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Stream: learning: questions

Topic: Algebras with more than one zero?

Eric M Downes (May 06 2020 at 20:27):

I am trying to categorify a certain algebra. It appears necessary to have an additive identity for each object, such that
$x + y = z \Rightarrow x + 0_y = z - y \Rightarrow x + 20_y = z - y + 0_y ..$
This teeters on the edge of madness, but when it is time to finish a calculation many (but not all) of these disappear under an equivalence class, and you are left with what appears to be a useful lower bound on error propagation in calculations performed with the resulting formula.

I feel like this must have been done before. If not in abstract algebra, then the design of floating point arithmetic, etc. Unfortunately I failed my Literature Search roll twice now, and can't attempt again until I've had a long rest. Any pointers on where I should look to educate myself would be very much appreciated! Thanks!

Cole Comfort (May 06 2020 at 20:59):

Eric M Downes said:

I am trying to categorify a certain algebra. It appears necessary to have an additive identity for each object, such that
$x + y = z \Rightarrow x + 0_y = z - y \Rightarrow x + 20_y = z - y + 0_y ..$
This teeters on the edge of madness, but when it is time to finish a calculation many (but not all) of these disappear under an equivalence class, and you are left with what appears to be a useful lower bound on error propagation in calculations performed with the resulting formula.

I feel like this must have been done before. If not in abstract algebra, then the design of floating point arithmetic, etc. Unfortunately I failed my Literature Search roll twice now, and can't attempt again until I've had a long rest. Any pointers on where I should look to educate myself would be very much appreciated! Thanks!

There is a related notion to what you are describing. Torsors are groups with no fixed additive identity so that the composition function, instead of being of type GxG->G is of type GxGxG->G where the middle element corresponds to a "fixed indentity".

Eric M Downes (May 06 2020 at 21:20):

Thanks Cole! Yes this is quite relevant... (and of course it looks like John Baez has written about them) I will read up more and post back about how it goes. I really appreciate your help!