# Re: What is an automorphism of a database instance?

Date: Fri, 28 Dec 2007 01:13:34 -0500

Message-ID: <2007122801133475249-kirakun_at_earthlinknet>

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?

-- -kiraReceived on Fri Dec 28 2007 - 07:13:34 CET