It is known that a semigroup with left identity element and left invertibility is a group.
I noticed that I can make an unnamed group-like structure with some tweaks. I call a semigroup $\langle G,*\rangle$ a demimonoid iff:
- There exists a unique right identity element.
Furthermore, I call a demimonoid a demigroup iff:
- For every element in $G$, there exists a unique left inverse.
The key motivation is from that the "one-sided definition" of groups derive that the left identity is also the right identity by using left invertibility.
Note that uniqueness matters. A notable non-example of a demimonoid is the left zero semigroup:
$$ \begin{array}{l|ll} * & x & y \\ \hline x & x & x \\ y & y & y \end{array} $$
Every two-element demimonoid is a group. An example of a three-element demimonoid which is not a monoid (nor a demigroup) is:
$$ \begin{array}{l|ll} * & x & y & z \\ \hline x & x & x & x \\ y & y & y & y \\ z & x & x & z \end{array} $$ where $z$ is the unique right identity element.
I see no example of a demigroup that is not a group. All three-element demigroup is a group, namely, the cyclic group. Is every demigroup a group? If not, what is the smallest demigroup which is not a group?