A pure monoid is a monoid $(M;*,1)$ where only the identity element $1$ is invertible, that is, has a two-sided inverse. Does there exist a pure monoid where there are non-identity elements that have a one-sided inverse? If so, can someone give me an example?
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We can just consider the universal example, namely the quotient of the free monoid on two generators $a, b$ by the relation $ab = e$. The elements of this monoid can be reduced uniquely to the normal form $b^n a^m$ and two such elements multiply by canceling factors of $ab$ in the middle. So we can give an explicit description of this monoid in terms of reduced words which shows in particular that no non-identity elements can have both a left and a right inverse.
If that isn't convincing then here's a faithful action of this monoid. We can use the example from the previous question and consider the action on $\mathbb{N}$ given by taking $b$ to be the function $b(k) = k + 1$ and $a$ to be the function $a(k) = \text{max}(0, k-1)$. Then $b^n a^m(k) = \text{max}(0, k-m) + n$ and we can recover $m$ and $n$ from this function: the function starts out being constant on an interval before becoming linear, that interval is $[0, m]$, and its constant value on that interval is $n$. So it fails to be injective unless $m = 0$ and it fails to be surjective unless $n = 0$.
Qiaochu Yuan
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