Two different generalizations of BrouwerZadeh posets (BZ posets) are intro- duced. The former (called pre-BZ poset) arises from topological spaces, whose.
Foundations of Physics, Vol. 30, No. 10, 2000
Pre-BZ and Degenerate BZ Posets: Applications to Fuzzy Sets and Unsharp Quantum Theories G. Cattaneo, 1 R. Giuntini, 2 and S. Pulmannova 3 Received March 12, 1999; revised January 12, 2000 Two different generalizations of BrouwerZadeh posets (BZ posets) are introduced. The former (called pre-BZ poset) arises from topological spaces, whose standard power set orthocomplemented complete atomic lattice can be enriched by another complementation associating with any subset the set theoretical complement of its topological closure. This complementation satisfies only some properties of the algebraic version of an intuitionistic negation, and can be considered as, a generalized form of a Brouwer negation. The latter (called degenerate BZ poset) arises from the so-called special effects on a Hilbert space. It is shown that the standard Brouwer negation for effect operators produces a degenerate BZ poset with respect to the order induced from the partial sum operation.
1. INTRODUCTION BrouwerZadeh (BZ) posets have been introduced in order to describe vagueness both in the classical context of generalized set theory (fuzzy sets on a universe set) (9) and in the quantum context of unsharp quantum theory (effect operators on a Hilbert space) (10) (and see also Ref. 11 for a general treatment of BZ theory). BZ posets consist of a partially ordered set 7 (which in general is not required to have a lattice structure) equipped with two unary operations $: 7 [ 7 and t: 7 [ 7. Both operations satisfy suitable weak (and unusual) forms of the conditions characterizing the standard orthocomplementation on posets. Distributive BZ lattices turn out to be a model of particular nonclassical logics (in general many-valued 1
Dipartimento di Informatica, Sistemistica e Comunicazione, Universita di Milano-Bicocca, Via Bicocca degli Arcimboldi 8, I-20126 Milano, Italy; e-mail: cattangdisco.unimib.it. 2 Dipartimento di Scienze Pedagogiche e Filosofiche, Universita di Cagliari, Via Is Mirrionis 1, I-01239 Cagliari, Italy; e-mail: giuntiniphilos.unifi.it. 3 Mathematical Institute, Slovak Academy of Sciences, SK-81473, Bratislava, Slovakia; e-mail: pulmannmau.savba.sk. 1765 0015-9018001000-1765 18.000 2000 Plenum Publishing Corporation
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logics). In this respect, the unusual complementations $ and t play the algebraic role of two nonclassical negations of Kleene (or Fukasiewicz) type and of Brouwer (or intuitionistic) type, respectively. (5, 7) In this paper we intend to generalize the notion of BZ poset, having in mind two concrete structures: topological spaces and special effects on a Hilbert space (for the latter see Refs. 15, 6). Both these structures can be equipped with two unary operations which retain most of the properties of the fuzzy-like complement $ and of the Brouwer-like complement t. However, these two concrete structures endowed with these two operations do not determine a BZ poset. In particular, in the topological space case, the operation t, differently from BZ posets, does not generally satisfy the weak double negation law (a a tt ). In the special effect case, instead, the BZ property a t a$ is violated. Thus, starting from these two (important) concrete examples, we have two different ways to generalize BZ poset. In this paper we will investigate such generalizations. In particular, in Sec. 2 we will introduce the notion of pre-BZ poset to model the structure arising from topological spaces. In Sec. 10, we will be concerned with degenerate BZ posets, i.e., the algebraic structures arising from special effects on a Hilbert space when the partial order relation is the one induced from the usual partial operation of sum. 2. PRE-BZ POSETS AND BZ POSETS In this section we investigate a weak form of BZ structure, which has as an interesting model the power set of a topological space. Definition 2.1. (i)
t
, 0) where:
( 7, , $, 0) is a Kleene poset, i.e., a poset, lower (rep., upper) bounded by the least (resp. greatest) element 0 (resp., 1 :=0$), equipped with a unary mapping $: 7 [ 7 (called Kleene complementation) satisfying for arbitrary a, b # 7 the following conditions: (K-1) (K-2) (K-3)
(ii)
A pre-BZ poset is a structure (7, , $,
a=a" a b implies b$ a$ a a$ and b$ b imply a b
t
is a unary operator on 7 satisfying the following conditions, for any a, b # 7: (preB-2) (preB-3)
a b implies b t a t a 7 a t =0
Pre-BZ and Degenerate BZ Posets
(iii)
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The two unusual complementations must satisfy the following weak interconnection rules, for any a # 7: (win-0) (win-1) (win-2)
1=0$=0 t a t a$ a$ t a$ t$ t
One can prove that the pre-BZ poset axioms are independent. Remark 1. The structure introduced in the above Definition 2.1 resembles, in a weak way, the definition of BZ poset since the following condition, involving the second complementation and characterizing this kind of structure, in general is not required to hold: (B-1)
a a tt.
Another type of weakness refers to the interconnection rules. Indeed, in a BZ poset conditions (win-0)(win-2) are replaced by the following stronger condition of interconnection: (in)
a t$=a tt [which under (B-1) implies (win-0), (win-1), and (win-2)].
In the following definition we formalize the standard definition of BZ poset structure which, owing to the previous remark, is in particular a pre-BZ poset (any BZ poset is a pre-BZ poset too). Definition 2.2. A BZ poset is a structure ( 7, , $, t, 0) consisting of a Kleene poset ( 7, , $, 0), equipped with a Brouwer complementation, i.e., a unary mapping t: 7 [ 7 satisfying the conditions, whatever be a, b # 7: (B-1)
a a tt
(B-2) (B-3)
a b implies b t a t a 7 a t =0
The two unusual complementations satisfy the following interconnection rule, whatever be a # 7: (in)
a t$=a tt
Example 2.1. Let M4 be the 4-element Boolean algebra. Let us define as follows: 0 t =1, 1 t =0, a t =a$, a$ t =0 (see Fig. 1). It turns out that M4 , equipped with t is a pre-BZ lattice which is not a BZ lattice since a 3 0=a t t and a tt =0{a=a t$. M4 is the smallest pre-BZ poset which is not a BZ poset. t
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Fig. 1.
M4 .
Differently from BZ posets, in a pre-BZ lattice 7 it may happen that there is an element a # 7 s.t. a 3 a tt and a tt 3 a. Example 2.2. In the pre-BZ lattice 7 8 of Fig. 2 We have that b 3 c=b tt and b tt =c 3 b. 7 8 is the smallest pre-BZ poset having this property. In Subsection 3.1 we will present another counterexample to this property, based on topological spaces. As pointed out in Remark 1, and comparing Definitions 2.1 and 2.2, both pre-BZ posets and BZ posets are based on a Kleene poset equipped with another unary map which in the latter case is a ``true'' Brouwerian complementation. We will make use in the sequel of the following definition. Definition 2.3. A Brouwer poset is a poset ( 7, , t, 0, 1) bounded by the least element 0 and the greatest element 1 and equipped with a
Fig. 2.
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Pre-BZ and Degenerate BZ Posets
Brouwer complementation, i.e., a unary map (B-1), (B-2), and (B-3).
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: 7 [ 7 satisfying conditions
In any pre-BZ poset (and so also in any BZ poset) another unary operation can be introduced, which in the case of BZ poset plays the role of anti-Brouwer complement. Proposition 2.1. In any pre-BZ poset the map : 7 [ 7 associating to every element a # 7 the element a :=a$ t$ satisfies the following conditions whatever be a, b # 7: (preAB-2)
a b implies b a
(preAB-3)
a 6 a =1
In the BZ poset case this is an anti-Brouwer complementation, i.e., the further condition holds: (AB-1)
a a
In the next subsections we briefly describe the two most interesting examples of BZ poset structure: generalized characteristic functionals as fuzzy sets based on a universe X and generalized projections as effect operators defined on a Hilbert space H. 2.1. BZ Distributive Lattice of Fuzzy Sets on a Universe A generalized characteristic functional or fuzzy set on the universe X is any mapping f : X [ [0, 1]; for any such fuzzy set one can single out some peculiar subsets of the universe X: (1)
the certainly-yes (also the necessity) domain: A 1( f ) :=[x # X : f (x)=1]=Ker(1& f )
(2)
the certainly-no (also the impossibility) domain: A 0( f ) :=[x # X : f (x)=0]=Ker( f )
(3)
the possibility domain: A p( f ) :=[x # X : f (x){0]=(A 0( f )) c
As particular fuzzy sets we have characteristic functionals / A of subsets A of X [defined as / A(x)=1 iff x # A, and =0 otherwise]. The identical zero
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map 0 :=/ < and the identical one map 1 :=/ X are fuzzy sets of this kind. The family [0, 1] X of all fuzzy sets is a BZ (distributive, complete, bounded by 0 and 1 ) lattice ( [0, 1] X, 7 , 6 , $,
t
, 0, 1 )
with respect to: v the lattice meet ( f 1 7 f 2 )(x) :=min[ f 1(x), f 2(x)] v the lattice join ( f 1 6 f 2 )(x) :=max[ f 1(x), f 2(x)] and the two unusual complementations: v the Kleene complement: f $ :=(1& f ) v the Brouwer complement: f t :=/ A0( f ) Note that the partial ordering induced on [0, 1] X by the above lattice operations is the pointwise one. For f, g # [0, 1] X, f g
iff
\x # X,
f (x) g(x)
(2.1)
The BZ lattice of all fuzzy sets [0, 1] X satisfies the ``dual'' de Morgan laws for the Brouwerian orthocomplementation: (dM-2B) f t 6 g t = ( f 7 g) t [this property in general is not required to hold in BZ lattices]. Moreover, there exists the fuzzy set ``\x # X, 12(x) :=12,'' called the half fuzzy set. Clearly, 12=(12)$. The noncontradiction law ``\f, f 7 f $=0,'' and the excluded middle law ``\f, f 6 f $=1,'' in general do not hold for the Kleene complementation: for instance, one has (12) 7 (12)$=(12) 6 (12)$=(12){0, 1. complementation satisfies the weak double negation The Brouwer principle (B-1), which algebraically expresses the fact that ``if f, then impossible-impossible f '' [i.e., f f tt ], but in general the vice versa does not hold: in particular, (12) t =0 leads to (12) 12 &.& 2
and
(10.1)
2. The set of all special effects is not empty since it (strictly) contains all orthogonal projections: 6(H)/Ese(H)
(10.2)
3. Ese(H), equipped with the restriction of the partial ordering and the complementations $ and t, determines a BZ poset in which both the excluded middle principle (\F # Ese(H), F 6 F$=I) and the noncontradiction principle (\F # Ese(H), F 7 F$=O) for the Kleene complementation hold. Let EKse(H) be the class of all K-sharp elements of Ese(H), i.e., the class of all special effects for which the excluded middle and the noncontradiction laws for the Kleene complementation hold. Let EBse(H) be the class of all B-sharp elements, i.e., the class of all special effects for which the strong double negation law for the Brouwer complementation holds. It turns out that: 6(H)=EBse(H)/EKse(H)=Ese(H) The set Ese(H) shows a particular ``pathological'' behavior with respect to the standard effect algebra structure of E(H). (P 1 )
Ese(H) is not closed with respect to the restriction of the partial sum operation Ä defined in E(H). To be precise, there exist two special effects F, G # Ese(H) such that their sum is an effect, F+G # E(H), but not a special effect.
Example 10.1. Let us consider the two dimensional Hilbert space C 2. Then the two effect operators
\
23 0
0 0
+
and
\
0 0
are special, but their sum (23) I is not special.
0 23
+
Pre-BZ and Degenerate BZ Posets
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In the three dimensional Hilbert space C 3 the two effect operators
\
0
0
0
F= 0 25 0 0 0 34
+
G=
and
\
34
0
0 0
0
25 0 0 0
+
are special effects since ( (0, 1, 0) | F(0, 1, 0)) =2512, and ( (1, 0, 0) | G(1, 0, 0)) =34>12 and ( (0, 1, 0) | F(0, 1, 0)) =25(12) &x& 2 ]. It is possible to avoid this bad behaviour by restricting the definition domain of the partial sum operation Ä. Let = se :=[(F, G) # Ese(H)_ Ese(H) : F+G # Ese(H)]. Then, the structure ( Ese(H), Ä se , $, O, I), where Äse is the usual sum of linear operator restricted to the set = se and $ is the standard orthocomplementation F $=I&F, is an effect algebra, (16, 13, 14) whose induced Kleene poset ( Ese(H), se , $, O, I) is based on the partial order relation: \F, G # Ese(H),
F se G
iff
_H # Ese(H) : F Äse H=G
(10.3)
Now we have the further pathological behavior: (P 2 )
The partial order (10.3) does not coincide with the restriction to Ese(H) of the natural partial order on E(H); we can only state that: \F, G # Ese(H),
F se G O F G
(10.4)
But in general the vice versa is not true. Example 10.2. On the two dimensional Hilbert space C 2, let us consider the two special effect operators G=
14 0
\ 0 1+
and
F=
0
0
\0 34+
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Then, we can state that F G since there exists the effect operator (G F ) :=
14
\0
0 14
+
such that FÄ (G F )=G, but (G F )=(14) I is not special. (P 3 )
The Kleene poset of all special effects is closed with respect to the mapping t: Ese(H) [ 6(H), associating with any special effect F the projection (and so the special effect) F t =P Ker(F ) [since all orthogonal projections are special effects: 6(H)/ Ese(H)]. However, the mapping t is no more a Brouwer complement with respect to the partial order (10.3), since in general it is not true that \F # Ese(H), F se F tt.
Example 10.3. Let P # 6(H) be a nontrivial projection and let : # (12, 1). (:= 12 cannot be considered, since from P I it follows that 1 1 1 2 P I& 2 P=( 2 P)$, i.e., non special). Let us consider the effect F=:P, 12
