Re: What is an automorphism of a database instance?

On 2007-12-28 00:15:49 -0500, Kira Yamato <kirakun_at_earthlink.net> said:

> I need help in understanding what is an automorphism of a database instance.
>
> The following definition is from the book Relational Database Theory by
> Atzeni and De Antonellis:
>
> Definition: An automorphism of a database instance r is a partial function
> h : D --> D
> where D is the domain of the database r such that
> 1) the partial function h is a permutation of the active domain D_r, and
> 2) when we extend its definition to tuples, relations, and database
> instances, we obtain a function on instances that is the identity on r,
> namely
> h(r) = r.
>
> I can understand 1), but I cannot understand 2).
>
> In mathematics, an automorphism is a 1-1 mapping that preserves the
> structure of an underlying set. For example, if in some set whose
> members x, y and z obeys
> z = x + y,
> then we expect an automorphism f on that set to also obey
> f(z) = f(x) + f(y).
> So, the structure of "addition" is preserved.
>
> Now, back to relational database theory, what "structure" is being
> preserved by 2)? Can someone explain the formalization in 2) more
> carefully?

On a different but related question, is there a notation of *isomorphism* between two database instances?

In another word, is there a way to formalize the notation that two databases are essentially containing the same information, except for a difference in the labeling of the attribute names and domain-value names?

-- 

-kira