SUO: Re: Relations And Their Divisitudes
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
RATD. Note 10
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
Tom,
I think I see another place where the translation
indicated in the Table might be losing something.
o---------------------------------------------------------------------o
| Table Talk in Translation, Presented as a Two-Column Table |
o-----------------------------o---------------------------------------o
| Standard Math Language, | Tom's translation into the language |
| A La Descartes | of relational databases |
o-----------------------------o---------------------------------------o
| k-adic relation | table with k columns |
o-----------------------------o---------------------------------------o
| k-tuple | row of a table with k columns |
o-----------------------------o---------------------------------------o
| relation(s) | table(s) in a relational database |
o-----------------------------o---------------------------------------o
| 1-adic projection | result of a relational PROJECT |
| | operation on a table, that |
| | leaves just one column |
o-----------------------------o---------------------------------------o
| a k-tuple is defined | a row of a table with k columns |
| to be determined by its | is defined to be determined by the |
| 1-adic projection data, | (ordered) set of relational PROJECT |
| but a k-adic relation in | operations, each of which results in |
| general is not determined | a single column instance of the table,|
| by its m-adic projection | but a table with k columns in general |
| data for any m < k. | is not determined by the (ordered) set|
| | of m PROJECT operations on that table,|
| | for any m > k. |
o-----------------------------o---------------------------------------o
In order to check this out, I will make up some really simple examples.
Start with a small domain, for instance, !A! = {a, b, c, ..., x, y, z},
and consider the 3-adic relations of the form L c X = X_1 x X_2 x X_3,
where X_1 = X_2 = X_3 = !A!.
For example, let L be the relation with the following ordered triples:
<a, b, c>
<a, b, d>
<a, c, d>
Then we have the following data:
For the 1-adic projections p_i : X -> X_i,
the induced projections p_i (L) are these:
p_1 (L) = {a}
p_2 (L) = {b, c}
p_3 (L) = {c, d}
For the 2-adic projections p_ij : X -> X_i x X_j,
the induced projections p_ij (L) are as follows:
p_12 (L) = {<a, b>, <a, c>}
p_13 (L) = {<a, c>, <a, d>}
p_23 (L) = {<b, c>, <b, d>, <c, d>}
What should be clear at this point is that an m-adic projection
is not the same thing as an ordered m-set of 1-adic projections.
The main thing about the m-adic projection of a k-adic relation,
for 1 =< m =< k, is that it always yields an m-adic relation as
the result.
Jon Awbrey
o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o