/sci/ - Science & Math

>>9041652
yes, that is what I found. But that falls apart quickly if you assume too much.
>>9041581
Distributivity matters once you assume + and * to be well-defined, but not invertible outside their original range. In that case you have to oppose further restrictions on the operations to make the connection unique. Since you however generally do not want to mess with the group structure, you demand 0 and 1 to be absorbing elements or something equivalent (as mentioned, distributivity implies the "inner" operation's unity to be an absorbing element). The issue is that trying to then fix everything you find that

1) you must drop both distributivities or
2) you must drop a group property
none of which I wanted to pursue further.

For example, one could set [math]0*a 0 a*0 = 0[/math], [math]1+a=a+1=1[/math] for all [math]K[/math] including the units.
Thus we can resolve all standard problems carried over by inversion (all equations containing 1+ or 0* follow special rules such as standard multiplication of the actual real number 0), but immediately run into issues if we only allow even one distributivity law, say [math]a*(b+c) = a*b + a*c [/math], because [math]a = a*(1+1) = a*1 + a*1 = a + a[/math].

So what could work is to build essentially a half-gorillion which is not distributive in either direction. I'm currently thinking of an instance.