Re: Notions of Type

Keith H Duggar wrote:
> 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 ask because in relational algebra, at least the rename
> > > operator involves a different type ("attribute name")
> > > than the "core type" (relation).
> >
> > Very true. Of the various relational operators that have
> > been identified over the years, only a few, like union,
> > are really algebraic. RESTRICT, PROJECT, etc. aren't.
> > Nonetheless people call it an algebra because it's an
> > algebra in spirit
>
> For certain your type theory knowledge is superior to mine,
> so I'm unsure what you are saying above. Because I'm almost
> certain that algebraic structures are not limited to a
> single set.
>
> For example, Linear Algebra is an algebraic structure over a
> vector space and a field. The entire family of multi-sorted
> algebras are defined over two or more different subsets of a
> particular set.
>
> In other words I thought
>
> algebraic structure : a set of function signatures and
> axioms defined over /one or more/ sets.
>
> Is this not the case? And if not, what does the above
> definition define?

Ah, yes; that.

Well, it turns out that the terminology is not alltogether nailed down in any kind of persistent way.

We have:

1. arithmetic with the addition of variables. (I remember factoring a lot of polynomials in junior high and high school.)
2. the study of things like rings, groups, fields, etc. Single sets with closed operations on elements of those sets.
3. a vector space over a field with multiplication.

All are called algebra in one context or another.

Now, when I was in high school and college, "algebra" mean 1 and the term "linear algebra" meant 2. As I understand it, nowadays the terminology is that 1 is called "arithmetic" and 2 is called "abstract algebra" or usually just "algebra." 3 today is called "linear algebra." As if that's not bad enough, most authors agree that linear algebra is not, techincally speaking, an algebra at all. However sometimes it appears that it is actually a substudy of abstract algebra.

Generally in this thread I'm talking about 2. If you generalize it sufficiently, the "or more" you mentioned above applies, but for relational algebra, we don't need to go there.

Marshall Received on Fri Aug 18 2006 - 04:31:33 CEST