I am playing with some questions concerning connections between 
certain poset partitions and their linear extensions. This is not
my usual playground, I just happened to stumble upon something.


When I was playing with these things, I came up with a very
simple construction that I cannot find anywhere.I suspect that this construction is a minor particular case of something well known
and much more general. It reminds of an equivariant map, but the group
is not fixed here. 


Let $n>2$, let $X$ be a finite set of cardinality $n$. Let $\oplus\_n$ be an action
of $Z\_n$ on $X$ such that the action of $1$ is a cycle of length $n$.
Pick $x\in X$ and let $\pi$ be a partition of the set $X$ such that
$\{x,x\oplus\_n 1\}$ is the only nonsingleton block of $\pi$.


In a very simple way, this gives rise to an action $\oplus\_{n-1}$ of $Z\_{n-1}$ on $\pi$
by ,,squeezing the action'' at $x$:


Put


* $\{x,x\oplus\_n 1\}\oplus\_{n-1} 1=\{x\oplus\_n 2\}$
* $\{x\oplus\_n(n-1)\}\oplus\_{n-1} 1=\{x,x\oplus\_n 1\}$
* $\{y\}\oplus\_{n-1} 1=\{y\oplus\_n 1\}$ for any $y\neq x\oplus\_n(n-1)$.


It can be visualized in a simple way by a digraph construction: if we
identify the action of $1$ on $X$ with an oriented cycle, this construction
corresponds to a contraction of an edge. 


Has anyone seen this construction before? 
Is there any name for it?



Actions of $Z_n$ and actions of $Z_{n-1}$
I might be missing something, but it seems to me that there is not much going on in your construction. In fact, your original action of $Z\_n$ on $X$ does nothing more than putting a cyclic ordering on your set $X$, and your complicated-looking construction creates a new cyclically ordered set $Y$ obtained from $X$ by removing one element (and retaining the ordering).


In any case, the construction is rather artificial from a group-theoretical point of view, so I don't expect a more general construction (e.g. for arbitrary group actions) in that sense. It is faintly reminiscent of the construction of a primitive action from an imprimitive permutation group by considering the induced action on the blocks of imprimitivity.


