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**Unformatted text preview: **th signature h 1 ; : : : ; n+1i is a function
: O( 1 )
O( n) ! O( n+1), where 1 ; : : : ; n+1 are any schemes. Similarly,
an n-ary operator on paraconsistent relations with signature h 1 ; : : : ; n+1i is a function
: P ( 1)
P ( n ) ! P ( n+1 ). De nition 4 An operator on paraconsistent relations with signature h 1; : : : ;
totality preserving if for any total relations R1 ; : : : ; Rn on schemes 1 ; : : : ;
(R1; : : : ; Rn) is also total. 2 4 n+1 i is n , respectively, De nition 5 A totality preserving operator on paraconsistent relations with signature
h 1 ; : : : ; n+1 i is a weak generalisation of an operator on ordinary relations with the
same signature, if for any total relations R1 ; : : : ; Rn on schemes 1 ; : : : ; n, respectively,
we have
(Rn)): 2
+1 ( (R1 ; : : : ; Rn )) = ( 1 (R1 ); : : : ;
n n The above de nition essentially requires to coincide with on total relations (which are
in one-one correspondence with the ordinary relations). In general, there may be many
operators on paraconsistent relations that are weak generalisations of a given operator
on ordinary relations. The behavior of the weak generalisations of on even just
the consistent relations may in general vary. We require a stronger notion of operator
generalisation under which, at least when restricted to consistent relations, the behavior
of all the generalised operators is the same. Before we can develop such a notion, we need
that of `completions' of a paraconsistent relation.
We associate with a consistent relation R the set of all (ordinary relations corresponding to) total relations obtainable from R by throwing in the missing tuples. Let the map
comps : C ( ) ! 2O( ) be given by
comps (R) = fQ 2 O( ) j R+ Q ( ) ? R?g:
The set comps (R) contains all ordinary relations that are `completions' of the consistent
relation R. Observe that comps is de ned only for consistent relations and produces
sets of ordinary relations. The following observation is immediate. Proposition 1 For any consistent relation R on scheme , comps (R) is the singleton
set fR+ g i R is total.
Figure 1 gives a pictorial view of the di erent kinds of relations and the maps and
comps .
We now need to extend operators on ordinary relations to sets of ordinary relations.
For any operator : O( 1 )
O( n) ! O( n+1 ) on ordinary relations, we let
O( 1 )
O( ) ! 2O( +1 ) be a map on sets of ordinary relations de ned as
S( ) : 2
2
follows. For any sets M1 ; : : : ; Mn of ordinary relations on schemes 1 ; : : : ; n, respectively,
S ( )(M1 ; : : : ; Mn) = f (R1; : : : ; Rn) j Ri 2 Mi ; for all i; 1 i ng:
In other words, S ( )(M1 ; : : : ; Mn) is the set of -images of all tuples in the cartesian
product M1
Mn. We are now ready to lead up to a stronger notion of operator
generalisation.
n n De nition 6 An operator on paraconsistent relations with signature h 1 ; : : : ;
is consistency preserving if for any consistent relations R1 ; : : : ; Rn on schemes
respectively, (R1; : : : ; Rn) is also consistent. 2
5 n+1 i...

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