What is an automorphism of a database instance?

#61

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

Quote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 inmy
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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Quote:

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

#62

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#63

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#64

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#65

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#66

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#67

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#68

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#69

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

Quote:

On Jan 9, 11:13*am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29*am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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 partialfunction

* * * * 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 identityon 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

-- Jan Hidders

#70

On Jan 9, 1:15 pm, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

Quote:

On 9 jan, 20:57, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 11:13 am, Jan Hidders <hidd... (AT) gmail (DOT) com> wrote:

On 9 jan, 19:10, Tegiri Nenashi <TegiriNena... (AT) gmail (DOT) com> wrote:

On Jan 9, 12:29 am, Kira Yamato <kira... (AT) earthlink (DOT) net> wrote:

On 2008-01-08 09:45:19 -0500, Jan Hidders <hidd... (AT) gmail (DOT) com> said:

On 28 dec 2007, 06:15, Kira Yamato <kira... (AT) earthlink (DOT) 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.

The difficulty is in formalizing the term "essentially" and "same information."

I suggest defining automorphism of database instance (where "database
instance" is understood to be a set of relations) algebraically as a
mapping f such that for any relations Q and R

f(Q /\ R ) = f(Q) /\ f(R)
f(Q \/ R ) = f(Q) \/ f(R)

this is general enough to cover both domain value permutations and
column/relation renamings.

How do you know that it is not too general?

I don't:-(

Btw. didn't you mean "homomorphism" rather than "automorphism"?

Automorphism is a homomorphism of a database instanse into itself,
isn't it?

It's usually defined as a kind of isomorphism.

OK, this amounts to additional requirement that f is bijective and
f^-1 is homomorphism as well. Alternatively, given that join and union
define order relation, one may borrow the order isomorphism
definition
http://en.wikipedia.org/wiki/Order_isomorphism
verbatim.