Re: What is an automorphism of a database instance?
From: Kira Yamato <kirakun_at_earthlink.net>
Date: Sat, 29 Dec 2007 01:04:57 -0500
Message-ID: <2007122901045716807-kirakun_at_earthlinknet>
>> 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?
Date: Sat, 29 Dec 2007 01:04:57 -0500
Message-ID: <2007122901045716807-kirakun_at_earthlinknet>
On 2007-12-28 13:07:54 -0500, Tegiri Nenashi <TegiriNenashi_at_gmail.com> said:
> On Dec 27, 9:15 pm, Kira Yamato <kira..._at_earthlink.net> wrote:
>> 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?
> > Well, in mathematics you rarely find an algebra with 7 (or 8?) > operations. Moreover, the operations are syntactically unattractive. > The elements of the algebra are relations, and yet > some operations like projection and selection take an additional > parameter, which is outside of the realm of > the relation objects. Some operations like union can't be applied to > any pair of relations. The explicit > renaming operation is like nothing else in mathematics, where renaming > variables has never been a big deal. > > If this line of thought resonates with you, please check up > http://arxiv.org/ftp/cs/papers/0603/0603044.pdf > There are 2 homomorhisms of relational algebra into boolean algebras > there.
Thank you for that link. It is certainly one direction to look into in defining morphisms between databases.
-- -kiraReceived on Sat Dec 29 2007 - 07:04:57 CET