An Extended Relational Model & SQL for Fuzzy Multidatabases - Advanced Database Query Systems - page 209

is m-ary), op f is an operation that returns an empty relation for any given input export fuzzy relation
op G = ( ) n.

where G i = merge ( L 1 i ,…, L ni )(i.e. global fuzzy relation G i is derived by combining export fuzzy relations
L 1 i ,…, L ni from DB 1 ,…, DB n respectively). We assume that for DB k that has no export fuzzy relation for
deriving G i will have L ki = f , and opMap op

( ,  , G ) merge op L ( ,  , L ),  , op L ( ,  , L )

G 1 m L 11 n 1 L 1 m nm

1 m

 . A set of one to one operation mapping
function from global fuzzy relational operations to export fuzzy relational operations is shown below:

( ) op , , op

=

G L L m

1

fs

DB ∈ DB

fs σ

opMap ( ) = op

1 

, , op where

p α β , i set

σ

p α ∧ ∈ ( s DB ), β n

op otherwise .

set

φ

fs fs

p

fs
p

opMap
opMap

( ) = , ...,

σ

σ σ

p α β , α β , α β ,

fs
A

f s

A

fs
A

(

sameDB

) , ...,

=

π π π

, ,

α β , α β , , α β ,

fs fs f s

sameDB

opMap ( ) = , ...,

DD DD

DD

p α β , p α β , p α β ,

fs sameDB fs fs

opMap
opMap

( ) , ...,

=

∪ ∪

∪

α β , α β , α β ,

fs sam fs fs

eDB

( ) =

=

, ...,

∩ ∩

∩

α β , α β , α β ,

fs sameDB fs fs

opMap ( ) , ...,

− − −

α β , α β , α , β

All these mapping functions shall be used to prove the consistency of the FTS relational operations
with the sameDB option as given below:

Proofs of Consistent Global Fuzzy Operations

The following two lemmas show that global FTS selection is consistent with or without source predicates.

Lemma 1 1 .

∀ L ,  , L G , = merge L ( ,  , L ),

1 j nj j 1 j nj

fs

) = merge op L (

1 1 

, , op L

σ

p s DB set

α ∧ ∈ ( ), β j n nj )

fs

DB ∈ DB

σ α β

p , i set

where

op otherwise .

φ

fs

Proof: g ∈ Gj ⇔

σ

p α ∧ ( . g s DBset ∈ ), β

g Gj ∈ ∧ p g ( ) ∧ µ ( )  g β ∧ ∃ DB ∈ DBs et g s DB , . = ⇔

α Gj i i

f

DB DBset g A , . L whereA Attr ( G ) Attr ( L ) for i 1 , ..., n

∃ ∈ ∈ = = − ⇔ σ αβ

i p ij j i j

g merge op L ∈ ( , ..., op L ).

1 1 j n nj

f

DB ∈ DBset

σ αβ

where op

p i

i

op

φ otherwise.

In the above proof, the source predicate constrains the result of global fuzzy selection to contain
only tuples with DB i as source values. This implies that the tuple attributes come from an export fuzzy
relation in DB i . Hence a global fuzzy selection operation produces result identical to that produced by