Generator of Subsemigroup/Examples/Positive Odd Numbers

Example of Generator of Subsemigroup

Let $\struct {\Z, +}$ be the additive group of integers.

Let $A$ be the set of positive odd integers.

The subsemigroup of $\struct {\Z, +}$ generated by $A$ is the semigroup of strictly positive integers under addition.

Proof

Let $\struct {S, +}$ be the subsemigroup of $\struct {\Z, +}$ generated by $A$.

First we note that $\struct {A, +}$ is not itself closed, as:

\(\ds 1\)

\(\in\)

\(\ds A\)

\(\ds 1 + 1\)

\(\notin\)

\(\ds A\)

Then we note that:

$\forall x \in \Z_{>0}: \begin {cases} x \in A & : \text {$x$ odd} \\ \exists y \in A: x = y + 1 & : \text {$x$ even} \end {cases}$

Hence $\Z_{>0}$ is the smallest subset of $\Z$ on which $+$ is a closed operation.

So:

$A + A = \Z_{>0}$

and from Non-Zero Natural Numbers under Addition form Semigroup, it follows that $\struct {\Z_{>0}, +}$ is a semigroup.

Hence the result by definition of subsemigroup of $\struct {\Z, +}$ generated by $A$.

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings