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

Date: Wed, 9 Jan 2008 03:29:36 -0500

```> On 28 dec 2007, 06:15, Kira Yamato <kira..._at_earthlink.net> wrote:
```

>> I need help in understanding what is an automorphism of a database instanc
```> e.
```

>>
>> 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?
```>
> I only just saw your posting so I wondered if you still needed help
> with this.

```

Thanks for the follow-up. The notion is still somewhat ambiguous in my mind. I sort of feel where I want to end up, but it is somewhat difficult to formulate it in rigorous formalism.

What I want to formalize is the notion that two databases are "essentially" containing the "same information" modulo a difference in labelings of the names of the relations/attributes/values.

```--

-kira
```
Received on Wed Jan 09 2008 - 09:29:36 CET

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