Extending Ordinals: Constructions & Inverses

[jostpuur]

Is there well studied constructions of some kind of extensions of the set of ordinal numbers, where each non zero number x also has the inverse x^(-1) so that x^(-1) x=1?

 

[Hurkyl]

Firstly, there are 'too many' ordinals to fit in a set, so you'd have to talk about the class of ordinals.

Now, the class of ordinals doesn't have any arithmetic operations on it -- which did you mean:
(1) You want to know if the multiplicative monoid of ordinals can be extended to a group.
(2) You want to see if there is any binary product on the class of ordinals (or an extension of them) that turns them into a group. (I assume you want associativity)


If you mean the former, then clearly no extension exists; the multiplicative monoid of ordinals is not right-cancellable.

 

[jostpuur]

Firstly, there are 'too many' ordinals to fit in a set, so you'd have to talk about the class of ordinals.

I've taken one course on the axiomatic set theory successfully, but I was lost during the entire course, and don't remember this stuff anymore even as badly as I did.

Now, the class of ordinals doesn't have any arithmetic operations on it -- which did you mean:
(1) You want to know if the multiplicative monoid of ordinals can be extended to a group.
(2) You want to see if there is any binary product on the class of ordinals (or an extension of them) that turns them into a group. (I assume you want associativity)


If you mean the former, then clearly no extension exists; the multiplicative monoid of ordinals is not right-cancellable.

I don't know what I want. I succeeded in avoiding calling this extension a "field extension", because I know that the addition and multiplication on ordinals don't work like in fields, but I was still thinking about some other kind of extension that would be similar.