[Torsors](http://math.ucr.edu/home/baez/torsors.html) are defined as a special kind of group action. I am wondering whether the analogous notion exists for monoid actions. Some references would be helpful.


In general I'm interesting in the notion of 'subtraction/division' induced by having a torsor. My application is in computer science. A monoid is used to capture modifications to a computer program and the monoid action corresponds to performing the modification on the program. If I have a torsor-like entity, I can take two software entities and produce the modification required to convert one into the other.


The answer is that any such monoid will automatically be a group. In my application, it seems that I will only get close to the notion of torsor if my modifications have inverses, which they do not.



Torsors for monoids
One common definition of torsor under a group $G$ is a (**Edit:** nonempty) set $X$ together with an action $act: G \times X \to X$, such that the map $(act,id): G \times X \to X \times X$ is a bijection of sets. The definition is still meaningful if you replace "group" with "monoid".


**Edit:** As Torsten has pointed out, replacing "group" with "monoid" doesn't give you any additional generality, since the bijection endows $G$ with inverses. We can instead ask that the map $(act,id)$ be injective. No, that doesn't specialize correctly to the setting of groups. I think I'll quit while I'm behind, and let this stand as a cautionary tale for others.


Allowing $X$ to be empty yields a notion of pseudotorsor, which seems kind of useless in a bare set-theoretic context. It seems to come up in algebraic geometry, though.


