residuated lattices. Trans. Amer. Math. Soc. 45 (1939), [25] J.Zhan, W.A.Dudek, Y.B.Jun: Interval valued (, q)-fuzzy filters of pseudo BL-algebras. Soft Comput. 13(2009), Authors addresses: J. Rachůnek, Department of Algebra and Geometry, Faculty of Sciences, Palacký University, 17. listopadu 12, Olomouc, Czech Republic, D. Šalounová, Department of Mathematical Methods in Economy, Faculty of Economics, VŠB-Technical University Ostrava, Sokolská 33, Ostrava, Czech Republic, 97