> Integer multiplication also forms a semigroup (and in fact a monoid, due to identity), but not a full group due to zero not having an inverse. Wait, what? V

> Integer multiplication also forms a semigroup (and in fact a monoid, due to identity), but not a full group due to zero not having an inverse. Wait, what? Very few integers have multiplicative inverses (in the integers). If you meant to speak of modular inverses, then even that is false for composite moduli like 2^n (with n > 1), which may be why im4w1 (https://news.ycombinator.com/item?id=12113934 https://news.ycombinator.com/item?id=12113934) asked if you meant to work modulo a prime.