Fuzzy Sets and Systems 108 (1999) 201–222www.elsevier.com/locate/fss

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

G. Cattaneoa, R. Giuntinib, R. Pillac

a Dipartimento di Scienze dell’Informazione, Universit�a di Milano, Milano, ItalybDipartimento 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 (BZMVdM-algebra). We study this structure and sketch its main properties. A BZMVdM-algebra is asystem endowed with a commutative and associative binary operator ⊕ and two unusual orthocomplementations: a Kleeneorthocomplementation (”) and a Brouwerian one (∼). As expected, every BZMVdM-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 BZMVdM-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 7→ {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 the

set-theoretic intersection, union and complement,respectively.

The set {0; 1}X of all characteristic functionals onX 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: S 0165 -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) = 0

0; i� �A(x) = 1:(oc′)

The mapping � : P(X ) 7→ {0; 1}X ; A 7→ �A is clearlya boolean lattice isomorphism since

�A∩ B = �A ∧ �B; (1.2a)

�A∪ B = �A ∨ �B; (1.2b)

�Ac =”�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 7→ [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 pairf1; f2 ∈ [0; 1]X and any x∈X by the following:

(f1�f2)(x) := max{0; f1(x) + f2(x) − 1}; (1.4a)

(f1⊕f2)(x) := min{1; f1(x) + f2(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 f1; f2 ∈ [0; 1]X

and any x∈X by the following:

(f1 ∧f2)(x) := min{f1(x); f2(x)}; (1.5a)

(f1 ∨f2)(x) := max{f1(x); f2(x)} (1.5b)

[the induced partial order is the usual pointwise order-ing f16f2 i� for all x∈X , f1(x)6f2(x)]. Of course,di�erently from characteristic functionals in {0; 1}X ,

it may happen that for some f1; f2 ∈ [0; 1]X one couldhave that

f1⊕f2 6=f1 ∨f2 or f1�f2 6=f1 ∧f2:

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) = 0

0; f(x) 6= 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) 6= 1

0; f(x) = 1= �{x∈ X :f(x) 6= 1}: (1.6c)

Trivially, ∼f6”f6[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 7→ [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 BZlattice 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 isjust (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. AnMV -algebra can be equivalentlyde�ned (with a slight modi�cation with respect to theaxiomatization proposed by Mangani in [21]) as a

system 〈A; ⊕ ;” ; 0〉 where A is a non-empty set; 0is 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. We will �rst prove the equivalence, 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 Aobeying 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 ispossible 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 the

binary join and meet operations ∨; ∧ de�nedby (2:3); and (2:4); the partial order relationinduced by these operators is

x6y i�def x∨y=y

i� x →L y :=”x⊕y= 1 (2.5a)

with respect to which the lattice A is boundedby the minimum element 0 and the maximumelement 1 :=”0:

∀x∈A; 06x61:

(2) The distributive lattice A is Kleene; since it canbe equipped with a unary operation” : A 7→ Aof 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∧”x6y∨”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=”0

is a consequence of (P5b) for the particular case ofy= 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) x6”x and ”y6y ⇒ x6y:

The Kleene condition (K3) is then assured by the(P5b) and the following monotony property of ⊕(Theorem 1.8 of [10]):

(Mo) x6y implies x⊕z6y⊕z:

In fact, an iterative application of (Mo) leads to thefurther property:

(∗Mo) x6y; h6k implies x⊕h6y⊕k:

Indeed, from x6y and the (Mo) we get x⊕h6y⊕hand y⊕h6y⊕k, and so by the transitivity of the par-tial order relation we conclude that x⊕h6y⊕k.

Now, by (∗Mo), x6”x and ”y6y imply

x⊕”y6”x⊕yfrom which

(∗∗ Mo) ”(x⊕”y)⊕”x⊕y= 1:

Then

1 = (x⊕”y)∨ (y⊕”x) (P5b)=” [”(x⊕”y)⊕y⊕”x]⊕y⊕”x (2:3)=”x⊕y (∗∗ Mo);

that is x6y.

Remark 1.Condition (K1) is the algebraic counterpartof the strong double negation law. From the algebraicpoint of view, let us notice that under condition (K1),the following are mutually equivalent, for arbitraryx; y∈X :(K2a) ”x∧”y=”(x∨y);(K2b) ”x∨”y=”(x∧y);(K2c) x6y implies ”y6”x:Conditions (K2a,b) are the de Morgan laws, whereas(K2c) is the contraposition law.

Lemma 3. In anyMV -algebra the following propertyholds:

x∨y6x⊕y:

Proof. Property x6x⊕y is a trivial consequenceof (P2) and (Mo) applied to the case 06y. Now,from x6x⊕y we get x∨y6(x⊕y)∨y and fromy6x⊕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

Ae := {e∈A : e⊕e= e}= {f∈A : f�f=f}

206 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

is closed under the operations ⊕ ; � and” and onehas that

∀e; f∈Ae; e⊕f= e∨f and e�f= e∧f :

Furthermore; the system 〈Ae; ⊕ ; � ;” ; 0〉 is not onlya subalgebra of A but is also the largest subalgebraof A which is at the same time a Boolean algebrawith respect to the same operations ⊕ ; � ;” .

The above result that ⊕ and � MV-operationsapplied to pairs of crisp elements coincide with thestandard ∨ and ∧ lattice operations can be extendedto the case in which only one of the two elements iscrisp.

Proposition 2.3. Let A be an MV-algebra. For anye∈Ae and any x∈A the following holds:(i) e⊕x= e∨ x;

(ii) e� x= e∧ x.

Proof. (i) By Lemma 3, one gets x∨y6 x⊕y soit su�ces to prove e⊕x6e∨ x. Now x6 x∨ e soby (Mo), x⊕e6(x∨ e)⊕e = ”(”x⊕e)⊕e⊕e =”(”x⊕e)⊕e= x∨ e.

(ii) Recalling that Ae is closed under the ”operation, part (ii) now follows by duality: e� x=”(”e⊕”x)= (for the now proved (i)) =”(”e∨”x) = e∧ x.

2.1. Di�erence and partial di�erence operationsinduced in any MV-algebra

Let 〈A; ⊕ ;” ; 0〉 be an MV-algebra, then the or-thogonality binary relation induced from the Kleeneorthocomplementation ” : A 7→ A is

x ⊥ y i�def x6”y i� x� y= 0: (2.1.1)

Proposition 2.4. With respect to this orthogonalityrelation the partial order relation (2.5a) can be equiv-alently stated as

x6y i� ∃z : x ⊥ z and x⊕z=y (2.5b)

and in this case z is uniquely determined by

z=”(x⊕”y)

and is called the orthogonal supplement of x relativeto y. Moreover; the orthogonal supplement is thesmallest supplement:

y= x⊕z= x⊕u and z ⊥ x ⇒ z6u:

Proof. Indeed

x�”(x⊕”y) = x� (”x� y) (C6)= (x�”x) � y (C2′)= 0 � y (C3′)=y� 0 (C1′)= 0 (C4′):

Thus, we have veri�ed that x ⊥ [”(x⊕”y)]: Be-sides, let x6y then

x⊕”(x⊕”y) =”(”y⊕x)⊕x (C1)=y∨ x (2:3)= x∨y (C9)=y (hp):

On the contrary, let y= x⊕z, then by the Lemma 3above x6y.

Lastly z is uniquely determined by z=”(x⊕”y).In fact, let z1 and z2 be two elements such that x ⊥ z1,x⊕z1 =y, and x ⊥ z2, x⊕z2 =y, then z1⊕x= z2⊕x,z16”x, and z26”x; using now the Theorem 1.14 of[10] we conclude that z1 = z2. Since z ⊥ xwe get z6u,so ”z= x⊕”(x⊕z) = x⊕”y. Therefore, ”z⊕u= x⊕”y⊕u=y⊕”y= 1, and z6u.

This result allows one to introduce in any MV-algebra A a new (partial) operation of di�erence. Firstof all, let us de�ne

D\(A× A) := {(x; y)∈A× A : x6y}:Then, we can introduce the mapping D\(A × A) 7→A partially de�ned in A × A by the law ∀(x; y)∈D\(A× A);

(y\x) :=”(x⊕”y)∈A: (2.1.2)

One can also de�ne the following (total) operation : A×A 7→ A, ∀x; y∈A:

xy :=y\(x∧y):

(Note that if x6y, then yx=y \ x.) In [12], itis proved that the structure 〈A; ∧; ∨; ; 0〉 is a dis-

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 207

tributive di�erence lattice (distributive D-lattice, forshort), which satis�es the following condition:

(xy)z= (xz)y (2.1.3)

On the contrary, Kopka and Chovanec proved that ifa D-lattice 〈A; ∧; ∨; ; 0〉 satis�es condition (2.1.3),then the structure 〈A; ⊕ ;” ; 0〉 is an MV algebra, oncede�ned ∀x; y∈A:

1 :=”0;”x := 1x;x⊕y :=”(”xy):

Thus, the class of all D-lattices satisfying condition(2.1.3) coincides with the class of all MV algebras.This result can be easily settled also in the frameworkof e�ect algebras (or unsharp orthoalgebras [19]),since every D-poset is an e�ect algebra and vice versa.

3. Distributive BZ lattices

In this section, we recall the de�nition and someresults on distributive BZ-lattices (for the general def-inition of BZ-poset see [8]).

De�nition 3.1. A distributive Brouwer–Zadeh (BZ)-lattice is a structure 〈�; ∨; ∧;” ;∼; 0〉, where(a) 〈�; ∨; ∧; 0〉 is a (nonempty) distributive lattice

with minimum element 0.(b) The mapping ” : �→ � is a Kleene orthocom-

plementation, that is(doc-1) ”(”a) = a;(doc-2) ”(a∨ b) =”a∧”b;

(re) a∧”a6b∨”b:(c) The mapping ∼: �→ � is a Brouwer orthocom-

plementation, that is(woc-1) a∧ ∼∼ a= a,(woc-2) ∼ (a∨ b) = ∼ a∧ ∼ b,(woc-3) a∧ ∼ a= 0.

(d) The two orthocomplementations are linked by thefollowing interconnection rule:(in) ” ∼ a= ∼∼ a:

The mapping ” is also called the Lukasiewicz [orfuzzy (Zadeh)] orthocomplementation while the map-ping ∼ is an intuitionistic-like orthocomplementation.The element 1 := ∼ 0 =”0 is the greatest elementof �.

Remark 1. Under condition (woc-1), the de Morganlaw (woc-2) is equivalent to the contraposition law forthe intuitionistic orthocomplementation “a6b implies∼ b6∼ a”. In general, in an abstract BZ-lattice thedual of the de Morgan law (woc-2) does not hold.

De�nition 3.2. A distributive de Morgan BZ-lattice(BZdM-lattice) is a distributive BZ-lattice for whichthe following hold:

∼ (a∧ b) = ∼ a∨ ∼ b:

Making use of the two unusual orthocomplemen-tations it is possible to de�ne the anti-intuitionisticorthocomplementation [ : � 7→ � by the law

[a :=” ∼”awhich satis�es the following conditions:(aoc-1) [[a6a;(aoc-2) [a∨ [c= [(a∧ c)

[equivalently, a6c implies [c6[a];(aoc-3) a∨ [a= 1.Trivially, for every a∈�, one has that

∼ a6”a6[a :In any distributive BZ-lattice � it is possible to singleout the set �e of exact (or sharp) elements, i.e., thoseelements which are closed with respect to the Brouw-erian (or, equivalently, to the anti-Brouwerian) ortho-complement (a=∼∼a i� a= [[a). The elementswhich are not exact are called fuzzy (or unsharp).

Theorem 3.1. Let 〈�; ∨; ∧;” ;∼; 0〉 be a distribu-tive BZ-lattice. Then the set of all sharp elements

�e = {�∈� : �=∼∼ �}= {�∈� : �= [[�}is closed under the operations ∨; ∧;” and ∼ sinceone has that

∀�; �∈�e; �∨e �= �∨ �; and �∧e �= �∧ �∀�∈�e; ”�= ∼ �∈�e:Further; the structure 〈�e; ∨; ∧;” ; 0〉 is a Booleansubalgebra of 〈�; ∨; ∧;” ;∼; 0〉

Given any BZ-lattice 〈�; ∨; ∧;” ;∼; 0〉, it is pos-sible to introduce two unary operators which can be

208 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

considered as algebraic versions of the “possibility”and the “necessity” connectives of modal logic andde�ned for any a∈� by the two sharp elements, re-spectively,

�(a) :=∼”a= [[a∈�e (necessity)

�(a) :=”∼ a= ∼∼ a∈�e (possibility):

In particular,

(mod-1) The necessity of an element “implies” theelement itself which in its turn “implies” thecorresponding possibility (if necessarily a;then a, which is the modal principle T, andif a, then possibly a):

�(a)6a6�(a):

(mod-2) Necessity and possibility are both idempo-tent: �(�(a)) = a and �(�(a)) = �(a).

(mod-3) Necessity is multiplicative and possibility isadditive

�(a∧ b) = �(a)∧ �(b)and�(a∨ b) = �(a)∨ �(b):

(mod-4) The modal principles of noncontradictionand excluded-middle ([Mo, 40], [Mo, 41])hold:

�(a)∧” �(a) = �(a)∧” �(a) = 0and�(a)∨” �(a) = �(a)∨” �(a) = 1:

The latter can be substituted by the weakerconditions:

a∧ �(”a) = a∧” �(a) = 0anda∨” �(a) = a∨ �(”a) = 1:

(mod-5) Operators � and � act on the exact elementsof �e as the identity operators:

∀�∈�e; �(�) = �(�) = �

[hence �e = �(�) = �(�)]. As a conse-quence, we have the modal principle 5:“necessarily a i� possibly necessarily a”and “possibly a i� necessarily possibly a”.Formally,

�(a) = �(�(a)) and �(a) = �(�(a)):

(mod-6) Necessity and possibility are linked bythe expected interconnection rules betweenmodal-like operators

�(a) =”�”(a)

(possibility = not−necessity−not),

�(a) =”�”(a)

(necessity = not−possibility−not).

(mod-7) An interconnection rule involving intuition-istic-like orthocomplementation and modal-like operators can be stated:

�(∼ a) =∼ �(a)

[in general, �(∼ a) 6= ∼ �(a)].(mod-8) The two unusual orthocomplementations,

both the intuitionistic and the anti-intuition-istic, can be expressed by means of modal-ities according to the following:

[a=”�(a) = ∼ �(a) = [�(a)

(contingency),

∼ a=”�(a) = ∼ �(a) = [�(a)

(impossibility).

3.1. Rough approximation by sharp elementsinduced from modalities

For any element a∈� the associated necessity andpossibility can be considered, respectively, as:(1) The lower (or inner) sharp approximation of a

(approximation of a from the bottom by sharpelements), since one can prove that

�(a) =∨

{�∈�e : �6a}; (3.1.1a)

�(a)∈{�∈�e : �6a}: (3.1.1b)

(2) The upper (or outer) sharp approximation of a(approximation of a from the top by sharp ele-ments), since one can prove that

�(a) =∧

{ ∈�e : a6 }; (3.1.2a)

�(a)∈{ ∈�e : a6 }: (3.1.2b)

a≡ b i� �(a) = �(b) is an equivalence relation on �.Any equivalence class modulo ≡ is called a property;

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 209

we shall denote by pr(a) the equivalence class gener-ated by the element a∈� and any element a∈pr(a)is said to be a representative of property pr(a). Thefollowing hold:

(i) the necessity �(a) belongs to property pr(a) andis the unique exact representative of this prop-erty [hence, all the other elements from the sameproperty are its fuzzy representatives];

(ii) �(a) minimizes the property pr(a): �(a) =∧pr(a);

(iii) �(a) is the best “sharp” approximation fromthe bottom of every fuzzy representativea of property pr(a): ∀a∈pr(a); �(a) =∨{x∈�e : x6a}:

Therefore, any property can be sharply identi�ed withits unique exact representative:

pr(a) ↔ �(a) (property)↔ (necessity):

a≡0 b i� �(a) = �(b) is an equivalence relation on�. The equivalence class generated by a∈� is denotedby pr0(a) and called noperty. �(a) is the unique exactrepresentative of noperty pr0(a), all other elementsfrom pr0(a) are its fuzzy representatives. There-fore, pr0(a) can be identi�ed with its unique exactrepresentative:

pr0(a) ↔ �(a) (noperty) ↔ (possibility):

From another point of view, pr0(a) can also be iden-ti�ed with the exact element”�(a) which, of course,does not belong to this class:

pr0(a) ↔”�(a) =∼ a(noperty) ↔ (impossibility):

The rough approximation of any a∈� by sharp ele-ments is the “necessity-possibility” ordered pair

r(a) := (�(a); �(a)) [with �(a)6�(a)] (3.1.3a)

pictured by the following diagram:

The rough approximation of a can be identi�ed withthe “necessity-impossibility” orthopair

rBZ(a) := (�(a);”�(a)) [with �(a) ⊥”�(a)]

(3.1.3b)

pictured by the diagram

Remark 2. In particular, the rough approximation of“not a”, is the impossibility-contingency pair:

r(”a) = (∼ a; [a);rBZ(”a) = (∼ a; �(a)):

4. BZMVdM-algebras

As mentioned in the previous sections, fuzzy setsdetermine both an MV-algebra and a distributive BZ-lattice. Accordingly, it seems quite natural to de�nean algebraic structure, which is rich enough to capturethe features of both structures.

De�nition 4.1. A BZMVdM-algebra is a system〈A;⊕;” ;∼; 0〉 where A is a non-empty set of ele-ments, 0 is a constant element of A, ” and ∼ areunary operations on A, ⊕ is a binary operation on A,obeying the following axioms:(MB1) (x ⊕ y) ⊕ z= (y ⊕ z) ⊕ x;(MB2) x ⊕ 0 = x;(MB3) ”(”x) = x;(MB4) ”(”x ⊕ y) ⊕ y=”(x ⊕”y) ⊕ x;(MB5) ∼ x⊕ ∼∼ x=”0;(MB6) x⊕ ∼∼ x= ∼∼ x;(MB7) ∼ ” [”(x ⊕ ”y) ⊕ ”y] =”(∼∼ x ⊕

” ∼∼ y) ⊕” ∼∼ y:Making use of BZMVdM-algebra structure, we cande�ne the element 1 =”0; some new operations canbe de�ned according to (2.1)–(2.4). In the followingtheorem we will show that from any BZMVdM-algebra

210 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

A= 〈A;⊕;” ;∼; 0; 1〉 it is possible to single outthe substructure AMV = 〈A;⊕;” ; 0; 1〉, which turnsout to be an MV-algebra, and the substructureABZ = 〈A;∧;∨;” ;∼; 0; 1〉, which turns out to be adistributive de Morgan BZ-lattice.

Theorem 4.1. If A is a BZMV dM-algebra; then thefollowing results are true:(1) x⊕y=y⊕ x;(2) (x⊕y) ⊕ z= x ⊕ (y⊕ z);(3) x⊕ 1 = 1;(4) x⊕” x= 1;(5) ”(x⊕ ∼∼ x)⊕ ∼∼ x= 1;(6) ”x⊕ ∼∼ x= 1;(7) x∧ ∼∼ x= x;(8) ” ∼ x= ∼∼ x;(9) ∼ (x∧y) = ∼ x∨ ∼ y;

(10) ∼ (x∨y) = ∼ x∧ ∼ y(equivalently: x6y implies ∼ y6 ∼ x);

(11) x∧ ∼ x= 0;(12) ∼ x= ∼∼∼ x;(13) ∼ x⊕ ∼ x= ∼ x;(14) ”0 = ∼ 0.

Proof. The equivalence between (MB1) and (1), (2)has been shown in Proposition 2.1:(3)

x ⊕ 1 = (x⊕ ∼∼ x)⊕ ∼ x (1); (2); (MB5)

= ∼∼ x⊕ ∼ x (MB6)

= 1 (MB5):

Let us notice that with the proof of (3) we have thatany BZMVdM-algebra is an MV-algebra according toDe�nition 2.1; therefore, in the sequel we can alsotake into account the structure of Kleene distributivelattice induced by the join and meet operations de�nedby (2.3) and (2.4).(4) See the proof of (M7) in Proposition 2.1.(5)

”(x⊕ ∼∼ x)⊕ ∼∼ x=” ∼∼ x⊕ ∼∼ x (MB6)

= ∼∼ x ⊕” ∼∼ x (1)

= 1 (4):

(6)

”x⊕ ∼∼ x =”x ⊕ x⊕ ∼∼ x (MB6), (2), (1)

= 1⊕ ∼∼ x (4)

= ∼∼ x ⊕ 1 (1)

= 1 (3):

(7)

x∧ ∼∼ x =” [”(x ⊕” ∼∼ x)⊕” ∼∼ x] (2:4)

=” [”(”x⊕ ∼∼ x) ⊕”x] (MB4)

=”(0⊕”x) (6); (2:1)

= x (1),(MB2),(MB3).

(8) If x=y then axiom (MB7) becomes

∼” [”(x⊕” x) ⊕”x]= ”(∼∼ x ⊕” ∼∼ x) ⊕” ∼∼ x

and thus

” ∼” [”(x ⊕”x) ⊕”x]= ” [”(∼∼ x ⊕” ∼∼ x) ⊕” ∼∼ x]

from which, applying (4), (2.1), (MB3), (1), (MB2),(MB3)

” ∼ x= ∼∼ x:(9) From (MB7) and from (8) we get

∼ {” [”(x ⊕”y) ⊕”y]}

= [”(” ∼ x⊕ ∼ y)⊕ ∼ y]; (4.0.1)

that is

(DM) ∼ (x∧y) = ∼ x∨ ∼ y:

(10) If x6y, where 6 is the usual partial order rela-tion for A with respect to which A is a lattice, we get

x= x∧yfrom which, applying (DM)

∼ x= ∼ (x∧y) = ∼ x∨∼y;

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 211

that is

∼ y6 ∼ x:[It is well known that under condition (7), ∀x x6∼∼ x, the now proved contraposition law is equivalentto the de Morgan law: ∼ (x∨y) = ∼ x∧ ∼ y].(11)

x∧∼ x =” [”(x ⊕” ∼ x) ⊕” ∼ x] (2:4)=” [”(x⊕ ∼∼ x)⊕ ∼∼ x] (8)= 0 (5), (2.1),

(MB3):

(12) It follows from (7) and (10).(13) It follows from (MB6) and (12).(14) We know that 1 =”0 is the lattice maximumelement, ∀x∈A, x61; in particular ∼ 06”0. Now,from ∀x∈A, 06 ∼ x, and using (7) and the contra-position law (10), it follows that ∀x∈A, x6 ∼∼ x6 ∼ 0; in particular, ”06 ∼ 0.

Remark 1. A BZMVdM-algebra is a de Morgan BZdistributive lattice, since it satis�es property (9). Letus notice that, using (13), the condition (MB5) isequivalent to the condition

(MB5a) ∼∼ x ⊕ (∼ x⊕ ∼ x) = 1:

We introduce now an equivalent de�nition ofBZMVdM-algebra, which is less economical froman axiomatic point of view, but more useful forcomputation.

Theorem 4.2. Let 〈A;⊕;” ;∼; 0〉 be a system whichveri�es (MB1)–(MB6) of De�nition 4:1; thencondition

(MB7) ∼” [”(x ⊕”y) ⊕”y]=”(∼∼ x ⊕” ∼∼ y) ⊕” ∼∼ y

is equivalent to the two conditions

(MB7a) ” ∼ x= ∼∼ x;(MB7b) ∼ {” [”(x ⊕”y) ⊕”y]}

= [”(” ∼ x⊕ ∼ y) ⊕ ∼y](i:e:; ∼ (x∧y) =∼ x∨∼y):

Proof. [(MB7)⇒ (MB7a), (MB7b)]: See the proofof properties (8) and (9) in Theorem 4.1.

[(MB7a), (MB7b)⇒ (MB7)]: Following the proofof Theorem 4.1, we can see that conditions (MB1)–(MB6) and (MB7a), (MB7b) are su�cient to provethe contraposition law (10), condition (12), and con-sequently condition (MB7).

Proposition 4.1. Let A be a BZMVdM -algebra; then(i) ∀x; y∈A; x∧y= 0 i� y6 ∼ x (equivalently;

i� x6 ∼ y);(ii) let x∈A be such that x⊕x= x; then ∀y∈A; x∧

y= 0 i� x6”y:

Proof. (i) Suppose x∧y= 0, then

y∧ ∼ x= (y∧ ∼ x) ∨ 0

= (y∧∼ x)∨ (y∧ ∼ y)

= y∧ (∼ x∨ ∼ y)

= y∧ ∼ (x∧y)

= y∧ ∼ 0 =y:

Conversely, suppose y6 ∼ x; then

x∧y= x ∧ (y∧ ∼ x) =y∧ (x∧ ∼ x) = 0:

(ii) Suppose x= x ⊕ x then, from the (i) of Propo-sition 2.2, x�y= x∧y, now from the (2.1.1) we getx6”y i� x∧y= 0.

The following proposition says that in anyBZMVdM-algebra the set of the ∼-exact elements ofthe BZ-lattice substructure

Ae;∼ = {x∈A : x= ∼∼ x}= {x∈A : x= [[x}coincides with the set of the ⊕-exact elements of theMV algebra substructure

Ae;⊕ = {x∈A : x= x ⊕ x}= {x∈A : x= x � x}:The coincidence between the two families of exact el-ements is a further insight supporting our claim thatBZMVdM-algebras can be regarded as a suitable al-gebraic structure to describe both MV-algebras anddistributive de Morgan BZ-lattices.

Theorem 4.3. In a BZMVdM -algebra the followingholds:

∼∼ x= x i� ∼ x ⊕ x= 1 i� x ⊕ x= x:

212 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

Proof. Under condition ∼∼ x= x,

(MB6) ⇒ x⊕ x= x;

(MB5) ⇒∼ x⊕ x= 1:

Suppose x= x⊕ x, then by the (ii) of Proposition 4.1in the particular case of y=”x we get x∧”x= 0;hence, by the (i) we have ”x6 ∼ x, which implies∼∼ x=”∼ x6x. But from the property (7) of The-orem 4.1 we have x∧∼∼ x= x; i.e. x6 ∼∼ x: There-fore, we can conclude, since 6 is an antisymmetricrelation, that x= ∼∼ x:

On the other hand, under condition ∼ x⊕ x= 1, weprove that x∧∼∼ x= ∼∼ x, i.e. ∼∼ x6x: In fact,

x∧ ∼∼ x=” [”(x ⊕” ∼∼ x)⊕” ∼∼ x] (2:4)

=” [”(x ⊕”” ∼ x)⊕”” ∼ x] (8)

=” [”(x⊕ ∼ x)⊕ ∼ x] (MB3)=”(0⊕ ∼ x) (1), (hp), (2.1),

(MB3)=” ∼ x (MB2)= ∼∼ x: (8):

Thus, similar to the above case we have x= ∼∼ x.

Corollary. In any BZMVdM -algebra one has

Ae;∼ =Ae;⊕:

Furthermore; this is a Boolean subalgebra of A withrespect to the same operations ⊕; �; and” (= ∼).

In the sequel, the Boolean subalgebra determined byAe;∼ ( =Ae;⊕) will be denoted by Ae and its elementsare the crisp (sharp, exact) elements of A.

The coincidence between Ae;∼ and Ae;⊕ is essen-tially due to condition (MB7). Indeed, in the follow-ing example we present a structure such that all theaxioms (MB1)–(MB6) are satis�ed, Ae;∼ ⊂Ae;⊕, but(MB7) fails.

Example 4.1. Let A= {0; a; b; 1} be a �nite set of dis-tinct elements. We de�ne

” ∼0 1 1a b 0b a 01 0 0

⊕0 0 00 a a0 b b0 1 1a 0 aa a aa b 1a 1 1b 0 bb a 1b b bb 1 11 0 11 a 11 b 11 1 1

Then the structure 〈{0; a; b; 1};” ;∼;⊕; 0; 1〉 satis�esthe conditions (MB1)–(MB6), but not (MB7) forx= a and y= b [in particular the de Morgan law (9)of Theorem 4.1 (also (MB7b)) does not hold since∼ (a∧”a) = 1 6= 0 =∼ a∨ ∼ ”a, whereas the in-terconnection law (MB7a) holds]. In this example, the⊕ operation correspond to the join operation ∨ withrespect to the lattice pictured in the following diagram:

In this case we have that Ae;∼ = {0; 1} and Ae;⊕ ={0; a; b; 1}.

We prove now some results about the BZMVdM-structure.

Proposition 4.2. Let A be a BZMVdM -algebra; thenfor any x; y; z ∈A we have

(x⊕y)∧ z= 0 i� x∧ z= 0 and y∧ z= 0:

Proof. Suppose (x⊕y)∧ z= 0; then triviallyx∧ z6(x⊕y)∧ z= 0 and y∧ z6(x⊕y)∧ z= 0.Therefore, x∧ z= 0 =y∧ z.

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 213

Conversely, suppose x∧ z= 0 and y∧ z= 0. By[10] (Theorem 1.15) x ⊕”z=”z=y ⊕”z. So,(x⊕y)⊕”z= x⊕ (y⊕”z) = x⊕”z=”z. Fromthis result we conclude that (x ⊕ y)∧ z= (2:4) =[(x ⊕ y) ⊕”z] �”z= 0 �”z= (C4′) = 0.

Proposition 4.3. Let A be a BZMVdM -algebra; thenfor any x; y; z ∈A we have

(x⊕y)∧ z= 0 i� z6 ∼ x �∼y:

Proof. Suppose (x⊕y)∧ z= 0; then clearly x∧ z= 0=y∧ z. Thus, by (i) Proposition 4.1, z6∼x andz6∼y; hence, z6 ∼ x∧ ∼ y. But ∼ x;∼y∈Ae,and therefore z6 ∼ x� ∼ y.

Conversely, suppose z6∼ x � ∼y, then triviallyz6{∼ x;∼ y} and, by (i) of Proposition 4.1, z ∧ x= 0= z ∧y; by Proposition 4.2 it follows (x⊕ y)∧ z= 0.

Making use of these results, we will prove now thatin any BZMVdM-algebra an analogous property of the(C6) De�nition 2.3 for the Brouwer orthocomplemen-tation holds.

Proposition 4.4. Let A be a BZMVdM -algebra; thenfor any x; y∈A we have

∼ (x⊕y) = ∼ x � ∼y:

Proof. From (11) in Theorem 4.1, (x⊕y)∧∼ (x ⊕y) = 0, and by Proposition 4.3 it follows ∼ (x⊕y)6∼ x � ∼y.

Moreover, x⊕y6∼∼ x ⊕ ∼∼y, from which weget ∼ (∼∼ x ⊕ ∼∼y)6∼ (x ⊕ y); but from thefact that ∼∼ x ⊕ ∼∼y∈Ae it holds ∼ (∼∼ x ⊕∼∼y) =”(∼∼ x ⊕ ∼∼y) =∼ x � ∼y. Thus∼ x � ∼y6∼ (x ⊕ y).

Remark 2. In general, the dual of the now provedproperty does not hold:

∼ (x�y) 6= ∼ x ⊕ ∼y:

Take x=y= (1=2), then∼ [(1=2)�(1=2)] =∼ 0 = 1 6=0 =∼ (1=2) ⊕ ∼ (1=2).

De�nition 4.2. Let A be a BZMVdM-algebra, then weintroduce two orthogonality binary relations:

x ⊥ y i� x6”y;x ⊥∼ y i� x6∼y:

Proposition 4.5. In any BZMVdM -algebra

x ⊥ y i� x � y= 0;x ⊥∼ y i� x∧y= 0:

Moreover; we have that

x ⊥∼ y implies x ⊥ y and x∨y= x⊕y:

Proof. The �rst statement is the (2.1.1) and the sec-ond the (i) of Proposition 4.1. The third statement isa consequence of the property ∀x∈A, ∼ x6”x. Letx6∼y; then 06x∧y6∼y∧y= 0, i.e., x∧y= 0,and by [10] Theorem 1.15 ”x ⊕ y=”x. Sox∨y=”(”x⊕y) ⊕ y=””x ⊕ y= x ⊕ y by(2.3) and (MB3).

We will now show that the class of all linear(i.e., totally ordered) BZMV-algebras is such that theBrouwer orthocomplementation is uniquely de�ned.

Proposition 4.6. Let A be a linear BZMVdM -algebra;then the Brouwer orthocomplementation ∼ isuniquely de�ned in the following way ∀x∈A:

∼ x=

{1 if x= 0;

0 otherwise:

Proof. If x= 0, then ∼ x=∼ 0 = 1. If x 6= 0, thenx∼ x since otherwise x= x∧∼ x= 0. Thus∼ x6x by the linearity of A, from which it follows∼ x=∼ x∧ x= 0.

4.1. Stonian MV algebras are BZMVdM -algebrasand vice versa

Let us recall the following de�nition of Belluce [1]:

De�nition 4.3. Let 〈A;⊕;” ; 0〉 be an MV-algebras;then A is said to be stonian i� the following holds:

∀x∈A; ∃ex ∈Ae;⊕ : {y∈A : x∧y= 0}= {y∈A :y6ex}: (4.1.1)

214 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

It is easy to prove that in the case of a stonian MV-algebra, for any x the crisp (⊕-idempotent) element ex(whose existence is assured by the above de�nition) isunique. An important result based on this uniquenessis the following theorem:

Theorem 4.4. Any stonian MV algebra 〈A;⊕;” ; 0〉equipped with the mapping

∼: A 7→A; x→∼ x := ex;

where ex is the unique crisp element associated to xby the (4:1:1); is a BZMVdM -algebra 〈A;⊕;” ; ∼; 0〉.

Proof. We list a sequence of results from [1] whichleads to this theorem

(1) ∀x∈A, x∧∼ x= 0 (noncontradiction law),(2) ∀x; y∈A, x6y implies ∼y6∼ x

(contraposition law),(3) ∀x; y∈A, ∼ (x∧y) =∼ x∨∼y

(condition (MB7b)),(4) ∀e∈Ae;⊕, ∼ e=”e (coherence condition).

Let us notice that from (4), applied to the element∼ x∈Ae;⊕, we have the interconnection law

(MB7a) ∀x∈A; ∼∼ x=” ∼ x:

Again from (4), recalling that Theorem 2.1 holds forany MV crisp elements e∈Ae;⊕ of an MV-algebra (inparticular e ⊕”e= e ∨”e= 1), we obtain, in theparticular case of the crisp element ∼ x

(MB5) ∼ x ⊕ ∼∼ x= 1.

The stonian condition (4.1.1) in the case of the crispelement ∼ x∈Ae;⊕ : {y∈A :y∧∼ x= 0}= {y∈A :y6∼∼ x}, taking into account the (1), implies thefollowing property:

(woc-1) ∀x∈A; x6∼∼ x:

Lastly, from (woc-1) we have x∨∼∼ x=∼∼ x, andapplying (ii) of Proposition 2.1 to ∼∼ x∈Ae weobtain

(MB6) x ⊕ ∼∼ x=∼∼ x:

Thus, we have obtained that any stonian MV-algebra is an BZMVdM-algebra; we now prove thatthe converse is also true.

Theorem 4.5. If 〈A;⊕;” ; ∼; 0〉 is a BZMVdM -algebra; then 〈A;⊕;” ; 0〉 is a stonian MV -algebra.

Proof. For any x∈A, condition (12) of Theo-rem 4.1 guarantees that ∼ x=∼∼ (∼ x), and thus, byTheorem 4.3, ∼ x is idempotent: ∼ x ⊕ ∼ x=∼ x,i.e., ∼ x∈Ae.

Now, let z ∈A, then by Proposition 4.1 we have thatx∧ z= 0 i� z6∼ x.

In conclusion, BZMV dM-algebras and stonian al-gebras are two equivalent ways of de�ning the samealgebraic structure; we only note that(1) The de�nition of BZMVdM-algebra is com-

pletely equational [also in the less economicalversion with (MB7a) and (MB7b)]; this will bevery useful for any application to many-valuedlogics and their algebraic semantics (Moisil al-gebras [22, 23], MV-algebras of Chang [10, 11], Lukasiewicz algebras of Cignoli and Monteiro[13, 14], BZ-algebras of Cattaneo et al. [6, 8]with the BZdM version [3, 5]). All these struc-tures can be recovered as particular substructuresfrom any BZMVdM-algebra.

(2) The BZMVdM equational version can be easilygeneralized to an algebraic structure of BZMV inwhich the (MB7b) is substituted by the weakerde Morgan condition (10) of Theorem 4.1 (i.e.,contraposition law for the Brouwerian orthocom-plementation). In this case we have always anMV-substructure and a BZ-substructure, whichis not a BZdM-algebra. Of course, owing to theabove results, no genuine BZMV-algebra can bea stonian algebra (an example of this algebra isgiven by Example 4.1), and applications alongthis nonstonian direction can be found in thelogic of unsharp quantum mechanics [5, 4].

4.2. BZMV 3-algebras

In this section we investigate an interesting subclassof BZMVdM.

De�nition 4.4. A BZMV3-algebra is a BZMVdM-algebra in which condition (MB5) is replaced by thefollowing (stronger) condition, for any x∈A:

(s-MB5) ∼ x ⊕ (x ⊕ x) = 1.

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 215

Trivially, any BZMV3-algebra is an BZMVdM-algebra once it is veri�ed that condition (s-MB5)applied to the element ∼ x, whatever be x∈A, yieldsjust the condition (MB5).

Proposition 4.7. Let A be a BZMVdM -algebra; thenthe following are equivalent:

(i) ∀x∈A; ∼ x ⊕ (x ⊕ x) = 1;(ii) ∀x∈A; x∨∼ x= x∨”x= x ⊕ ∼ x.

Proof. (i)⇒ (ii): By Proposition 2.2(ii) and ∼ x=∼ x ⊕ ∼ x we get

x ∨ ∼ x= x ⊕ ∼ x:Since x∨ ∼ x6x∨”x, it remains to show x∨”x6x ∨ ∼ x, i.e., (x∧”x) ⊕ (x ∨ ∼ x) = 1:

(x∧”x) ⊕ (x ∨ ∼ x)= (x∧”x) ⊕ x ⊕ ∼ x= [(x ⊕ ∼ x) ⊕ x]∧ [x ⊕ ∼ x ⊕”x]= [∼ x ⊕ (x ⊕ x)]∧ 1 = 1;

where the last equality follows from the (s-MB5).(ii)⇒ (i): Let now the (ii) be true, then ”x6x ∨

”x= x ⊕ ∼ x; thus ∼ x ⊕ (x ⊕ x) = 1.

Theorem 4.6. Any BZMV 3-algebra A is a BZ3 dis-tributive lattice [3, 5], i.e.; the following is true:

∀x; y∈A; x∧ �(y)6�(x)∨y[which we recall is equivalent to the property that theBZ order is determined by modalities (x6�(y) and�(x)6y imply x6y)].

Proof. We want to show that in any BZMV3-algebra∀x; y∈A; ”(x∧∼∼y) ⊕ (∼”x ∨ y) = 1. Indeed,

”(x∧∼∼y) ⊕ (∼”x∨y)

= (”x ∨ ∼y) ⊕ (∼”x∨y)

= ”x ⊕ ∼y ⊕ ∼”x ⊕ y Proposition 2:2

= (”x∨ x) ⊕ (y∨”y) Proposition 4:7:

By the regularity condition (re) of De�nition 3.1we have ”y∧y6x∨”x and therefore (y ∨”y)⊕ (x ∨”x) = 1.

We will now characterize in the category of alllinear BZMVdM-algebras the subcategory of allBZMV3-algebras.

Proposition 4.8. A linear BZMVdM -algebra is aBZMV 3-algebra i� it is of one of the two formsA2 := {0; 1} or A3 := {0; 1=2; 1} [where 1=2 is a(unique) �xed point of the Kleene orthocomple-mentation ” ].

Proof. If A is {0; 1} or {0; 1=2; 1}, then it is linear. ByProposition 4.7 the identity ∼ x ⊕ (x ⊕ x) = 1 is triv-ially veri�ed. Conversely, suppose {0; 1}⊂A. Then∃x∈A such that x 6= 0; 1. By Lemma 1, ∼ x= 0.By Hypothesis 1 =∼ x ⊕ (x ⊕ x) =”(”x) ⊕ x,from which we get ”x6x. But x 6= 0; 1 implies”x 6= 0; 1 and thus ∼”x= 0. From this it followsthat 1 =∼”x ⊕ (”x ⊕ ”x) =”x ⊕ ”x, i.e.,x6”x. In conclusion, x=”x and owing to theuniqueness of the half element of a BZ structure weget that A= {0; x; 1}.

Example 4.2. The particular case of the three-valuedlinear BZMVdM-algebra A3 = {0; 1=2; 1} is character-ized by the tables:

” ∼0 1 11=2 1=2 01 0 0

⊕ 0 1=2 10 0 1=2 11=2 1=2 1 11 1 1 1

where the second table is uniquely determined by con-ditions (4) and (5) of De�nition 2.3:

∀x; x ⊕ 0 = x; x ⊕ 1 = 1

and the fact that (1=2) =”(1=2), by De�nition 2.3(3),implies (1=2)⊕(1=2) = 1. Let us notice that the “sum”operation ⊕ of this example is the join operation ∨ ofthe induced Kleene lattice substructure, for any pairof elements except:

(1=2) ⊕ (1=2) = 1 and (1=2)∨ (1=2) = (1=2):

5. Direct products of BZMVdM-algebras and arepresentation theorem

Before presenting the next results, we introducethe following de�nitions. Let A and B be BZMVdM-algebras. We say that the function ’ :A→B is a

216 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

homomorphism of A onto B if ’ is such that ’(0)= 0 and ’ preserves the three operations ⊕;” ; ∼ :

’(0) = 0;

’(x⊕y) =’(x)⊕’(y);

’(”x) =”’(x);

’(∼ x) = ∼’(x):

Trivially, if ’ is a homomorphism we have

’(1) = 1;

’(x�y) =’(x)�’(y):

If the function ’ is one-to-one and onto, then ’ iscalled an isomorphism of A onto B.

Given a collection of BZMVdM-algebras Ax : x∈X ,their cartesian (or direct) product is the structure⟨∏x∈X

Ax;⊕;” ; ∼; 0⟩;

where∏x∈X Ax is the cartesian (or direct) product of

the sets Ax, and the operations are de�ned accordingto the following:

0 = (0x)x∈X ;

(ax)x∈X ⊕ (bx)x∈X = (ax ⊕ bx)x∈X ;”(ax)x∈X = (”ax)x∈X ;∼ (ax) = (∼ ax)x∈X :The identity element 1 :=”0 and the product opera-tion � are given by

1 = (1x)x∈X ;

(ax)x∈X � (bx)x∈X = (ax � bx)x∈X :

Remark 1. In the case of a countable family ofBZMVdM-algebras (indexed as Aj, j∈N) the aboveoperations can be written in sequential notation as

0 = (01; 02; : : : ; 0j; : : :);

(a1; a2; : : : ; aj; : : :)⊕ (b1; b2; : : : ; bj; : : :)

= (a1 ⊕ b1; a2 ⊕ b2; : : : ; aj ⊕ bj; : : :);”(a1; a2; : : : ; aj; : : :) = (”a1;”a2; : : : ;”aj; : : :);∼ (a1; a2; : : : ; aj; : : :) = (∼ a1; ∼ a2; : : : ; ∼ aj; : : :):

In the case of a family of BZMVdM-algebras Ax, x∈X ,under the condition that there exists a BZMVdM-algebra A such that ∀x∈X , Ax =A we have that anyelement (ax)x∈X ∈

∏x∈X Ax( =A) can be written in

functional notation as f :X 7→A, x→f(x) := ax ∈A,i.e.,∏x∈X

Ax =AX :

In functional notation the above operations are written,∀f; g∈AX and ∀x∈X , as follows:

0(x) = 0;

(f⊕ g)(x) =f(x)⊕ g(x);(”f)(x) =”f(x);

(∼f)(x) = ∼f(x);

where the operations in the second member ofthese equalities are performed inside the BZMVdM-algebra A.

Due to the form of the axioms (MB1)–(MB7), witha slight modi�cation of Theorem 1.18 in [10] (also byBirkho�’s HSP theorem), we immediately get:

Theorem 5.1. A homomorphic image of a BZMV dM-algebra is a BZMV dM-algebra and the direct productof BZMV dM-algebras is a BZMV dM-algebra.

De�nition 5.1. Let {Ax : x∈X } be a collectionof BZMVdM algebras. A subdirect product of{Ax : x∈X } is any BZMVdM-subalgebra B of∏x∈X Ax such that ∀x∈X , the projection on the

x-component prx :B→Ax is onto.

In the sequel, we will prove that every BZMVdM-algebra is the subdirect product of totally orderedBZMVdM-algebras.

De�nition 5.2. Let A be a BZMVdM-algebra. An idealof A is any subset I of A such that the followingconditions are satis�ed:

(i) 0∈ I ,(ii) if x; y∈ I , then x⊕y∈ I ,

(iii) if x∈ I and y∈A, then x�y∈ I .Remark 2. One can easily see that condition (iii)above is equivalent to the following:

(iiia) if x∈ I and y6x, then y∈ I .

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 217

Suppose (iii) holds, x∈ I and y6x. By (iii), weget y=y ∧ x= (y⊕”x)� x∈ I . Conversely, sup-pose (iiia) holds and x∈ I . Since x�y6x, we get, by(iiia): x�y∈ I .

Let us notice that from x6x⊕y we immediatelyget the following property:

x⊕y∈ I ⇒ x; y∈ I:

De�nition 5.3. Let A be a BZMVdM-algebra. A ∼ -ideal of A is an ideal, which satis�es the followingcondition:(iv) ∀x; y∈A, if x�y∈ I , then ∼∼ x� ∼”y∈ I .

It should be noticed that not every ideal is a ∼ -ideal. A counterexample can be constructed in theChang algebra C (see [10]). C is a stonian MV-algebra (being linear). Thus, by Proposition 4.6, isalso a MVBZdM-algebra.

Theorem 5.2. Let A be a BZMV dM-algebra and letI be a ∼ -ideal. The relationx≡I y i� (x�”y)⊕ (”x�y)∈ Iis a congruence relation on A.

Proof. By Chang [10] Theorem 4.3, it su�ces toshow that ∀x; y∈ I , if x≡I y, then ∼ x≡I ∼y.Let us suppose x≡I y. By de�nition, x�”y∈ Iand ”x�y∈ I . Since I is ∼ -ideal, we haveI 3∼∼ x� ∼””y=∼∼ x�∼y and I 3∼∼y� ∼””x= ∼ ∼y� ∼ x. Thus, ∼ x≡I ∼y.

Theorem 5.3. Let A be a BZMV dM-algebra and let≡ be a congruence relation. The set I := {x : x≡ 0}is a ∼ -ideal.

Proof. I is clearly an ideal since ≡ is a congruencerelation. We prove that I is a ∼ -ideal. Let us sup-pose x�y∈ I , i.e., x�y≡ 0. We have to prove that∼ ∼ x� ∼”y≡ 0. By hypothesis, x�y≡ 0 andtherefore ”y⊕ x�y≡”y. Thus, x∨”y≡”y.Hence, ∼ (x∨”y) ≡ ∼”y. Therefore, ∼ x ∧∼”y≡ ∼”y, so that ∼”y ∧ ∼ x≡ ∼”y.Hence, ∼ ∼ x� (∼”y ∧ ∼ x) ≡ ∼ ∼ x� ∼”y.Now, ∼ ∼ x� (∼”y ∧ ∼ x) = (∼”y⊕ ∼ ∼ x)�∼ x�∼ ∼ x= 0. Thus, ∼ ∼ x� ∼”y≡ 0.

Clearly, {0} is a ∼ -ideal. It is easy to see that thecorrespondence I 7→≡I is a bijection from the set ofall ∼-ideals of A onto the set of all congruences onA. Given an element x∈ I , the equivalence class ofx determined by ≡I , will be denoted by [x]I and thequotient algebra by A=≡I .

De�nition 5.4. Let A be a BZMVdM-algebra. An idealon A is said to be prime i� ∀x; y∈A, x�”y∈ I or”x�y∈ I .

Theorem 5.4. Let A be a BZMV dM-algebra. ∀x∈A;if x 6= 0; then there exists a prime ∼ -ideal such thatx =∈ I .

Proof. Since {0} is a ∼ -ideal, there exists a ∼ -idealI which is maximal w.r.t. the property a =∈ I . By Chang[11], I is prime.

Let A be a BZMVdM-algebra. Chang [11] provedthat an ideal I on A is prime i� A=≡I is totally ordered.As a consequence of this result and Theorem 5.4, oneobtains the following:

Theorem 5.5. Every BZMV dM-algebra can be rep-resented as the subdirect product of totally orderedBZMV dM-algebras.

6. Two interesting examples of BZMVdM-algebras

In the present section we discuss two interestingexamples of BZMVdM-algebras.

Example 6.1. In the real unit interval [0,1] oncede�ned the operations

a⊕ b :={a+ b i� a+ b61,

1 otherwise,”a := 1 − a;∼ a :=

{1 i� a= 0,

0 otherwise,

we get the algebraic structure 〈[0; 1];⊕;” ; ∼ ; 0〉,which is a BZMVdM-algebra whose set [0; 1]e of allexact elements is the boolean pair {0; 1}, since

a= ∼ ∼ a i� a∈{0; 1}:

218 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

It should be noticed that [0; 1] is an example of aBZMVdM-algebra which is not a BZMV3-algebra:

∼ 13 ⊕ ( 1

3 ⊕ 13 ) = 0⊕ 2

3 = 23 6= 1:

The subset {0; (1=2); 1}⊆ [0; 1] is closed with respectto the operations ⊕;” ; ∼ and is a (linear) MVBZ3-subalgebra of [0; 1] (see Example 4.2).

An Open Problem: In [11], Chang proved that ev-ery equation which holds in the MV-algebra based on[0,1] holds in every MV-algebra. We do not knowwhether such a theorem is still true for BZMVdM-algebras. It should be noticed that the answer is posi-tive for any MVBZ3 and the concrete MVBZ3 basedon {0; (1=2); 1}; this is a direct consequence of Propo-sition 4.8 and Theorem 5.5.

Example 6.2. Let [0; 1]X be the set of all general-ized characteristic functionals (fuzzy sets) on thereference space X ; the structure 〈[0; 1]X ;⊕;” ; ∼ ; 0〉is a BZMVdM-algebra with respect to the opera-tions de�ned pointwise for any f; g∈ [0; 1]X and anyx∈X as follows (taking into account that ∀x∈X , thevalues f(x); g(x)∈ [0; 1] and recalling [0; 1] can beequipped with the standard operations introduced inExample 6.1):

(f⊕ g)(x) :=f(x)⊕ g(x)

={f(x) + g(x) i� f(x) + g(x)61,

1 otherwise,(”f)(x) :=”f(x) = 1 − f(x);

(∼f)(x) := ∼f(x) ={

1 i� f(x) = 0,

0 otherwise.

The null element is the function 0 :X 7→ [0; 1], associ-ating to any point x∈X the constant value 0∈ [0; 1].The unit element 1 =”0 is the function associatingto any element x∈X the constant number 1(x) = 1.Characteristic functionals �� of subsets � of X [de-�ned as ��(x) = 1 i� x∈�, and = 0 otherwise] arethe fuzzy sets ranging on {0; 1}; in particular 0 := �∅and 1 := �X are sets of this kind.

Remark 1. The operation ⊕, and its dual �, are the“truncated” standard operations of sum and producton [0; 1], and they can be expressed in the following

compact forms [see (1.4a,b)]:

(f⊕ g)(x) = min{f(x) + g(x); 1};

(f� g)(x) = max{f(x) + g(x) − 1; 0}:

Let us notice that the BZMVdM-algebra of all fuzzysets is the direct product indexed by the family x∈Xof the BZMVdM-algebras [0; 1]x = [0; 1]:

[0; 1]X =∏x∈X

[0; 1]x:

Moreover, all the results of the present section can beextended to the general case of generalized A-valuedcharacteristic functions (A-valued fuzzy sets), for Alinear BZMVdM-algebra:

AX =∏x∈X

Ax;

where ∀x∈X , Ax =A.

In the context of MV-algebraic structures, the joinand meet operations of the corresponding Kleene lat-tice [see (2.3) and (2.4)] are the following:

(f ∨ g)(x) = max{f(x); g(x)}; (6.1)

(f ∧ g)(x) = min{f(x); g(x)}: (6.2)

The partial ordering relation induced from these lat-tice operations [see (2.5a)] is the standard pointwiseordering relation on fuzzy sets:

f6g i� ∀x∈X; f(x)6g(x): (6.3)

The binary operator → L introduced in the (2.5a), inthe present case de�nes a new fuzzy set starting fromany pair f and g of fuzzy sets according to

(f→L g)(x) = (”f⊕ g)(x)

={

1 i� f(x)6g(x),

1 − f(x) + g(x) i� g(x)¡f(x).

Remark 2. This fuzzy set, using the truncated sum for-malism of Remark 1, is the truth value functional forthe implication connective introduced by Lukasiewicz

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 219

in his in�nite valued logic L∞ (see [25]):

(f→L g)(x) = min{1 − f(x) + g(x); 1}:

Trivially, by de�nition,

(f →L g) = 1 i� f6g:

Moreover, the Kleene orthocomplement of any fuzzyset f can be recovered in the following way:

(f→L 0) =”f:One can also consider the truth value functional of

the intuitionistic implication connective introduced byG�odel in his in�nite-valued logic G∞ (see [25]):

(f→G g)(x) =∨

{h(x) :f(x)∧ h(x)6g(x)}

={

1 i� f(x)6g(x),

g(x) i� g(x)¡f(x).

The fuzzy set f→G g is the relative pseudo-complementation (also Brouwerian orthocomple-mentation, see [15]), of f relative to g (or in g),i.e., it is the unique element of [0; 1]X satisfying theclassical implicative condition:

z6(f→G g) i� f ∧ z6g;which in the particular case of z= 1 assumes the form

(f →G g) = 1 i� f6g:

In this way, the complete lattice [0; 1]X is a pseudo-boolean lattice [2] or Brouwerian lattice [24]. Bymeans of this relative pseudo-orthocomplementationwe can induce the pseudo-complement (or Brou-werian-complement) of any fuzzy set f, which justturns out to be the intuitionistic orthocomplementof f:

(f→G 0) = ∼f:Let us remark the following interesting relations:

∀j=L; G; (1→j f) =f and (f→j f) = 1:

Note that in the case of two exact sets represented bythe characteristic functionals �H and �K we have that

(�H →j �K) = �(H\K)c = �(H c ∪K)

=”�H ∨ �K :

Therefore, the pseudo-complement of an exact set,represented by the characteristic function of H , is justthe characteristic function of the set H c, which repre-sents the set-theoretic complement of H :

(�H →j 0) = �H c :

The exact elements of the BZMVdM-algebra [0; 1]X

are the {0; 1}-valued characteristic functionals (crispor sharp sets):([0; 1]X

)e = {�� :X 7→ {0; 1} |�∈P(X )} :

Thus, the exact part of [0; 1]X can be identi�ed withthe power set P(X ) of X by the Boolean algebrasisomorphism

〈P(X );∩;∪; c; ∅〉 ≡ ⟨{0; 1}X ;∧; ∨;” ; 0⟩

�↔ ��

Given any fuzzy set f∈ [0; 1]X it is possible toassociate with f some peculiar subsets of the universeX in the following way:• the certainly-yes (also the necessity) domain,�1(f) := {x∈X :f(x) = 1};

• the certainly-no (also the impossibility) domain,�0(f) := {x∈X :f(x) = 0};

• the contingency domain, �c(f) := {x∈X :f(x)6= 1};

• the possibility domain,�p(f) := {x∈X :f(x) 6= 0}.The necessity �(f) =∼”(f) of a fuzzy set f is thecharacteristic functional of the certainly-yes domainof f:

�(f) = ��1(f) ={

1 if f(x) = 1

0 otherwise(necessity):

Therefore, two fuzzy sets de�ne a unique property i�they have the same necessity domain; in this way, wecan associate to every property of fuzzy sets the com-mon necessity domain �1(f) of any of its elements.This property is interpreted as: “the point necessarilybelongs to the subset�1(f) of X ” and thus it is sharplyrepresented by the characteristic functional ��1(f).

The possibility �(f) =” ∼ (f) of a fuzzy set f is

�(f) = ��0(f)c ={

1 if f(x) 6= 0

0 otherwise(possibility);

220 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

and so the impossibility operator is given by the char-acteristic functional of the impossibility domain of f:

”�(f) = ∼ f= ��0(f) (impossibility):

Thus, two fuzzy sets de�ne the same noperty i� theyhave the same impossibility domain. To every nopertywe can associate the unique subset �0(f) of X whichrepresents the noperty: “it is impossible that the pointbelongs to the subset �0(f) of X ” and this nopertyis sharply represented by the characteristic functional��0(f).

The rough approximation of a fuzzy set f by sharpsets is the “necessity-possibility” pair, identi�ed withthe pair of ordered subsets of X :

r(f) = (��1(f); ��p(f))≡ (�1(f); �p(f))

[with �1(f)⊆�p(f)]:

From another point of view, one can also construct therough approximation of a fuzzy set as the “necessity-impossibility” pair, identi�ed with the pair of disjointsubsets of X :

rBZ(f) = (��1(f); ��0(f)) ≡ (�1(f); �0(f))

[with �1(f) ∩ �0(f) = ∅]:

All this can be summarized in the following diagram:

Appendix A: Independence of (MB1)–(MB7)conditions

We will now show that the axioms (MB1)–(MB7)are independent in both cases of MVBZdM andMVBZ3 algebras. Note that since (s-MB5) implies(MB5) if we show the independence of (MB5) thenwe automatically have the independence of (s-MB5).Independence of (MB1): Let A= {0; a; b; c; d; 1} be

a �nite set of distinct elements. We de�ne

” ∼0 1 1a b bb a ac d dd c c1 1 1

⊕0 0 00 a a0 b b0 c c0 d d0 1 1a 0 aa a aa b 1a c aa d da 1 1b 0 bb a 1b b bb c bb d db 1 1c 0 cc a ac b bc c cc d 1c 1 1d 0 dd a dd b dd c 1d d dd 1 11 0 11 a 11 b 11 c 11 d 11 1 1

(MB1) is not satis�ed, since

(a⊕ c)⊕d= a⊕d=d

and

(c⊕d)⊕ a= 1⊕ a= 1:

Independence of (MB2): Let A= {0; a; b; 1}. Wede�ne

G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222 221

” ∼0 1 1a b 1b a 11 0 1

x⊕y= 1; ∀x; y∈A:(MB2) is not satis�ed; in fact,

a⊕ 0 = 1 6= a:Independence of (MB3): Let A= {0; 1}. We de�ne

” ∼0 1 11 1 1

⊕0 0 00 1 11 0 11 1 1

(MB3) is not satis�ed; in fact

””0 = 1 6= 0:

Independence of (MB4): Let A= {0; a; b; 1}. Wede�ne

” ∼0 1 1a b 0b a 01 0 0

⊕0 0 00 a a0 b b0 1 1a 0 aa a 1a b 1a 1 1b 0 bb a 1b b 1b 1 11 0 11 a 11 b 11 1 1

(MB4) is not satis�ed; in fact,

”(”a⊕ b)⊕ b= b and ”(a⊕”b)⊕ a= a:

Independence of (MB5) and thus of (s-MB5): LetA= {0; a; 1}. We de�ne

” ∼0 1 1a a a1 0 0

⊕0 0 00 a a0 1 1a 0 aa a aa 1 a1 0 11 a a1 1 1

(MB5) is not satis�ed; in fact,

∼ a⊕ ∼ ∼ a= a 6= 1:

Independence of (MB6): Let A= {0; a; 1}. Wede�ne

” ∼0 1 1a a a1 0 0

⊕0 0 00 a a0 1 1a 0 aa a 1a 1 11 0 11 a 11 1 1

(MB6) is not satis�ed; in fact,

a⊕ ∼ ∼ a= 1 6= a:Independence of (MB7): See Example 4.1.

References

[1] L.P. Belluce, Generalized fuzzy connectives on MV-algebras,J. Math. Anal. Appl., to appear.

[2] G. Birkho�, Lattice Theory, vol. XXV, AmericanMathematical Society, Providence, RI, 1940.

[3] G. Cattaneo, Fuzzy quantum logic II. The logic of unsharpquantum mechanics, Internat. J. Theoret. Phys. 32 (1993)1709–1734.

[4] G. Cattaneo, A uni�ed framework for the algebra of unsharpquantum mechanics, Internat. J. Theoret. Phys. 36 (1997)3083–3115.

[5] G. Cattaneo, R. Giuntini, Some results on BZ structuresfrom Hilbertian unsharp quantum mechanics, Found. Phys.25 (1995) 1147–1183.

[6] G. Cattaneo, G. Marino, Brouwer–Zadeh posets and fuzzyset theory, in: A. di Nola, A.G.S. Ventre (Eds.), Proc. ofMathematics of Fuzzy Systems, Napoli, 1984.

222 G. Cattaneo et al. / Fuzzy Sets and Systems 108 (1999) 201–222

[7] G. Cattaneo, G. Marino, Non-usual orthocomplementations onpartially ordered sets and fuzziness, Fuzzy Sets and Systems25 (1988) 107–123.

[8] G. Cattaneo, G. Nistic�o, Brouwer–Zadeh posets and three-valued Lukasiewicz posets, Fuzzy Sets and Systems 33 (1989)165–190.

[9] G. Cattaneo, R. Pilla, Algebraic structures for distributivemany valued logics, DSI preprint, 1995.

[10] C.C. Chang, Algebraic analysis of many valued logics, Trans.Am. Math. Soc. 88 (1958) 467–490.

[11] C.C. Chang, A new proof of the completeness of the Lukasiewicz axioms, Trans. Am. Math. Soc. 95 (1959)74–80.

[12] F. Chovanec, F. Chopka, D-lattices, Internat. J. Theoret. Phys.34 (1995) 1297–1302.

[13] R. Cignoli, A. Monteiro, Boolean elements in Lukasiewiczalgebras. II, Proc. Japan Acad. 41 (1965) 676–680.

[14] R. Cignoli, Moisil Algebras, Notas de Logica Matematica,vol. 27, Instituto de Matematica, Univ. Nacional del Sur,Bahia Blanca, Argentina, 1970.

[15] A. De Luca, S. Termini, Algebraic properties of fuzzy sets,J. Math. Anal. Appl. 40 (1972) 373.

[16] R. Giuntini, Unsharp orthoalgebras and quantum MValgebras, in: C. Garola, A. Rossi (Eds.), The Foundations ofQuantum Mechanics, Kluwer, Dordrecht, 1995, pp. 325–334.

[17] R. Giuntini, Quasilinear QMV algebra, Internat. J. Theoret.Phys. 34 (1995) 1–11.

[18] R. Giuntini, Quantum MV algebras, Stud. Logica 56 (1966)393–417.

[19] G. Giuntini, H. Greuling, Toward a formal language forunsharp properties, Found. Phys. 19 (1989) 931–945.

[20] F. Guzm�an, C. Squier, Subdirectly irreducible and freeKleene–Stone algebras, Algebra Universalis 31 (1994)266–273.

[21] P. Mangani, On certain algebras related to many valued logics,Boll. Un. Mat. Ital. 8 (1973) 68–78.

[22] G.C. Moisil, Recherches sur les logiques non-chrysippiennes,Ann. Sci. Univ. Jassy 26 (1940) 431–466.

[23] G.C. Moisil, Notes sur les logiques non-chrysippiennes, Ann.Sci. Univ. Jassy 27 (1941) 86–98.

[24] H. Rasiowa, R. Sikorski, The Mathematics of Metamathe-matics, PWN-Polish Scienti�c Pub., Warszawa, 1970.

[25] N. Rescher, Many-valued Logic, McGraw-Hill, New York,1969.

[26] H.P. Sankappanavar, Heyting algebras with dual pseudo-complementation, Paci�c J. Math. 117 (1985) 341–349.

[27] H.P. Sankappanavar, Principal congruences of pseudo-complemented de Morgan algebras, Z. Math. LogikGrundlagen Math. 33 (1987) 3–11.