Relations with signature h 1 n1i is a function

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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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This note was uploaded on 09/28/2013 for the course CSC 8710 taught by Professor Staff during the Fall '08 term at Georgia State University, Atlanta.

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