-
Notifications
You must be signed in to change notification settings - Fork 189
New issue
Have a question about this project? Sign up for a free GitHub account to open an issue and contact its maintainers and the community.
By clicking “Sign up for GitHub”, you agree to our terms of service and privacy statement. We’ll occasionally send you account related emails.
Already on GitHub? Sign in to your account
Opposite ring #1921
Comments
It occurs to me that if we want |
After some quick experimentation I noticed that a ring
This is obviously not ideal and needs to be cleaned up. |
Right, the issue is that About R^op^op being definitionally R, it's possible that it won't come up as often as opposite categories do, but it's worth thinking about. One can also reverse groups, semigroups, monoids, etc, and it's probably not worth adding extra associativity clauses for all of these. |
Hmm it's a lot more work to change up the definitions in mathclasses. I've wanted to refine the hierarchy for a while now, but I don't have to motivation to do it here. I managed to get opposites working by carefully picking apart the data and making sure the correct data gets put back. This should be good enough for now. |
If one were designing an algebraic hierarchy from the ground up, one might consider abstracting out a notion of "involutive equality", proving general properties about it, and then just phrasing all the algebraic definitions in terms of it. |
If I were designing things from the ground up, I would even consider making everything a setoid and having equality as a special case. There is would make sense to have "involutive equality". It would have a lot of unifying potential with non-set based structures. However I am not particular keen on doing this. At the end of the day, the real motivation behind any algebra that I do is to do some actual calculations in homotopy theory. :) |
Why do you need to change things there? Can't you just drop the abelian group structure from the Ring Record you defined, and then define a coercion to AbGroup? Carrying around two abelian group structures seems like a recipe for problems. (Edit: changed "here" to "there" in first sentence.) |
Oh I see what you mean now. Maybe that is possible. |
This is finished now, but I'll make an issue about removing some of the redundant fields from Ring and AbGroup. |
We should define the opposite ring and remove the duplication in the theory of ideals.
The text was updated successfully, but these errors were encountered: