Talk:Abstract algebra

Is it common usage to call a group "an abstract algebra"? I think of abstract algebra as a field of mathematics which studies algebraic structures such as groups. And the term "abstract" is only used if there is a need to distinguish it from elementary or college algebra. Maybe there's also a confusion with universal algebra? --AxelBoldt

I thought it was a bit odd too. I would have used the term "algebraic system". In universal algebra they would just be called "algebras" (except for modules and vector spaces, which don't qualify because of the external multiplication). I'm not sure what to do about it at the moment. In any case, we need an article on universal algebra. --Zundark, 2001-09-04

From the (principal so far) author of the article on universal algebra: Modules and vector spaces are indeed covered under universal algebra if you fix the ring R that the modules are over. But scalar multiplication is not a binary operation of course; instead, for each element r of R, you have a unary operation "scalar multiplication by r". There's no rule that says, for example, that you must have only finitely many operations!

Somebody should probably explain this in the article on universal algebra, or maybe on modules, but I'm not sure how to organise it now. --Toby, 2002/04/03

I think it fits in universal algebra, as an additional example. And all the structures that are covered by universal algebra should have a link to universal algebra. Can you also deal with topological groups in universal algebra? If not, then that should be mentioned as well. AxelBoldt

Well, I'll put it in there, but I don't know when I'll get to it -- I'm having lots of fun looking around here.

As for topological spaces, universal algebra doesn't handle them as such. However, if you start with topological spaces as given, then you can define universal algebra in the category of topological spaces analogously to defining universal algebra in the category of sets, just as you define a topological group (that is a group in the category of topological spaces) analogously to defining a group (that is a group in the category of sets). Every time that the definition of an algebraic system calls for a set, you replace that with a topological space; and every time that the definition calls for a function, you replace that with a continuous map. So topological groups aren't covered by universal algebra any more than they're covered by group theory, but they are covered by topological universal algebra, just as they are (obviously) covered by topological group theory.

So I suppose that I should add a comment to that effect.

-- Toby