The integers with addition build a group $\langle \mathbb{Z},+,0\rangle$.
The functions $\operatorname{succ}:\mathbb{Z} \rightarrow \mathbb{Z}$, $\operatorname{pred}:\mathbb{Z} \rightarrow \mathbb{Z}$ with $\operatorname{succ}(x) = x +1$, $\operatorname{pred}(x) = x -1$ are the generators of the (free) group $\langle \{\operatorname{succ},\operatorname{pred}\}^+,\circ,\operatorname{id}\rangle$ with $\{\operatorname{succ},\operatorname{pred}\}^+$ the set of finite concatenations of $\operatorname{succ}$, $\operatorname{pred}$ and $\operatorname{succ}\circ \operatorname{pred} = \operatorname{pred}\circ \operatorname{succ} = \operatorname{id}$. This group is supposed to act on $\mathbb{Z}$.
There is a natural one-to-one mapping between $\langle \mathbb{Z},+,0\rangle$ and $\langle \{\operatorname{succ},\operatorname{pred}\}^+,\circ,\operatorname{id}\rangle$:
- $+ \leftrightarrow \circ$
- $0 \leftrightarrow \operatorname{id}$
- $k > 0 \leftrightarrow \operatorname{succ}^k$
- $k < 0 \leftrightarrow \operatorname{pred}^{-k}$
What's the official name of the group $\langle \{\operatorname{succ},\operatorname{pred}\}^+,\circ,\operatorname{id}\rangle$?
Compare with the monoid $\langle \mathbb{N},+,0\rangle$.
Consider the functions $\operatorname{succ}':\mathbb{N} \rightarrow \mathbb{N}$, $\operatorname{pred}':\mathbb{N} \rightarrow \mathbb{N}$ with $\operatorname{succ}'(x) = x +1$, $\operatorname{pred}'(0) = 0$ and $\operatorname{pred}'(x) = x -1$ if $x > 0$. The functions $\operatorname{succ}'$ and $\operatorname{pred}'$ are freely composable and $\operatorname{pred}'\circ \operatorname{succ}' = \operatorname{id}$.
But $\operatorname{succ}'\circ \operatorname{pred}' \neq \operatorname{id}$ because $\operatorname{succ}'\circ \operatorname{pred}'(0) = 1$. On the other side $\operatorname{succ}'\circ \operatorname{pred}'$ is somehow "almost" the identity because $\operatorname{succ}'\circ \operatorname{pred}'(n) = n$ for every other $n \neq 0$.
This means: $\operatorname{pred}'$ has a ("strong") right-inverse and $\operatorname{succ}'$ has a ("strong") left-inverse. In turn $\operatorname{pred}'$ has only a "weak" left-inverse and $\operatorname{succ}'$ has only a "weak" right-inverse.
Is this distinction between "strong" and "weak" inverses of any (and in any) use?
What's the name of the kind of algebraic structure $\langle \{\operatorname{succ}',\operatorname{pred}'\}^+,\circ,\operatorname{id}\rangle$ – which is supposed to act on $\mathbb{N}$ – is an example of?
If I understand the definition correctly, the structure $\langle \{\operatorname{succ}',\operatorname{pred}'\}^+,\circ,\operatorname{id}\rangle$ is an inverse monoid, i.e. an inverse semigroup with identity, since
$\operatorname{succ}' = \operatorname{succ}' \circ\operatorname{pred}'\circ \operatorname{succ}'$
$\operatorname{pred}' = \operatorname{pred}' \circ\operatorname{succ}'\circ \operatorname{pred}'$