Monoids are a set of elements, and a binary combining operation (op) that returns another member of the set.

Properties

Associativity

1. Given a binary joining operation •,
2. and all a, b, and c of the set,
3. then: a • (b • c) == (a • b) • c

Identity element

• Unique element (id, sometimes called the ‘zero’ of the set)

• Behaves as an identity with op

  identity = 0 op = &(&1 + &2) # Integer addition append(34, identity) == 34

  identity = 1 append = &(&1 * &2) # Integer multiplication append(42, identity) == 42

Counter-Example

Integer division is not a monoid. Because you cannot divide by zero, the property does not hold for all values in the set.