ONT Re: Relations And Their Divisitudes
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RATD. Note 9
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JA = Jon Awbrey
TJ = Tom Johnston
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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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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:
| 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 v
| o<--------o-------->o
| X p X x Y q 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.
|
| Mac Lane, 'Cat Work Math', p. 1.
|
| Saunders Mac Lane,
|'Categories for the Working Mathematician',
| 2nd edition, Springer, New York, NY, 1997.
|
| http://suo.ieee.org/ontology/msg04790.html
Jon Awbrey
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