Articles & News Stories

BZMVdM algebras and stonian MV-algebras (applications to fuzzy sets and rough approximations)

Description
BZMVdM algebras and stonian MV-algebras (applications to fuzzy sets and rough approximations)
Published
of 22
All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.
Related Documents
Share
Transcript
  Fuzzy Sets and Systems 108 (1999) 201–222www.elsevier.com/locate/fss BZMV dM algebrasandstonianMV-algebras(applicationstofuzzysetsandroughapproximations) G. Cattaneo a , R. Giuntini  b , R. Pilla c a Dipartimento di Scienze dell’Informazione, Universita di Milano, Milano, Italy  b Dipartimento di Filosoa, Universita di Firenze, Firenze, Italy c G.R.T.I. Dipartimento di Informatica e Sistemistica, Universita di Pavia, Pavia, Italy Received November 1995; received in revised form October 1997 Abstract The natural algebraic structure of fuzzy sets suggests the introduction of an abstract algebraic structure called de MorganBZMV-algebra (BZMV dM -algebra). We study this structure and sketch its main properties. A BZMV dM -algebra is asystem endowed with a commutative and associative binary operator   ⊕  and two unusual orthocomplementations: a Kleeneorthocomplementation ( ” ) and a Brouwerian one ( ∼ ). As expected, every BZMV dM -algebra is both an MV-algebra anda distributive de Morgan BZ-lattice. The set of all  ∼ -closed elements (which coincides with the set of all  ⊕ -idempotentelements) turns out to be a Boolean algebra (the Boolean algebra of sharp or crisp elements). By means of  ”  and  ∼ , twomodal-like unary operators (   for necessity and    for possibility) can be introduced in such a way that   ( a ) (resp.,   ( a ))can be regarded as the sharp approximation from the bottom (resp., top) of   a . This gives rise to the rough approximation(  ( a ) ; ( a )) of   a . Finally, we prove that BZMV dM -algebras (which are equationally characterized) are the same as the StonianMV-algebras and a rst representation theorem is proved. c   1999 Elsevier Science B.V. All rights reserved. 1. Introduction As is well known, the power set of a  universe  X  (denoted by P (  X  )) and the family of all characteris-tic functionals ( { 0 ; 1 } -valued functions on  X  ) are ina one-to-one correspondence with respect to the map- ping, which associates to any subset  A  of   X   the func-tion    A  :  X   → { 0 ; 1 }  dened as   A (  x ):=  1 ;  i   x ∈  A; 0 ;  i   x =  ∈  A: (1.1)The structure   P (  X  ) ; ∩ ; ∪ ;  c ; ∅ ;X    is an atomicBoolean (complete) lattice, where  ∩ ,  ∪  and  c are theset-theoretic intersection, union and complement,respectively.The set  { 0 ; 1 }  X  of all characteristic functionals on  X   determines an atomic Boolean (complete) lattice  { 0 ; 1 }  X  ;  ∧ ;  ∨ ; ” ; 0 ; 1  , where 0 and 1 are the char-acteristic functionals of the empty set and of the wholeuniverse, respectively; the operations ∧ , ∨  and ” aredened  ∀  x ∈  X   by the laws(   A ∧   B )(  x )=min {   A (  x ) ;  B (  x ) }  (me-a)= max { 0 ;  A (  x ) +    B (  x ) − 1 } ;  (me-b)(   A ∨   B )(  x )=max {   A (  x ) ;  B (  x ) }  (jo-a)= min { 1 ;  A (  x ) +    B (  x ) } ;  (jo-b) 0165-0114/99/$ – see front matter  c  1999 Elsevier Science B.V. All rights reserved.PII: S0165-0114(97)00328-X  202  G. Cattaneo et al./Fuzzy Sets and Systems 108 (1999) 201–222 ( ”   A )(  x )=(1 −   A )(  x ) (oc)=  1 ;  i     A (  x )=00 ;  i     A (  x )=1 : (oc ′ )The mapping    : P (  X  )  → { 0 ; 1 }  X  ; A  →    A  is clearlya boolean lattice isomorphism since   A ∩  B  =   A ∧   B ;  (1.2a)   A ∪  B  =   A ∨   B ;  (1.2b)   A c  = ”   A :  (1.2c)The most direct generalization of the notion of char-acteristic functional on the universe  X   is the notionof   generalized characteristic functional   (or   fuzzy set )dened as a [0 ; 1]-valued function on  X  : f  :  X   →  [0 ; 1] :  (1.3)In this paper we will be concerned with the problemof introducing an appropriate class of abstract alge- braic structures, having the set [0 ; 1]  X  of all gener-alized characteristic functionals on  X   as a concretemodel. Such a structure has to take into account thecorresponding behavior of the characteristic function-als (which are the crisp or sharp elements of [0 ; 1]  X  ).From (me-b) and (jo-b) above, it follows that arst possible choice consists in considering a struc-ture equipped with the two operations of “truncated” product ( ⊙ ) and sum ( ⊕ ) dened for any pair  f 1 ;f 2 ∈ [0 ; 1]  X  and any  x ∈  X   by the following:( f 1 ⊙ f 2 )(  x ):= max { 0 ;f 1 (  x ) +  f 2 (  x ) − 1 } ;  (1.4a)( f 1 ⊕ f 2 )(  x ):= min { 1 ;f 1 (  x ) +  f 2 (  x ) } :  (1.4b)A second possible choice consists in considering astructure equipped with two lattice operations of meet( ∧ ) and join ( ∨ ), which are the natural extensions of (me-a) and (jo-a) dened for any pair   f 1 ;f 2 ∈ [0 ; 1]  X  and any  x ∈  X   by the following:( f 1 ∧ f 2 )(  x ):= min { f 1 (  x ) ;f 2 (  x ) } ;  (1.5a)( f 1 ∨ f 2 )(  x ):= max { f 1 (  x ) ;f 2 (  x ) }  (1.5b)[the induced partial order is the usual pointwise order-ing  f 1 6 f 2  i for all  x ∈  X  ,  f 1 (  x ) 6 f 2 (  x )]. Of course,dierently from characteristic functionals in  { 0 ; 1 }  X  ,it may happen that for some  f 1 ;f 2 ∈ [0 ; 1]  X  one couldhave that f 1 ⊕ f 2  = f 1 ∨ f 2  or   f 1 ⊙ f 2  = f 1 ∧ f 2 : Three possible generalizations of the orthocomple-ment can be dened as follows:(a) The  diametrical   (Lukasiewicz or Kleene) ortho-complement (which is an extension of (oc ′ ))( ” f )(  x ):=(1 − f )(  x ) :  (1.6a)(b) The  intuitionistic  (Brouwer) orthocomplement(which is an extension of (oc ′ ))( ∼ f )(  x ):=  1 ; f (  x )=00 ; f (  x )  =0=   {  x ∈  X  : f (  x )=0 } :  (1.6b)(c) The  anti-intuitionistic  (anti-Brouwer) orthocom- plement (which is a second extension of (oc ′ ))( [f )(  x ):=  1 ; f (  x )  =10 ; f (  x )=1=   {  x ∈  X  : f (  x )  =1 } :  (1.6c)Trivially, ∼ f 6 ” f 6 [f . The point to be underlinedis that the denitions of the binary and unary opera-tions given above are not independent. For instance,in the structure(MV-f)  [0 ; 1]  X  ; ⊕ ; ” ; 0  ; we have that:1= ” 0 ;  (1.7a) f ⊙ g = ” ( ” f ⊕ ” g ) ;  (1.7b) f ∨ g = ” ( ” f ⊕ g ) ⊕ g =( f ⊙ ” g ) ⊕ g;  (1.7c) f ∧ g = ” [ ” ( f ⊕ ” g ) ⊕ ” g ]=( f ⊕ ” g ) ⊙ g: (1.7d) Remark 1.  The Kleene orthocomplementation is aone-to-one mapping  ”  : [0 ; 1]  X  →  [0 ; 1]  X  whichinstitute a “de Morgan” duality between the binaryoperations: f ⊙ g = ” ( ” f ⊕ ” g ) and  f ⊕ g = ” ( ” f ⊙ ” g ) ;  G. Cattaneo et al./Fuzzy Sets and Systems 108 (1999) 201–222  203 f ∧ g  = ” ( ” f ∨ ” g )and f ∨ g = ” ( ” f ∧ ” g ) : The intuitionistic and the anti-intuitionistic negationsare linked by the duality relations [f = ” ∼ ” f  and  ∼  f = ” [ ” f:  (1.8)According to the above considerations, it seems thata “good” algebraic structure for fuzzy sets should beone of the following kind:(BZMV-f)  [0 ; 1]  X  ; ⊕ ; ” ; ∼ ;  0  : The point is that in literature one can nd two dier-ent, and presently “disjoint”, approaches to the alge- bra of many-valued logics: the MV-algebra approachintroduced by Chang [10,11], and the structure of BZ lattice of Cattaneo et al. [7,8]. The former consists inan algebraic structure in which the above (MV-f) is anexample and the latter is a lattice equipped with twonegations ”  and  ∼ . The aim of this paper is the in-vestigation of an abstract algebraic structure contain-ing both these features; a model of these structure is just (BZMV-f). 2. MV-Algebras We begin this section introducing a simplied def-inition of MV-algebra: Denition 2.1.  An MV-algebra is a system    A; ⊕ ; ” ; 0  where  A  is a non-empty set, 0 is a constant ele-ment of   A , ⊕ is a binary operation on  A , ” is a unaryoperator, obeying the following axioms:(P1) (  x ⊕ y ) ⊕  z   =( y ⊕  z  ) ⊕  x; (P2)  x ⊕ 0=  x; (P3)  x ⊕ ” 0= ” 0 ; (P4)  ” ( ” 0)=0 ; (P5)  ” ( ”  x ⊕ y ) ⊕ y = ” (  x ⊕ ” y ) ⊕  x: In [9] we have shown that axioms (P1)–(P5) areindependent. Proposition 2.1.  An  MV  -algebra can be equivalentlydened   ( with a slight modication with respect to theaxiomatization proposed by Mangani in  [21])  as asystem    A; ⊕ ; ” ; 0   where  A  is a non-empty set ;  0 is a constant element of   A;  ⊕  is a binary operationon  A; ”  is a unary operator ;  obeying the followingaxioms :(M1)  x ⊕ y = y ⊕  x; (M2) (  x ⊕ y ) ⊕  z   =  x ⊕ ( y ⊕  z  ) ; (M3)  x ⊕ 0=  x; (M4)  x ⊕ ” 0= ” 0 ; (M5)  ” ( ”  x )=  x; (M6)  ” ( ”  x ⊕ y ) ⊕ y = ” (  x ⊕ ” y ) ⊕  x; (M7)  x ⊕ ”  x = ” 0 : Proof.  Wewillrstprovetheequivalence,under(P2)[i.e., (M3)], between (P1) and (M1),(M2): rst of all,let (P2) and (P1) be true(M1)  x ⊕ y  =(  x ⊕ y ) ⊕ 0 (P2)=( y ⊕ 0) ⊕  x  (P1)= y ⊕  x  (P2) ; (M2)(  x ⊕ y ) ⊕  z   =( y ⊕  z  ) ⊕  x  (P1)=  x ⊕ ( y ⊕  z  ) (M1) : Trivially, (P1) follows from (M1) and (M2).(M5) Applying (P2) to the element ”  x  we get ”  x = ”  x ⊕ 0 (P2)from which it follows that ””  x  = ” ( ”  x ⊕ 0)= ” ( ”  x ⊕ 0) ⊕ 0 (P2)= ” (  x ⊕ ” 0) ⊕  x  (P5)= ”” 0 ⊕  x  (P3)=  x  (P4) ; (M1) ; (P2) ; (M7)  x ⊕ ”  x  = ””  x ⊕ ”  x  (M5)= ” ( ”  x ⊕ 0) ⊕ ”  x  (P2)= ” (0 ⊕ ”  x ) ⊕ ”  x  (M1)= ” (  x ⊕ ” 0) ⊕ ” 0 (P5)= ” 0 (P3) : Let us remark that in the proof of the (M7) all theconditions (P1)–(P5) are used.Let us introduce the following new operations:1:= ” 0 ;  (2.1)  204  G. Cattaneo et al./Fuzzy Sets and Systems 108 (1999) 201–222  x ⊙ y = ” ( ”  x ⊕ ” y ) ;  (2.2)  x ∨ y :=(  x ⊙ ” y ) ⊕ y = ” ( ”  x ⊕ y ) ⊕ y;  (2.3)  x ∧ y :=(  x ⊕ ” y ) ⊙ y = ” [ ” (  x ⊕ ” y ) ⊕ ” y ] : (2.4)Using the Mangani equivalent denition of MV-algebra proved in the above Proposition 2.1, the fol-lowing can be easily proved (for the technical details,see [21]). Proposition 2.2.  An  MV  -algebra can be equiva-lently described   ( according to the original de-nition introduced by Chang in  [10])  as a system   A; ⊕ ; ⊙ ;  ∨ ;  ∧ ; ” ; 0 ; 1   where  A  is a non empty setof elements ;  0  and   1  are distinct constant elementsof   A;  ⊕  and   ⊙  are binary operations on elementsof   A;  and  ”  is a unary operation on elements of   A obeying the following axioms :(C1)  x ⊕ y = y ⊕  x; (C2)  x ⊕ ( y ⊕  z  )=(  x ⊕ y ) ⊕  z; (C3)  x ⊕ ”  x =1 ; (C4)  x ⊕ 1=1 ; (C5)  x ⊕ 0=  x; (C6)  ” (  x ⊕ y )= ”  x ⊙ ” y; (C7)  ” ( ”  x )=  x; (C9)  x ∨ y = y ∨  x; (C10)  x ∨ ( y ∨  z  )=(  x ∨ y ) ∨  z; (C11)  x ⊕ ( y ∧  z  )=(  x ⊕ y ) ∧ (  x ⊕  z  ) ; (C1 ′ )  x ⊙ y = y ⊙  x; (C2 ′ )  x ⊙ ( y ⊙  z  )=(  x ⊙ y ) ⊙  z; (C3 ′ )  x ⊙ ”  x =0 ; (C4 ′ )  x ⊙ 0=0 ; (C5 ′ )  x ⊙ 1=  x; (C6 ′ )  ” (  x ⊙ y )= ”  x ⊕ ” y; (C8)  ” 0=1 ; (C9 ′ )  x ∧ y = y ∧  x; (C10 ′ )  x ∧ ( y ∧  z  )=(  x ∧ y ) ∧  z; (C11 ′ )  x ⊙ ( y ∨  z  )=(  x ⊙ y ) ∨ (  x ⊙  z  ) : It is well known that from any MV-algebra it is possible to induce a lattice structure according to theresult which we present now. Before its introductionwe present some results about MV-structures: Lemma 1.  In any  MV  -algebra the following holds :  x ∨ y = y  i   ”  x ⊕ y =1 : Proof.  From y  =  x ∨ y  (2 : 3)= ” ( ”  x ⊕ y ) ⊕ y  ( ∗ )we get ”  x ⊕ y  = ”  x ⊕ [ ” ( ”  x ⊕ y ) ⊕ y ] ( ∗ )=( ”  x ⊕ y ) ⊕ ” ( ”  x ⊕ y ) (M1) ;  (M2)=1 (M7) : Conversely, suppose ”  x ⊕ y =1; then, trivially  x ∨ y  = ” ( ”  x ⊕ y ) ⊕ y  (2 : 3)= y  (hp) ; (2 : 1) ; (P4) ; (P2) : The second lemma contains three results which arestrongly depending from the crucial axiom (P5) Lemma 2.  In any  MV  -algebra the following proper-ties hold  :(P5a)  x ⊕ ( y ∧  z  )=(  x ⊕ y ) ∧ (  x ⊕  z  ) ; (P5b) (  x ⊕ ” y ) ∨ ( y ⊕ ”  x )=1 ; (P5c)  x ∧ ( y ∨  z  )=(  z  ∧  x ) ∨ ( y ∧  x ) . The following theorem shows that any  MV  -algebrainduces a Kleene (distributive) lattice. Theorem 2.1.  Let    A; ⊕ ; ” ; 0   be an  MV  -algebra ; then the structure    A;  ∨ ;  ∧ ; ” ; 0   turns out to be adistributive Kleene algebra ;  i.e. ; (1)  A  is a distributive lattice with respect to thebinary join and meet operations  ∨ ;  ∧  dened by  (2 : 3) ;  and   (2 : 4);  the partial order relationinduced by these operators is  x 6 y  i  def   x ∨ y = y i   x  →  L  y := ”  x ⊕ y =1 (2.5a) with respect to which the lattice  A  is bounded by the minimum element  0  and the maximumelement  1:= ” 0: ∀  x ∈  A;  0 6  x 6 1 : (2)  The distributive lattice  A  is Kleene ;  since it canbe equipped with a unary operation ” :  A  →  A of Kleene orthocomplementation such that (K1)  ” ( ”  x )=  x;  G. Cattaneo et al./Fuzzy Sets and Systems 108 (1999) 201–222  205 (K2)  ” (  x ∧ y )= ”  x ∨ ” y; (K3)  x ∧ ”  x 6 y ∨ ” y: Proof.  In order to have a distributive lattice, the fol-lowing two properties are sucient:(i)  x ∧ (  x ∨ y )=  x; (P5c)  x ∧ ( y ∨  z  )=(  z  ∧  x ) ∨ ( y ∧  x ) :  Now we prove that (i) holds in any MV-algebra. From( y ∨  x ) ∧  x  = [ ” ( ” y ⊕  x ) ⊕  x ] ∧  x = ” [ ” [ ” ( ” y ⊕  x ) ⊕  x ⊕ ”  x ] ⊕ ”  x ]=  x (which is true for arbitrary  x;y ∈  A ) applying DeMorgan, we get( y ∧  x ) ∨  x =  x: From (P5c)  x ∧ (  x ∨ y )=( y ∧  x ) ∨ (  x ∧  x )=( y ∧  x ) ∨  x =  x: Once proved the lattice structure of   A , let us noticethat condition(M7)  x ⊕ ”  x = ” 0is a consequence of (P5b) for the particular case of  y =  x .For the Kleene conditions, note that (K1) is the(M5) and the (K2) is the (vi) of Theorem 1.4 in [10].In order to prove (K3), let us notice that in a latticewith unary operation satisfying (K1) and (K2) condi-tion (K3) is equivalent to the implication:(K3a)  x 6 ”  x  and  ” y 6 y  ⇒  x 6 y: The Kleene condition (K3) is then assured by the(P5b) and the following monotony property of   ⊕ (Theorem 1.8 of  [10]):(Mo)  x 6 y  implies  x ⊕  z  6 y ⊕  z: In fact, an iterative application of (Mo) leads to thefurther property:( ∗ Mo)  x 6 y; h 6 k   implies  x ⊕ h 6 y ⊕ k: Indeed, from  x 6 y  and the (Mo) we get  x ⊕ h 6 y ⊕ h and  y ⊕ h 6 y ⊕ k  , and so by the transitivity of the par-tial order relation we conclude that  x ⊕ h 6 y ⊕ k  . Now, by ( ∗ Mo),  x 6 ”  x  and ” y 6 y  imply  x ⊕ ” y 6 ”  x ⊕ y from which( ∗∗ Mo)  ” (  x ⊕ ” y ) ⊕ ”  x ⊕ y =1 : Then1 =(  x ⊕ ” y ) ∨ ( y ⊕ ”  x ) (P5b)= ” [ ” (  x ⊕ ” y ) ⊕ y ⊕ ”  x ] ⊕ y ⊕ ”  x  (2 : 3)= ”  x ⊕ y  ( ∗∗ Mo) ; that is  x 6 y . Remark1. Condition(K1)isthealgebraiccounterpartof the  strong double negation  law. From the algebraic point of view, let us notice that under condition (K1),the following are mutually equivalent, for arbitrary  x;y ∈  X  :(K2a)  ”  x ∧ ” y = ” (  x ∨ y ) ; (K2b)  ”  x ∨ ” y = ” (  x ∧ y ) ; (K2c)  x 6 y  implies ” y 6 ”  x: Conditions (K2a,b) are the  de Morgan  laws, whereas(K2c) is the  contraposition  law. Lemma 3.  In any  MV  -algebra the following propertyholds :  x ∨ y 6  x ⊕ y: Proof.  Property  x 6  x ⊕ y  is a trivial consequenceof (P2) and (Mo) applied to the case 0 6 y . Now,from  x 6  x ⊕ y  we get  x ∨ y 6 (  x ⊕ y ) ∨ y  and from y 6  x ⊕ y , written as (  x ⊕ y ) ∨ y =  x ⊕ y , we obtainthe thesis.In any  MV  -algebra the elements which are idem- potent with respect to the operation  ⊕  (equivalently, ⊙ ) are exactly those which satisfy the law of the “ex-cluded middle” with respect to the lattice operation ∨ (equivalently, the law of “noncontradiction” with re-spect to the lattice operation  ∧ ). This result leads tothe following [10]: Theorem 2.2.  Let  A  be an MV-algebra. Then ;  theset of all crisp  ( exact ;  sharp )  elements  A e  := { e ∈  A  :  e ⊕ e = e } = { f ∈  A  :  f ⊙ f = f }
Search
Tags
Related Search
We Need Your Support
Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks
SAVE OUR EARTH

We need your sign to support Project to invent "SMART AND CONTROLLABLE REFLECTIVE BALLOONS" to cover the Sun and Save Our Earth.

More details...

Sign Now!

We are very appreciated for your Prompt Action!

x