Can relvars be dissymetrically decomposed? (vadim and x insight demanded on that subject)
Date: 7 Jul 2006 01:01:47 -0700
Message-ID: <1152259307.758906.172460_at_75g2000cwc.googlegroups.com>
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.
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).
The question that arose from that observation is whether the domain DoR1 is equivalent to the ensemble consitued by the some adjonction of 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
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.
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.
The admission of this axiom may have interesting consequences in
operation characterization.
I would like to get some insight, advice and reading pointers from