SUO: Re: Relations And Their Divisitudes
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RATD. Note 9
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TJ = Tom Johnston
Tom,
Continuing with your meta-table and other questions.
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| Metaboly |
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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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TJ: One problem is "determined by", appearing (twice) in Jon's
statement and in my paraphrase. What does it mean?
I almost always speak of "determination" in the same sense that
one says "two points determine a line". In the first instance
above, where a k-tuple is defined as being determined by its
1-adic projections, I am referring specifically to the usual
definition of cartesian products in category theory:
| Introduction (cont.)
|
| Many properties of mathematical constructions may
| be represented by universal properties of diagrams.
| Consider the cartesian product X x Y of two sets,
| consisting as usual of all ordered pairs <x, y>
| of elements x in X and y in Y. The projections
| <x, y> ~> x, <x, y> ~> y of the product on its
| "axes" X and Y are functions p : X x Y -> X,
| q : X x Y -> Y. Any function h : W -> X x Y
| from a third set W is uniquely determined by
| its composites p o h and q o h. Conversely,
| given W and two functions f and g as in the
| diagram below, there is a unique function h
| which makes the diagram commute; namely,
| h w = <f w, g w> for each w in W.
|
| W
| o
| /|\
| / | \
| f / | \ g
| / | \
| v | v
| o<----o---->o
| X X x Y Y
|
| Thus, given X and Y, <p, q> is "universal" among pairs of
| functions from some set to X and Y, because any other such
| pair <f, g> factors uniquely (via h) through the pair <p, q>.
| This property describes the cartesian product X x Y uniquely
| (up to a bijection); the same diagram, read in the category
| of topological spaces or of groups, describes uniquely the
| cartesian product of spaces or the direct product of groups.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
|
| http://suo.ieee.org/ontology/msg04466.html
Have to break here ---
Jon Awbrey
TJ: Second problem is the purported demonstration of the claim.
I just don't follow it.
TJ: My own thoughts about the difference between a row of
a table (a tuple) and the table itself (a relation) is
that the former is part of the extension of the relation,
while the latter (more specifically, the set membership
conditions which define it) represents the intension of
the relation. Next, that one cannot always infer the
intensional rules from the extensional instances because,
at any given moment, the set of all those instances may
not define the boundary conditions of all those rules.
To take a simple example, if one column of the table is
a status-code column, whose domain is the letters A - X
inclusive, it might be that the status-code value of P
is not instantiated in any tuple. Or: a column might be
defined as nullable, although all of the current rows
have a value in that column. In short: that intensional
rules might be inferrable from the full Cartesian Product
of a set of columns (if we knew it was the full Cartesian
Product), but are not inferrable from any subset of the
full Cartesian Product.
TJ: So Jon, can you re-cast your points demonstrating that
last claim of yours listed above, in my language of
working databases? If not, can you give it another
try in your own language?
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