Definition:Product in Naturally Ordered Semigroup

Definition

Let $\left({S, \circ, \preceq}\right)$ be a naturally ordered semigroup.

Let $*$ be the binary operation on $S$ defined using the Principle of Recursive Definition by:

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

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

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

The product of $n$ and $m$ is defined as $n * m \in S$ and the operation $*$ is called multiplication.

Thus we can define $n * m$ as follows:

$n * m := \circ^n m = \underbrace{m \circ m \circ \cdots \circ m}_{n \text{ copies of } m}$

It can be seen that this definition is a special case of a Recursive Mapping to Semigroup.

Sources

• Seth Warner: Modern Algebra (1965)... (previous)... (next): $\S 16$