Matthew West wrote:
> Dear Mitch
>
>>  Any (relational) database schema describes a set of relations/an
>> ontology and most are extremely multiary.
>
> [MW] Yes but these are mostly denormalisations to overcome the costs of
> table joins. (01)
The whole aspect of arity of tables is bound up with the theory of
"normal forms". Among other things, a table of 5 columns might still
only represent a binary relation, and in "4th normal form" that is what
it represents. The problem is how many columns it takes to provide the
key values that identify each of the two participating individuals. So
4th normal form requires a table of n columns to express a function F of
the form:
F(c\1, ..., c\k) = (c\k+1, ..., c\n) (02)
So the table representing the total price of each line item of a
purchase order may have four columns (order#, line#, amount, currency)
where the function "total price" is:
total_price (order#, line#) = (amount, currency) [4th NF]
or
(totalPrice lineItem monetaryAmount) [binary relation] (03)
It is only in "5th normal form" that all unary relations are onecolumn
tables and all binary relations are twocolumn tables. And this is
accomplished by creating artificial "identifiers" that correspond to
"covariant vectors"  the "compound keys" for the individuals. (04)
The highly prized "3rd normal form" has the property Matthew observes:
For each column c\i where i > k,
c\i = F\i(c\1, ..., c\k)
i.e., each data column is a function of _all_ the key columns, but there
is no requirement for the data columns to have any other relationship to
each other. All the functions are different. Thus the 3rd normal form
table is the "natural join" of a set of "4th normal form" tables. (05)
Relational algebra is just yet another mathematical theory that has some
relationships to knowledge engineering. (06)
Ed (07)

Edward J. Barkmeyer Email: edbark@xxxxxxxx
National Institute of Standards & Technology
Manufacturing Systems Integration Division
100 Bureau Drive, Stop 8263 Tel: +1 3019753528
Gaithersburg, MD 208998263 FAX: +1 3019754694 (08)
"The opinions expressed above do not reflect consensus of NIST,
and have not been reviewed by any Government authority." (09)
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