Definition of DB Category - Big Data Integration Theory - page 113

Remark In order to simplify the presentation, we extend the structure-operators also

to unary cases: for example, 1 ≤ i ≤ 1 A i = (A 1 ) A 1 , 1 ≤ i ≤ 1 f i = (f 1 )

= = f 1 .

We extend the application of also to (unordered) sets, so that for a given

nonempty set (of sets) S , S represents an application of to any ordered list

(tuple) of sets in S . It will be demonstrated that is a commutative and associative
binary operator (up to isomorphism) so that, given any ordered list (a permutation)
(s 1 ,...,s n ) of all elements (which are the sets) in S , the following isomorphism is
valid in DB :
S (s 1 ,...,s n )

(i,a)

s i .

= (s 1 ··· s n ) = | a ∈

1 ≤ i ≤ n

Thus, for example, we have the following cases for a composition of complex mor-
phisms obtained by using n -ary separation-compositions:

1. For 1 ≤ i ≤ n f i : 1 ≤ i ≤ n A i → 1 ≤ i ≤ n B i and 1 ≤ i ≤ n g i : 1 ≤ i ≤ n B i →

1 ≤ i ≤ n C i ,

g i f i ◦ = (g i ◦ f i ) : A i → C i ;

1 ≤ i ≤ n 1 ≤ i ≤ n 1 ≤ i ≤ n 1 ≤ i ≤ n 1 ≤ i ≤ n

→ 1 ≤ i ≤ n B i and 1 ≤ i ≤ n g i : 1 ≤ i ≤ n B i → 1 ≤ i ≤ n C i , 2. For f 1 ,...,f n : A

g i ◦ f 1 ,...,f n = g 1 ◦ f 1 ,...,g n ◦ f n : A → C i ;

1 ≤ i ≤ n 1 ≤ i ≤ n

3. For 1 ≤ i ≤ n f i : 1 ≤ i ≤ n A i → 1 ≤ i ≤ n B i and g 1 ,...,g n ]: 1 ≤ i ≤ n B i →

[ C ,

f i

[ g 1 ,...,g n ]◦ =[ g 1 ◦ f 1 ,...,g n ◦ f n ]: A i → C ;

1 ≤ i ≤ n 1 ≤ i ≤ n

→ 1 ≤ i ≤ n B i and g 1 ,...,g n ]: 1 ≤ i ≤ n B i →

4. For f 1 ,...,f n : A [ C ,

1

[ g 1 ,...,g n ]◦ f 1 ,...,f n = S : A → C if | S |≥ 1 ;⊥ : A → C otherwise

f i | g i ◦ g i ∩ f i =⊥

0 , 1

(where S ={ g i ◦ f i = ≤ i ≤ n } );

→ 1 ≤ i ≤ n C i ,

5. For f : A → B and g 1 ,...,g n : B

g 1 ,...,g n ◦ f = g 1 ◦ f,...,g n ◦ f : A → C i ;

1 ≤ i ≤ n

f 1 ,...,f n ]: 1 ≤ i ≤ n A i →

6. For [ B and g : B → C ,

g ◦[ f 1 ,...,f n ]=[ g ◦ f 1 ,...,g ◦ f n ]: A i → C ;

1 ≤ i ≤ n