The following seems to be the "official" notion of torsion-freeness in the context of semigroups: 



> 
> 
> > 
> > **TF1.** A (multiplicatively written) semigroup $\mathfrak A$ is torsion-free if there do not exist $a,b \in \mathfrak A$ and $n \in \mathbb N^+$ such that $a \ne b$ and $a^n = b^n$.
> > 
> > 
> > 
> 
> 
> 


On another hand, I recently ended up with the following alternative idea:



> 
> 
> > 
> > **TF2.** A semigroup $\mathfrak A$ is torsion-free if, given $a \in \mathfrak A$, $a^m = a^n$ for some $m,n \in \mathbb N^+$ with $m \ne n$ only if $a$ is idempotent.
> > 
> > 
> > 
> 
> 
> 


Both of these generalize the usual notion of torsion-freeness for groups. Also, it is not difficult to check that TF1 implies TF2, but not viceversa. So, my questions are:



> 
> 
> > 
> > **Q1.** What about existing literature concerning torsion-free semigroups in the sense of the second definition? **Q2.** Could you point out some reasons why the former definition should be preferred to the latter?
> > 
> > 
> > 
> 
> 
> 


For the record, this is somehow related to [question 105851](https://mathoverflow.net/questions/105851/strictly-totally-ordered-semigroups-looking-for-references).



On the notion of torsion-freeness in semigroup theory
I believe your first definition is primarily used by people in commutative semigroup theory. Most people in non-commutative semigroup theory that I know prefer the second. An important example is the free profinite monoid. Every periodic element is an idempotent so it is torsion-free in the second sense. This is a non-trivial theorem due to Rhodes and myself and relies on projectivity of maximal subgroups and torsion-freeness of projective profinite groups. 


I think the same maybe true for Stone-Čech compactifications of free monoids, but perhaps they have some nilpotent cyclic subsemigroups. I believe their maximal subgroups are torsion-free. 


