#readwise # Quasigroups  ## Metadata - Author: [[Mark Seemann]] - Full Title: Quasigroups - URL: https://blog.ploeh.dk/2017/12/18/quasigroups/ ## Highlights - normal arithmetic operations like addition and multiplication form monoids. Perhaps you've been wondering where subtraction fits in. **Subtraction forms a quasigroup. What's a quasigroup? It's an invertible binary operation.** - Inversion # **What does it mean for a binary operation to be invertible? It means that for any two elements a and b, there must exist two other elements x and y that turns a into b.** - No identity # No identity element exists for integer subtraction. For instance, 3 - 0 is 3, but 0 - 3 is not 3. Therefore, subtraction can't be a monoid. No associativity # Likewise, subtraction is not an associative operation. You can easily convince yourself of that by coming up with a counter-example, such as (3 - 2) - 1, which is 0, but different from 3 - (2 - 1), which is 2. Therefore, it can't be a semigroup either.