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.