Re: Notions of Type

Marshall wrote:
> erk wrote:
> >
> > Sorry if this is obvious to everyone else, but does an algebra include
> > only operations defined on values of the type in question?
>
> Yes.

I read elsewhere (and I'm not saying it's right) that an algebra could be defined as a closure over a set of types. Is that untrue? If it's not true, then is there an analogous term for this algebra-like closure, but over multiple types?

> Note the Tropashko algebra is a true algebra, and is complete.

I've read his excellent paper, but don't have it handy; how does he allow what we've come to expect from relations without reference to attributes and their types? Or does he? While I was extremely impressed with what he said about joins, does the nature of an algebra imply that the relational model, as we know it, can never rely on an algebra alone?

> > As division by zero is undefined, either its denominator type is
> > restricted to nonzero, or its range includes "undefined" as a value. If
> > there's another option, I can't think of it.
>
> There is another option, and I would claim it's a better one than
> either of the more popular ones you've identified: partial functions.
> Division is a partial function. There are bazillions of partial
> function out there, and they require at least as much attention
> as the total ones.

Agreed. Can multiple partial functions over the same domain then complete an otherwise "impossible" algebra? In other words, in an alternate reality I could define division by zero as always yielding 42, regardless of the domain value.

• erk