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

Tony D wrote:
> Cimode wrote:
> > No. Mathematics of ensembles (known as Ensemblist mathematics) is a
> > totally independent area from set theory (they of course exist
> > relationship between the two). While set theory main focus is
> > operation definitions between sets of values, ensemble mathematics main
> > focus is characterization and definition of ensembles of values at
> > higher level of abstraction. For simplification purposes, I guess one
> > could consider ensemblist math as a macro view and set theory as a
> > micro view of the same problem.
>
> Unfortunately every Google on "théorie des ensembles" returned
> definitions of set theory. I Amazon'd for the Bourbaki book, and the
> English translation is "Elements of Mathematics I : The Theory of
> Sets". However, from the table of contents I think I know what you're
> getting at. Just to check; are "ensembles of parties" the "families of
> sets" described in sections 3 & 4 of chapter II ? That would give me
> something to go searching on.
After doing some research it seems that the book *Théorie des ensembles* was published in US under the pseudonym of Nicolas BOURBAKI and the name you provided ...(In fact there are several mathematicians constituing the BOURBAKI association)... Here is the book I have found on Amazon. http://www.amazon.fr/exec/obidos/ASIN/3540225250/403-4729874-1286004

>From additional research, I discovered also that the axiom of the
*ensemble of parties* (French litteral translation) would be known as *Axiom of power set*. I recommend that reading for continuing this discussion in a useful manner.

> > As stated, my belief is that set theory might not be *sufficient* to
> > characterize nature of relvars and that much work still needs to be
> > done using math to clarify issues that are unclear in Codd's and Date's
> > work.
> >
>
> As an aside, what are the issues you consider unclear in the work of
> Codd & Date ? (Not argumentative; I'd like to know what areas are
> problematic to you.)
In general, relvars themselves, as opposed to their projections . For instance, relvar are multidimensional variables that have not been to my knowledge defined one according to the others in simple terms. For instance, no operators have been defined for handling relvars...such as R1 = r1 op r2 op r3 (r1, r2 and r3 being attributes and R1 being the global relvar) --> what operator is defined at relvar level as opposed to operators defined on their projection as R tables. This is what I came to the conclusion that relvars have no been characterized sufficiently.

> > No. The question is not really about how are defined the constraints or
> > according to *what* they are applied. The question is about the relvar
> > characterization knowing how they have been already defined in Codd's
> > and Date's work thanks to the use of other mathematical tools.
> >
>
> What do you mean by "relvar characterization" here ?
I mean specifying the characteristics of relvars as opposed to their projections.

> > I am sorry but that is really off topic. The question is not about
> > constraints but about the relvar itself.
> >
>
> It's sneakily relevant in a way; if the domains are different, then the
> sets of values are different; if the domains are the same, then the
> sets of values are the same, but the restrictions occur elsewhere.
How are restrictions defined mathematically on relvars themselves? The formula proposed reveals some assymetrical nature which would be a consequence of restrictions applied on extractions from domains.

> > Accepting the difference between domain and data type is not really
> > relevant to this topic because such difference is not the main focus of
> > trying to launch discussion about the nature of relvars.
> >
>
> > *Ensemble of parties* is a the abstract ensemble that is defined as a
> > ensemble to which necessarily belong N sub ensembles. It has been
> > proven mathematically that such ensemble ALWAYS exists when some sub
> > ensemble of values exist. Such ensemble is also called Ensemble of
> > parties (noted B(DoR1) --> read *Beta of DoR1*). As you may have
> > guessed domains here are used as ensembles...
> >
>
> That's why the difference drawn between data type and domain is
> important; as I mentioned above, it potentially changes the sets of
> values available.
Yes. I won't argue with something obvious that has been defined and redefined over and over. But consider the problem on a different perspective: what if the difference between data type and domain is not the main focus. What about the relvar itself? What are the characteristics of a relvar R1 (operators) in relationship to the elementary relvars that constitute it? How one can *evaluate* operators that apply on a specific relvar based on the operators that apply on the attributes that constitute the relvar?

> > That's a valid point. But be aware that we are here in unknown ground
> > in RM (which in my perspective should be the purpose of
> > comp.database.theory). Keep also in mind, that it may be interesting
> > to get some better understanding of ensemblist math theory before
> > getting any further.
>
> We're in unknown territory for me anyway, that's for sure. However, the
> terrain is quite interesting ...
In that case, I reitterate my encouragement to do some reading first. Received on Sun Jul 16 2006 - 21:03:40 CEST