# Re: Can relvars be dissymetrically decomposed? (vadim and x insight demanded on that subject)

Date: Sun, 09 Jul 2006 05:20:14 GMT

Cimode wrote:
> As far as I could observe, relational variables are repetitively
> confused with their projections as tables. On the last few years, I
> have focused some efforts into searching mathematical tools to help
> characterize more precisely relvars.
>
> On such perspective, I have found ensemblist mathematics useful.

Can you provide some pointers to 'ensemblist mathematics'? Google didn't seem to help.

> Once defined according to predicate theory requirements and RM, relvar
> have the characteristics of constituing themselves new domain of
> arbitrarily complex values. For instance, relvar R1{A1, A2} draws and
> restricts values from domains Da1 and Da2 (for attributes A1 and A2).
> As soon as defined, all occurrences of relvar R1 populate a domain of
> value DoR1. (Such domain may be utilized for instance to define a new
> data type DaR1).

'value DoR1'? Or of 'type DoR1' or 'domain DoR1'?

> The question that arose from that observation is whether the domain
> DoR1 is equivalent to the ensemble consitued by the some adjonction of

constituted? And 'adjunction'? (Are these typos, or some new words?)

> all attribute domains D1, D2. So far, assuming they are different
> indeed has some advantage: it allows differentiation of domains which
> simplifies deduction about relvar characterization. As a consequence,
> I expressed the relvar R1 as equal to the intersection between the
> domain of value it would constitute, and the ensemble consituted by the
> attributes domains.
>
> In Zermelo-Fraenkel approach, this would be expressed as

Googling 'Zermelo-Fraenkel' picks up reasonably comprehensible definitions at Wikipedia and Wolfram Mathworld.

> Assuming DoR1 there is an ensemble of values constituted by R1 occuring
> values, there exists an ensemble of parties B(DoR1) that represents
> their attributes as element of the DoR1 group.
>
> Considering B(DoR1) as an ensemble of values different from DoR1, R1
> could be defined by the intersect of the 2 ensembles.
>
> Stated in math language...
>
> R1 = DoR1 INTERSECT B(DoR1)
>
> One observation that arises from this axiom is the dissimetrical nature
> of relvar definition due to restriction of drawing value by data type
> definition. One advantage would be formal expression of the difference
> between domain and data type.
>
> A consequence is that the ensemble of parties made by all attributes
> are constituted as a product of belonging and restrictions.
>
> For all R1{A1, A2, A3} relvars drawn from domain DoR1, there exists an
> ensemble of parties that allows dissymetrical decomposition and
> characterization of relvars.

Google couldn't help on a definition of dissymetric, though it turned up a number of references to the word.

> The admission of this axiom may have interesting consequences in
> operation characterization.
>
> I would like to get some insight, advice and reading pointers from
> vadim and x on that point but anybody else that has constructive
> motivation is welcome as well.
>

```--
Jonathan Leffler                   #include <disclaimer.h>