Groups acting on semigroups Suppose $G$ is a group and $S$ is a semigroup (a set with an associative binary operation). Just as in Groups acting on groups, say a group action of $G$ on $S$ is sensible if $g * (ss') = (g * s)(g * s')$ for all $g \in G$ and $s, s' \in S$.

What are some properties of groups acting (sensibly) on semigroups that are not always true of groups acting on sets?


Here is one idea I stumbled upon. Suppose that $F$ is a field and $G$ is a group acting on a semigroup $S$. There is a corresponding linear permutation representation of $G$ on the semigroup ring $F[S]$ ($G$ acts by permuting the basis vectors: $g * \sum_{s \in S} c_s \vec{e}_s := \sum_s c_s \vec{e}_{g * s} = \sum_s c_{g^{-1} * s} \vec{e}_s$). If $G$ acts sensibly on $S$, then the linear representation also respects multiplication: $g * (v w) = (g * v)(g * w)$ for all vectors $v, w \in F[S]$.
There is a neat consequence of this result. $S_n$ plainly acts on $[n] := \{ 1, 2, \dots, n\}$, so there is a corresponding linear permutation representation of $S_n$ on $\mathbb{R}^n$: $\sigma * (x_1, x_2, \dots, x_n) := (x_{\sigma^{-1}(1)}, x_{\sigma^{-1}(2)}, \dots, x_{\sigma^{-1}(n)})$. Of course, $S_n$ acts in the same way on $\mathbb{N}^n$ and continues to respect the addition operation of $\mathbb{N}^n$, so there is a corresponding linear permutation representation of $S_n$ on $F[x_1, x_2, \dots, x_n]$ as $\sigma * h(x_1, x_2, \dots, x_n) = h(x_{\sigma(1)}, x_{\sigma(2)}, \dots, x_{\sigma(n)})$. (Although it is not immediately clear in this last case, I am talking about the exact same construction of a linear representation from a group action every time I say "corresponding linear permutation representation." I think it's awesome that this works out.) Thanks to the preceding paragraph, $\sigma * (fh) = (\sigma * f)(\sigma * h)$ for all $f, h \in F[x_1, x_2, \dots, x_n]$. Thus $\sigma$ corresponds not only to an invertible linear operator on $F[x_1, x_2, \dots, x_n]$, but also a ring automorphism that fixes $F$. So by the universal property of fraction fields, $S_n$ acts linearly on $F(x_1, x_2, \dots, x_n)$ as field automorphisms fixing $F$.
 A: I think that what you are calling a "sensible action" would more usually be called an "action by semigroup automorphisms". Actually, in most contexts, when a group acts on a semigroup this property would be implicitly assumed (or else why mention the semigroup structure at all?).
Anyway, at the level of general semigroups, there is probably not much to be said. Given any set $S$ and a choice of a point $0 \in S$, defining $xy=0$ for all $x,y \in S$ makes $S$ into a semigroup. A bijection $\phi : S \to S$ will be a semigroup automorphism (i.e. will satisfy $\phi(xy)=\phi(x)\phi(y)$ for all $x,y\in S$) if and only if $\phi(0)=0$. So the semigroup automorphisms of $S$ are just the permutations of $S$ which fix the point $0$.
Obviously there is not much of a difference between general group actions on sets, and group actions on pointed sets which preserve the base point. There is a straightforward recipe for going between the two (add/remove the basepoint).
So, one response to your question "what properties do group actions on semigroups have that group actions on sets lack?" would be "not much". Without restricting attention to some special type of semigroups, basically any group action on a set can be put into this context.
Another comment: given a set $X$, you can still form a vector space $F[X]$, the vector space having the elements of $X$ as a basis. It doesn't have an interesting algebra structure, like it would in the semigroup case, but let's not worry about that. Now, if $G$ is acting on the set $X$, it also acts on the vector space $F[X]$ by linear transformations (namely ones that permute the basis). So you still get a permutation representation from a group action on a set.
