Given a transformation $t$ from the transformation semigroup $T\_{n}$, if you take powers of $t$ under composition you get a length $s$ stem followed by a cycle. Permutations by definition have a length zero stem.


Are the terms "stem" and "cycle" common, or is some other terminology used in the literature?



Transformation terminology question
This question seems a bit ill-posed to me. There are two possible interpretations here.


(1) If $s$ is an element of a finite semigroup, then $s^i=s^{i+p}$ where there is a minimal such $i$ and $p$. What is the common terminology for $i,p$? 


The "official" terminology is that $i$ is the index and $p$ is the period. Or one might call the elements $\lbrace s,..,s^{i-1}\rbrace$ the nilpotent part and $\lbrace s^i,..,s^{i+p-1}\rbrace$ the maximal subgroup or minimal ideal. 


This in any event has nothing to do with transformations.


(2) You are interested in the analogue of cycle decomposition for a transformation. In this case, the situation is more complicated than a stem and cycle. 


The transformation $f$ has a set of recurrent points (also called the eventual image) on which $f$ acts like a permutation and hence decomposes into cycles. There may be several of them. Attached to each cycle are some number of trees which are directed toward the cycle. There is a only a stem followed by a cycle if there is a point from which every other point can be reached. The terminology is not so standard for this setup. I think there is a good chapter in Peter Higgins book Techniques of semigroups on this sort of thing.


