Ele troni Notes in Theoreti al Computer S ien e 82 No. 4 (2003)URL: http://www.elsevier.nl/lo ate/ent s/volume82.html 12 pagesAn Algebrai Approa h to Shadowed SetsGianpiero Cattaneo 1;2 and Davide Ciu i 1;2Dipartimento Di Informati a, Sistemisti a e Comuni azioneUniversit�a di Milano { Bi o aVia Bi o a degli Ar imboldi 8, I{20126 Milano, ItaliaAbstra tAfter the introdu tion of shadowed sets and the investigation of their relation withfuzzy sets, we present BZMVdM algebras as an abstra t environment for both shad-owed and fuzzy sets. Then, we introdu e the weaker notion of pre-BZMVdM alge-bra. This stru ture enables us to algebrai ally de�ne a mapping from fuzzy sets toshadowed sets.Key words: Shadowed sets, fuzzy sets, BZMV algebras, roughapproximations.1 Fuzzy and Shadowed SetsIn this se tion, we introdu e the basi notions of fuzzy and shadowed sets,and outline the relation existing between them. First, we give the de�nitionof fuzzy sets. The reader interested in the, widely studied, �elds of fuzzy setsand fuzzy logi , an refer to some lassi al text (see, for example, [5,10℄).De�nition 1.1 Let X be a set of obje ts, alled the Universe. A fuzzy set onX is any mapping f : X ! [0; 1℄. We denote the olle tion of all fuzzy setson X as [0; 1℄X or sometimes simply by F .The role of a fuzzy set is to des ribe vagueness: given a vague on eptf on a universe X, the value f(x) indi ates the degree to whi h x belongsto the on ept f . One feature of su h an approa h is the des ription of avague on ept through an exa t numeri al quantity. A di�erent approa hto vagueness has been proposed by Pedry z ([7,8,9℄). His intention was \tointrodu e a model whi h does not lend itself to pre ise numeri al membershipvalues but relies on basi on epts of truth values (yes - no) and on entire1 This work has been supported by MIURnCOFIN proje t "Formal Languages and Au-tomata: Theory and Appli ation"2 Email: attang�dis o.unimib.it, iu i�dis o.unimib.it 2003 Published by Elsevier S ien e B. V.

Cattaneo and Ciu iunit interval per eived as a zone of un ertainty" ([7℄). This idea of modelingvagueness through vague (i.e., not purely numeri ) information, lead him tothe de�nition of shadowed sets.De�nition 1.2 Let X be a set of obje ts, alled the Universe. A shadowedset on X is any mapping s : X ! f0; 1; (0; 1)g. We denote the olle tion ofall shadowed sets on X as f0; 1; (0; 1)gX.In the sequel we will indi ate (0; 1) with the value 12 . This will simplifyour algebrai approa h from a synta ti al point of view, without losing thesemanti of \total un ertainty" of the value (0; 1). In fa t, if 1 orresponds totruth, 0 to falseness, then 12 is halfway between true and false, i.e., it representsa really un ertain situation.From a fuzzy set it is possible to obtain a shadowed set. Let f be afuzzy set; then, it is suÆ ient to de�ne a value � 2 [0; 12) and set to 0 themembership values f(x) whi h are less than or equal to � and set to 1 thosegreater than (1 � �). The membership values belonging to (�; 1 � �) arethose hara terized by a great un ertainty or la k of knowledge and they are onsequently onsidered the \shadow" of the indu ed shadowed set, i.e., theyare set to 12 .In a more formal way, on e �xed a value �, we an de�ne the �{approximationfun tion of a fuzzy set f , denoted by s�(f), as the following shadowed set:s�(f)(x) := 8><>:0 f(x) � �1 f(x) � 1� �12 otherwise (1)In Figure 1 it is represented a fuzzy set and the indu ed shadowed set.a b c d0

α

1−α1 1

a b c d

1/2

0Fig. 1. A fuzzy set and its orresponding shadowed set2 An algebrai frameworkAs an algebrai approa h to both fuzzy and shadowed sets we propose BZMVdMalgebras ([2,3℄).De�nition 2.1 A de Morgan Brouwer Zadeh Many Valued (BZMVdM) alge-bra is a system hA;�;:;�; 0i, where A is a non empty set, � is a binaryoperator, : and � are unary operators, 0 is a onstant, obeying the followingaxioms: 2

Cattaneo and Ciu i(BZMV 1) (a� b)� = (b� )� a(BZMV 2) a� 0 = a(BZMV 3) :(:a) = a(BZMV 4) :(:a� b)� b = :(a� :b)� a(BZMV 5) � a� �� a = :0(BZMV 6) a� �� a =�� a(BZMV 7) � :[(:(a� :b)� b)℄ = :(�� a� : �� b)� : �� bOn a BZMVdM algebra, it is possible to derive the following further oper-ators: a� b := :(:a� :b)a _ b := :(:a� b)� ba ^ b := :(:(a� :b)� :b)Conne tives _ and ^ are the algebrai realization of logi al disjun tion and onjun tion of a distributive latti e; in parti ular, they are idempotent op-erators. Conne tives � and � are the well known MV disjun tion and MV onjun tion operators, whi h are not idempotent ([10℄). A partial order anbe naturally indu ed by the latti e operators as:a � b i� a ^ b = a (equivalently, a _ b = b)Let us noti e that, sin e it is possible to prove that � 0 = :0, in the sequelwe set 1 :=� 0 = :0. With respe t to the just de�ned partial order we havethat the latti e is bounded: 8a 2 A, 0 � a � 1.The unary operation : : A 7! A is a Kleene (or Zadeh) ortho omplemen-tation (negation). In other words, it satis�es the properties:(K1) :(:a) = a(K2) :(a _ b) = :a ^ :b(K3) a ^ :a � b _ :bLet us re all that under (K1), ondition (K2) is equivalent to the dual deMorgan law. In general neither the non- ontradi tion law, 8a : a ^ :a = 0,nor the ex luded middle law, 8a : a _ :a = 1, are satis�ed by this negation.The unary operation�: A 7! A is a Brouwer ortho omplementation (nega-tion). In other words, it satis�es the properties:(B1) a^ �� a = a (equivalently, a ��� a)(B2) � (a _ b) =� a^ � b(B3) a^ � a = 0In general from (B1){(B3) neither the ex luded middle law 8a; a_ � a = 1nor the dual de Morgan law � (a ^ b) =� a_ � b an be dedu ed.Using the above de�nitions, we an justify the quali� ation of de Morgangiven to BZMV algebras in De�nition 2.1. In fa t, it an be proved that3

Cattaneo and Ciu iBZMVdM algebras satisfy all de Morgan properties::(a ^ b) = :a _ :b :(a _ b) = :a ^ :b� (a ^ b) =� a_ � b � (a _ b) =� a^ � bBesides, it is possible to de�ne, through the intera tion of the two unaryoperations : and �, the modal operators of ne essity, �(a) :=� :a, andpossibility, �(a) := : � a = :�(:a).These modal operators turn out to have an S5-like behavior based on aKleene algebra, instead of on a Boolean one ([4℄).Proposition 2.2 In any BZMVdM algebra the following onditions hold:(1) �(a) � a � �(a). In other words: ne essity implies a tuality and a tualityimplies possibility (a hara teristi prin iple of the modal system T ).(2) �(�(a)) = �(a), �(�(a)) = �(a). Ne essity of ne essity is equal to ne es-sity; similarly for possibility (a hara teristi S4-prin iple).(3) a � �(�(a)). A tuality implies ne essity of possibility (a hara teristi B-prin iple).(4) �(a) = �(�(a)) �(a) = �(�(a)) Possibility is equal to the ne essity ofpossibility; whereas ne essity is equal to the possibility of ne essity (a hara teristi S5-prin iple).As a onsequen e of the above de�nitions we have that � a = :�(a), thatis the Brouwer omplement an be interpreted as the negation of possibilityor impossibility.As stated in Proposition 2.2, for any element of a BZMVdM algebra the or-der hain �(a) � a � �(a) holds. We are, now, interested to those elementswhi h satisfy the strongest ondition �(e) = e (equivalently, e = �(e)), i.e., tothose elements whi h present the lassi al feature that a tuality oin ide withne essity and possibility. These elements are alled sharp (exa t, risp) ele-ments (in ontraposition to the elements whi h are fuzzy) and their olle tionis denoted by Ae.Remark 2.3 This is not the only way to de�ne sharp elements. In fa t, sin ein general x ^ :x 6= 0 (equivalently, x _ :x 6= 1) it is possible to onsideras Kleene sharp (K-sharp) the elements whi h satisfy the non ontradi tion(or equivalently the ex luded middle) law with respe t to the Kleene negation:Ae;: := fe 2 A : e^:e = 0g = fe 2 A : e_:e = 1g. Alternatively, onsideringthe Brouwer negation we have that, in general, the double negation law doesnot hold (see the (B1)). So, we an introdu e a further de�nition of Brouwersharp (B-sharp) elements: Ae;B := fe 2 A :�� e = eg. Finally, as said before� is not an idempotent operator. So the �-sharp elements are: Ae;� = fe 2A : e� e = eg. However, it an be proved that all this notions are equivalent([2,3℄). Let A be a BZMV dM algebra, then Ae = Ae;B = Ae;� = Ae;:.Consequently, we simply talk of sharp elements and write Ae to denote their4

Cattaneo and Ciu i olle tion.Given an element a of a BZMVdM algebra, modal operators, � and �, an be used to give a rough approximation of a by sharp de�nable elements.In fa t, �(a) (resp., �(a)) turns out to be the best approximation from thebottom (resp., top) of a by sharp elements. To be pre ise, for any elementa 2 A the following holds:(I1) �(a) is sharp (�(a) 2 Ae).(I2) �(a) is an inner (lower) approximation of a (�(a) � a).(I3) �(a) is the best inner approximation of a by sharp elements (let e 2 Aebe su h that e � a, then e � �(a)).Analogously(O1) �(a) is sharp (�(a) 2 Ae).(O2) �(a) is an outer (upper) approximation of a (a � �(a)).(O3) �(a) is the best outer approximation of a by sharp elements (letf 2 Ae be su h that a � f , then �(a) � f).De�nition 2.4 Given a BZMVdM algebra hA;�;:;�; 0i, the indu ed roughapproximation spa e a ording to [1℄ is the stru ture hA;Ae; �; �i onsisting ofthe set A of all approximable elements, the set Ae of all de�nable (or sharp)elements, and the inner (resp., outer) approximation map � : A ! Ae (resp.,� : A! Ae).For any element a 2 A, its rough approximation is de�ned as the pair of sharpelements: r(a) := h�(a); �(a)i [with �(a) � a � �(a)℄.So the map r : A ! Ae � Ae approximates an unsharp (fuzzy) element bya pair of exa t ones representing its inner and outer sharp approximation,respe tively. Clearly, sharp elements are hara terized by the property thatthey oin ide with their rough approximations: e 2 Ae i� r(e) = he; ei:An equivalent way to de�ne a rough approximation spa e is to use the im-possibility operator instead of the possibility one. So, given a fuzzy ele-ment its approximation is given by the map ri : A ! Ae � Ae de�ned asri(a) := h�(a);:�(a)i = h�(a);� ai.We return now to fuzzy and shadowed sets, and we show how it is possibleto give them the stru ture of BZMVdM algebras.Proposition 2.5 Let F = [0; 1℄X be the olle tion of fuzzy sets on the uni-verse X. On e de�ned the operators:(f � g)(x) := minf1; f(x) + g(x)g:f(x) := 1� f(x)� f(x) := (1 if f(x) = 00 otherwiseand the identi ally zero fuzzy set: 0(x) := 0; then, the stru ture5

Cattaneo and Ciu ihF ;�;:;�; 0i is a BZMVdM algebra.Similarly, it is possible to give the stru ture of BZMVdM algebra to the olle tion of shadowed sets S = f0; 12 ; 1gX on the universe X. The operationssynta ti ally are exa tly as in Proposition 2.5, but they are de�ned on thedomain of shadowed sets.Proposition 2.6 Let S = f0; 12 ; 1gX be the olle tion of shadowed sets on theuniverse X. Then, the stru ture hS;�;:;�; 0i, where �;:;�; 0 are de�nedas in proposition 2.5, is a BZMVdM algebra.Now let us onsider a fuzzy set f 2 F . Then, the possibility and ne essityof f are de�ned, respe tively, as�(f)(x) = (0 f(x) = 01 f(x) 6= 0 �(f)(x) = (1 f(x) = 10 f(x) 6= 1So, when f is a fuzzy set, the abstra t rough approximation as the pair ri(f) =h�(f);� (f)i singles out a shadowed set. In fa t, �(f) an be interpreted asthe hara teristi fun tion of the elements whi h have value 1 in the indu edshadowed set; and � f as the hara teristi fun tion of the elements whi hhave value 0. The other elements of the universe X represents the shadow ofthe shadowed set.Pre isely, the shadowed set s� de�ned in equation (1) an be obtained, inthe ase that � = 0, through a ombination of these modal operators:s0(f) := �(f)� (�(f)� 12) s0(f)(x) = 8><>:0 f(x) = 01 f(x) = 112 otherwise (2)where 12 is the fuzzy set identi ally equal to 12 , i.e., for all x 2 X, 12(x) := 12 .Remark 2.7 The mapping s0 : F ! S; f ! s0(f), is not a bije tion nor anhomomorphism between BZMV dM algebras, as an be seen in the following ounterexample. Let us onsider the fuzzy sets f1; f2 : [0; 1℄ 7! [0; 1℄ de�nedas: f1(x) := 0:2 if x = 0 and 0 otherwise, and f2(x) := 0:3 if x = 0 and 0otherwise. So, f1 6= f2 but s0(f1) = s0(f2) = 0:5 if x = 0 and 0 otherwise, andthis proves that s0 is not a bije tion. Furthermore (stressing with symbols �Sand �F the \trun ated" sum operation a ting on S and F respe tively),[s0(f1)�S s0(f2)℄(x) = (1 f(x) = 00 otherwise 6= (12 f(x) = 00 otherwise = [s0(f1 �F f2)℄(x)and so s0 is neither an homomorphism of BZMVdM algebras.Of ourse, s0 gives only the indu ed shadowed set in the parti ular ase of� = 0, and it does not apture all the possible ones that an be obtained from6

Cattaneo and Ciu ia fuzzy set by equation (1). In order to onsider all that possibilities, it isne essary (and suÆ ient) to generalize the intuitionisti negation as follows:8� 2 [0; 12) (�� f)(x) := (1� f(x) if f(x) � �0 otherwise (3)Clearly, this is a generalization of�, in fa t when � = 0 we obtain�0 f =� f .In Figure 2, it is represented a fuzzy set and its � - impossibility (i.e., ��).0

α

1−α1

0

1

Fig. 2. Generalized ��: � 2 (0; 1=2) and � = 0The derived operators, �� and �� then be ome:��(f)(x) := (: �� f)(x) = (f(x) f(x) � �1 f(x) > ���(f)(x) := (�� :f)(x) = (f(x) f(x) � (1� �)0 f(x) < (1� �)Let us introdu e the shadowed set s�(f), indu ed by the fuzzy set f andde�ned analogously to the (2). This oin ide with the shadowed set previouslyde�ned by the (1):s�(f) := ��(f)� (��(f)� 12) s�(f)(x) = 8><>:0 f(x) � �1 f(x) � 1� �12 otherwiseSo, given a fuzzy set f , on one side we an obtain the rough approximationr�(f) = h��(f); ��(f)i, and on the other side we an indu e the shadowedset s�(f). The relation between the two fun tions r� and s� is given by themapping r�(f)! (r�(f)) de�ned as: (r�(f)) = (h��(f); ��(f)i) = ��(f)� (��(f)� 12) = s�(f)In the following diagram all the three fun tions, r�, s� and are drawn,7

Cattaneo and Ciu ishowing the relations among them:f r�s� r�(f) s�(f)We remark that the fun tion is not a bije tion as an be seen in the following ounterexample.Example 2.8 Let 1 be the identi ally one fuzzy set and 23 the fuzzy set onstantly equal to 23 . We have r0:4(23) = h23 ; 1i and r0:4(1) = h1; 1i. Then,we obtain (r0:4(23)) = 1 = (r0:4(1)).Coming ba k to our algebrai stru ture, we have that, by a substitutionof � by ��, the system h[0; 1℄X;�;:;��; 0i is no more a BZMV dM algebra.In fa t, for example, axiom BZMV 6 is not satis�ed. Given a fuzzy set f , wehave:f(x)� (���� f)(x) = (1 f(x) > �f(x) f(x) � � 6= (1 f(x) > �0 f(x) � � = (���� f)(x):Next se tion is devoted to the study of an algebrization of the stru turehF ;�;:;��; 0i ontaining this new operator ��.3 pre-BZMVdM algebrasWe, now, introdu e a new algebra, whi h turns out to be weaker than BZMVdMalgebra. The advantage of this new stru ture is that it admits as a model the olle tion of fuzzy sets endowed with the operator ��.De�nition 3.1 A stru ture hA;�;:;�w; 0i is a pre-BZMVdM algebra, if thefollowing are satis�ed:(i) The substru ture hA;�;:; 0i is a MV algebra, whose indu ed latti eoperators are de�ned as a_b := :(:a�b)�b, a^b := :(:(a�:b)�:b),and the partial order as a � b i� a ^ b = a.(ii) The following properties are satis�ed:(a) a� �w�w a = : �w a(b) �w a � :a( ) �w a^ �w b = �w (a _ b)(d) �w a_ �w b = �w (a ^ b)(e) �w :a � �w : �w :aThe olle tion of all fuzzy sets an be equipped with a stru ture of pre-BZMVdM algebra, a ording to the following result.Proposition 3.2 Let F be the olle tion of fuzzy sets based on the universeX and let � 2 [0; 12). On e de�ned the standard � and : operators on F , and8

Cattaneo and Ciu ithe �� negation as in Equation (3), then the stru ture F� = hF ;�;:;��; 0iis a pre-BZMVdM algebra, whi h is not a BZMVdM algebra.In general, it is possible to show that any BZMVdM algebra is a pre-BZMVdM algebra. In fa t, in [3℄ it is shown that all axioms of de�nition 3.1are true in any BZMVdM algebra. In general, the vi e versa does not hold. Forexample, let us onsider the stru ture F� with � = 0:4 and X = R, and de�nethe fuzzy set f(x) = 0:3 for all x 2 R. Then, �� f(x)� ���� f(x) = 0:7 forall x. So, axiom (BZMV5) is not satis�ed.Proposition 3.3 If hA;�;:;�w; 0i is a pre-BZMVdM algebra satisfying �w�wa = : �w a, then it is a BZMVdM algebra.Proposition 3.4 Let hA;�;:;�w; 0i be a pre-BZMVdM algebra. Then, thefollowing properties hold:(i) �w 0 = :0. In the sequel we set 1 :=�w 0 = :0.(ii) If a � b then �w b ��w a ( ontraposition law).So �w is a unary operator satisfying both de Morgan laws and the on-traposition law (ii). However, it is not an intuitionisti negation, in fa t, ingeneral, it satis�es neither the non ontradi tion law (property B3), nor theweak double negation law (property B1), nor the Brouwer law (i.e., the law8a, �w a =�w�w�w a is not satis�ed).Anyway, also in a pre-BZMVdM algebra, it is possible to introdu e modaloperators of ne essity, �w(a) :=�w :a and possibility �w(a) := : �w a.However, in this stru ture �w and �w do not have an S5 - like behavior butonly an S4 - like one (always based on a Kleene latti e instead of on a Booleanone).Proposition 3.5 Let hA;�;:;�w; 0i be a pre-BZMVdM algebra. Then, forevery a 2 A the following properties are satis�ed:(i) �w(a) � a � �w(a) [T prin iple℄(ii) �w(�w(a)) = �w(a) �w(�w(a)) = �w(a) [S4 prin iple℄In general the properties: a � �w(�w(a)), �w(a) = �w(�w(a)), and �w(a) =�w(�w(a)) do not hold. As an example, let us onsider the algebra F0:4 , withX = [0; 1℄, and de�ne the fuzzy set f(x) = 0:3 if x < 12 and f(x) = 0:7otherwise. We have:��(f(x)) = (0:3 x < 121 x � 12 6= (0 x < 121 x � 12 = ��(��(f(x))):��(f(x)) = (0 x < 120:7 x � 12 6= (0 x < 121 x � 12 = ��(��(f(x))):Finally, f(x) is in omparable with ��(��(f(x))).9

Cattaneo and Ciu iEven if the ne essity and possibity mappings have a weaker modal behaviorin pre-BZVMdM algebras than in BZMVdM algebras, they an still be used tode�ne a lower and upper approximation, and it turns out that �w is an interioroperator and �w is a losure operator.Proposition 3.6 Let hA;�;:;�w; 0i be a pre-BZMVdM algebra. Then themap �w : A ! A su h that �w(a) := : �w a is a losure operator. That is,the following are satis�ed:(C0) 0 = �w(0) (normalized)(C1) a � �w(a) (in reasing)(C2) �w(a) = �w(�w(a)) (idempotent)(C3) a � b implies �w(a) � �w(b) (monotone)The olle tion of all losed sets is then de�ned asC (A) = fa 2 A : a = �w(a)gProposition 3.7 Let hA;�;:;�w; 0i be a pre-BZMVdM algebra. Then themap �w : A! A su h that �w(a) :=�w :a is an interior operator,i.e.:(I0) 1 = �w(1) (normalized)(I1) �w(a) � a (de reasing)(I2) �w(a) = �w(�w(a)) (idempotent)(I3) a � b implies �w(a) � �w(b) (monotone)The olle tion of all open sets is then de�ned asO (A) = fa 2 A : a = �w(a)g:It is possible to show that, in general, these subsets of A do not oin ide,neither one is a subset of the other. So, it is worthwhile to onsider alsothe set of all lopen elements, i.e. elements whi h are both losed and open:C O (A) = C (A) \ O (A).The above onsiderations lead to the de�nition of an abstra t approxima-tion spa e generated by a pre-BZMVdM algebra.De�nition 3.8 Let A be a pre-BZMVdM algebra. The indu ed rough ap-proximation spa e is the stru ture hA;O (A); C (A); �w ; �wi, where A is theset of approximable elements; O (A) � A is the set of innerde�nable elements,su h that 0 and 1 2 O (A); C (A) � A is the set of outerde�nable elements,su h that 0 and 1 2 C (A); �w : A ! O (A) is the inner approximation map;10

Cattaneo and Ciu i�w : A ! C (A) is the outer approximation map. For any element a 2 A, itsrough approximation is de�ned as the pair:rw(a) := h�w(a); �w(a)i [with �w(a) � a � �w(a)℄drawn in the following diagram: a 2 A�w �wrw�w(a) 2 O (A) �w(a) 2 C (A)h�w(a); �w(a)iThis approximation is the best approximation by open (resp. losed) ele-ments that it is possible to de�ne on a pre-BZMVdM stru ture, i.e., there holdproperties similar to (I1){(I3) and (O1){(O3), the only di�eren e is that herewe have to distinguish between open-exa t and losed-exa t elements.In the ontext of the fuzzy sets pre{BZMVdM algebra of proposition 3.2the olle tion of open and losed elements are respe tively:C (F) = ff 2 F : f(x) > � i� f(x) = 1gO (F) = ff 2 F : f(x) < 1� � i� f(x) = 0gThe lopen sets are the 0{1 valued fuzzy sets, C O (F) = f0; 1gX . In theuniverse [0; 1℄, on e set � = 0:4, an example of open element is f1(x) = 0 ifx < 12 and 0:7 otherwise and an example of losed set is f2(x) = 0:3 if x < 12and 1 otherwise. The fuzzy sets f1 and f2 are drawn in Figure 3.0 x

f1(x)

1/2 1

α1−α

1

0.7

0

1

x

f2(x)

1/2 1

0.3α

1−αFig. 3. Example of open fuzzy set, f1, and losed fuzzy set, f2.Finally, we remark that we an also enri h the olle tion of shadowed setsf0; 12 ; 1gX with the operation ��, in order to give it a pre-BZMVdM stru ture.However, it an be easily proved that in this ase �� is equivalent to �0 forall � 2 [0; 12) and so we again obtain a BZMV dM algebra.4 Con lusionsIn this paper we analyzed shadowed sets from the algebrai point of view. Asa �rst result we have seen that on e properly de�ned the operators on the11

Cattaneo and Ciu i olle tion of shadowed sets of a given universe, they result to be a BZMVdMalgebra. The same an be proved about the olle tion of fuzzy sets. Moreover,in su h a stru ture it is possible to algebrai ally de�ne an operator whi hgiven a fuzzy set returns a parti ular indu ed shadowed set. In order togeneralize su h an operator it was ne essary to introdu e the new stru tureof pre-BZMVdM algebras. Finally, it was shown that the olle tion of fuzzysets with a generalized notion of intuitionisti negation is a model of pre-BZMVdM algebras. A possible development of the present work is a deepertheoreti al analysis of pre-BZMVdM algebras, whi h involves the study of theindependen e of its axioms, the proof of a representation and a ompletenesstheorem. On the other hand, it would also be interesting to analyze theimpli ations of su h a stru ture in an appli ation ontext.Referen es[1℄ Cattaneo, G., Abstra t approximation spa es for rough theories, in: L. Polkowskiand A. Skowron, editors, Rough Sets in Knowledge Dis overy 1: Methodologyand Appli ations, Studies in Fuzziness and Soft Computing, Physi a{Verlag,Heidelberg, 1998 pp. 59{98.[2℄ Cattaneo, G., M. Dalla Chiara and R. Giuntini, Some algebrai stru turesfor many-valued logi s, Tatra Mountains Mathemati al Publi ation 15 (1998),pp. 173{196, spe ial Issue: Quantum Stru tures II, Dedi ated to GudrunKalmba h.[3℄ Cattaneo, G., R. Giuntini and R. Pilla, BZMVdM and Stonian MV algebras(appli ations to fuzzy sets and rough approximations), Fuzzy Sets Syst. 108(1999), pp. 201{222.[4℄ Chellas, B. F., \Modal Logi , An Introdu tion," Cambridge University Press,Cambridge, MA, 1988.[5℄ Dubois, D. and H. Prade, \Fuzzy Sets and Systems. Theory and Appli ations,"A ademi Press, New York, 1980.[6℄ H�ajek, P., \Metamathemati s of Fuzzy Logi ," Kluwer, Dordre ht, 1998.[7℄ Pedry z, W., Shadowed sets: Representing and pro essing fuzzy sets, IEEETransa tion on Systems, Man and Cyberneti s - PART B: Cyberneti s 28(1998), pp. 103{109.[8℄ Pedry z, W., Shadowed sets: Bridging fuzzy and rough sets, in: S. Pal andA. Skowron, editors, Rough Fuzzy Hybridization, Springer{Verlag, Singapore,1999 pp. 179{199.[9℄ Pedry z, W. and G. Vukovi h, Granular omputing with shadowed sets,International Journal of Intelligent Systems 17 (2002), pp. 173{197.[10℄ Turunen, E., \Mathemati s Behind Fuzzy Logi ," Physi a{Verlag, Heidelberg,1999. 12