By the way, while one indeed needs the ability to "cancel out"/invert, we can derive similar swaps in weaker algebraic structures (and don't need full Abelian g

By the way, while one indeed needs the ability to "cancel out"/invert, we can derive similar swaps in weaker algebraic structures (and don't need full Abelian groups). For example, non-zero rationals can be divide-swapped: a = a / b; b = a / (1 / b); a = b / a; or similar for subtraction-swap. In general, a quasigroup seems to suffice: a = a * b; b = a / b; a = b \ a (where * is the group operation and / and \ are the right and left divisions, respectively).