Definition of DB Category - Big Data Integration Theory - page 112

1. For k : A 1 → B 1 ,l : A 2 → B 2 ,f : B 1 → C 1 and g : B 2 → C 2 ,

(f g) ◦ (k l) = (f ◦ k) (g ◦ l) : A 1 A 2 → C 1 C 2 ;

2. For k : A → B 1 ,l : A → B 2 ,f : B 1 → C 1 and g : B 2 → C 2 ,

(f g) ◦ k,l = f ◦ k,g ◦ l : A → C 1 C 2 ;

3. For k : A 1 → B 1 ,l : A 2 → B 2 ,f : B 1 → C and g : B 2 → C ,

[ f,g ]◦ (k l) =[ f ◦ k,g ◦ l ]: A 1 A 2 → C ;

4. For k : A → B 1 ,l : A → B 2 ,f : B 1 → C and g : B 2 → C , [ f,g ]◦ k,l = f ◦

1 ,g 1 ); g 1 ,g 1 ); f

◦ ◦ =⊥ ◦ =⊥ ◦ ◦ =⊥ ◦ =⊥ ◦

k,g l (if f k l l (if f k l k (if

1 ,g 1 ); 1

f ◦ k =⊥ ◦ l =⊥ ⊥ otherwise (an arrow from A to C );

5. For k : A 1 → B,f : B → C 1 and g : B → C 2

f,g ◦ k = f ◦ k,g ◦ k : A 1 → C 1 C 2 ;

6. For k : A 1 → B,l : A 2 → B and f : B → C ,

f ◦[ k,l ]=[ f ◦ k,g ◦ k ]: A 1 A 2 → C ;

7. For k : A → B,l : A → B , f : B → C 1 , and g : B → C 2

= : f,g ◦ k,l f ◦ k,f ◦ l , g ◦ k,g ◦ l A → C 1 C 1 ;

:

A 1 →

:

A 2 →

: → : →

8. For k B,l B,f B C and g B C ,

]= :

f,g ◦[ k,l f ◦ k,g ◦ k , f ◦ l,g ◦ l A 1 A 2 → C ;

9. For k : A → B,l : A → B,f : B → C and g : B → C ,

f,g ◦ k,l = f ◦ k,g ◦ k,f ◦ l,g ◦ l : A → C.

Now we can extend these binary structural-operators in Definition 20 to n -ary oper-
ators:

Definition 21 We generalize the binary structure-operators in Definition 20 into
n -ary operators, for any n 2, by:

1. 1 ≤ i ≤ n f i = (f 1 ,...,f n )

≥

: 1 ≤ i ≤ n A i → 1 ≤ i ≤ n B i where f i : A i → B i for

each 1 ≤ i ≤ n ;

f 1 ,...,f n ]: 1 ≤ i ≤ n A i → 2. [ B , where f i : A i → B for each 1 ≤ i ≤ n ;

→ 1 ≤ i ≤ n B i , where f i :

3. f 1 ,...,f n : A A → B i for each 1 ≤ i ≤ n ;

f 1 ,...,f n : A → B , where f i =⊥

0

4. for each 1 ≤ i ≤ n .