ONT Re: Relations And Their Divisitudes
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RATD. Note 10
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JA = Jon Awbrey
TJ = Tom Johnston
I think I see another place where the translation
indicated by the Table might be losing something.
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| Table Talk in Translation, Presented as a Two-Column Table |
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| JA: | TJ: |
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| Standard Math Language, | Tom's translation into the language |
| A La Descartes | of relational databases |
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| k-adic relation | table with k columns |
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| k-tuple | row of a table with k columns |
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| relation(s) | table(s) in a relational database |
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| 1-adic projection | result of a relational PROJECT |
| | operation on a table, that |
| | leaves just one column |
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| 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. |
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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.
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RATD. Note 11
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Let us see if we can bring some visual intuitions --
the most fallible but still strangely useful kind --
into play. After all, this is where all that biz
about projections first came into our table talk.
Here is a picture of the initial corner of the 3-dimensional space
X = X_1 x X_2 x X_3 = !A!^3, where !A! = {a, b, c, ..., x, y, z},
as before. Here I have plotted the point p = <a, b, c> in X and
further illustrated its projections on the three planes in view.
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| |
| X_3 |
| |
| c |
| /|\ |
| / | \ |
| <a,c> . b \ |
| /|\ | \ |
| / | \| \ |
| / | \ . <b,c> |
| / | |\ /|\ |
| / | | \ / | \ |
| * = <a,b,c> o | o * | o |
| | | / \ | | | |
| | |/ \| | | |
| | a | | | |
| | / \ |\ | | |
| | / \ | \| | |
| | b \ | b | |
| | / \ | / \ | |
| |/ \|/ \| |
| c . c |
| X_1 \ <a,b> / X_2 |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| \ / |
| o |
| |
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Figure 1. Plane Projections of the Point <a, b, c>
Reading off the Figure, we have this picture of the data:
With respect to the 1-adic projections of p = <a, b, c>:
p_1 (p) = p_1 <a, b, c> = a in X_1
p_2 (p) = p_2 <a, b, c> = b in X_2
p_3 (p) = p_3 <a, b, c> = c in X_3
With respect to the 2-adic projections of p = <a, b, c>:
p_12 (p) = p_12 <a, b, c> = <a, b> in X_1 x X_2
p_13 (p) = p_13 <a, b, c> = <a, c> in X_1 x X_3
p_23 (p) = p_23 <a, b, c> = <b, c> in X_2 x X_3
To finish up one of the questions about determination,
we can say that p = <a, b, c> is uniquely determined by
its 1-adic projections, simply because all that we really
mean by <a, b, c> is a thing that is uniquely determined by
its possessing the 1-adic projections a, b, c, respectively.
With this in mind, we have gained an edge on understanding
all that abstract nonsense in mathematical category theory
about "universal properties of diagrams". This is nothing
but a way of formalizing to the max the following insight:
If all we mean by "<a, b, c>" is a thing x that is uniquely
determined, within the proper context of discussion, by the
data p_1 (x) = a, p_2 (x) = b, p_3 (x) = c, then anything
else that is uniquely determined, in that same context, by
that same data, whether we call it by any other word, say,
"ALPHABETSOUPSPOONFULL0123", or anything else, is really
nothing other than <a, b, c>, in no matter what strange
disguise some may fancy it for a particular occasion.
So let's watch out for that.
Jon Awbrey
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