Thanks John!

The problem with my approach on the boolean example seems to be that it's a bit hard to know what to do with -1 + -1. How do the negative elements behave when not directly combined with their inverse or id? I drew up a truth table, and -1 + -1 = -1 is the only one that doesn't kill associativity. Extracting ourselves from booleans, we could express the general rule as (-x + -x) = -(x + x), which feels nice, but I don't remember groups having a distributive law. I haven't done the working, but maybe this is too arbitrary to qualify. The general case also needs (-x + y) defined for the new elements...

My reasoning with integers was similarly wrong: integers are a free group from a single generator {.}, not the (N, +, 0) monoid. The integer behaviour for (-x + -x) or (-x + y) here uses special integer knowledge, not just general monoid structure, which we must exclusively use to construct the group.

My next hunch is that it might require the alternating red, black, red black thing we saw in Chapter 1, although I haven't thought this through at all. I've taken my intuition as far as it can go now, and really need to do some calculations.

The problem with my approach on the boolean example seems to be that it's a bit hard to know what to do with -1 + -1. How do the negative elements behave when not directly combined with their inverse or id? I drew up a truth table, and -1 + -1 = -1 is the only one that doesn't kill associativity. Extracting ourselves from booleans, we could express the general rule as (-x + -x) = -(x + x), which feels nice, but I don't remember groups having a distributive law. I haven't done the working, but maybe this is too arbitrary to qualify. The general case also needs (-x + y) defined for the new elements...

My reasoning with integers was similarly wrong: integers are a free group from a single generator {.}, not the (N, +, 0) monoid. The integer behaviour for (-x + -x) or (-x + y) here uses special integer knowledge, not just general monoid structure, which we must exclusively use to construct the group.

My next hunch is that it might require the alternating red, black, red black thing we saw in Chapter 1, although I haven't thought this through at all. I've taken my intuition as far as it can go now, and really need to do some calculations.