Definition:Multiplication

See product.

Multiplication is an operation on a naturally ordered semigroup $$\left({S, \circ; \preceq}\right)$$ which can be defined using the Principle of Recursive Definition as follows:

$$\forall m, n \in S: n \ast m = g_m \left({n}\right)$$

where $$g_m: S \to S$$ is the unique mapping that satisfies:

$$ \forall m \in S: g_m \left({n}\right) = \begin{cases} 0: n = 0 \\ g_m \left({r}\right) \circ m: n = r \circ 1 \end{cases} $$