Is there a name for a group-like structure under a unary operation? It seems to me that the set,
$$
S=\{ \sin, \cos, -\sin, -\cos \}
$$
forms something like a cyclic group under differentiation. But I understand groups to be defined to have a binary operation.
Is there a name for a group-like structure under a unary operation?
 A: To me the right context seems to be that the set $M$ consisting of the differentiation operator and its powers is a monoid, and the set you gave is a set with the monoid $M$ acting upon it.
So I would advise you to see semigroup and monoid actions.
A: If $f$ is a function, and $D$ means "differentiate", and I change your set notation for an ordered 4-tuple (because I'll bet you want order to matter), then what you've got is
$$(D^0 f, D^1 f, D^2 f, D^3 f)$$
where $f$ is $\sin$. A dynamicist would say that you've identified a 4-cycle for the action of $D$ on (differentiable) functions. And as others have noted, there's a monoid structure at play here, because $D^n D^m = D^{n+m}$.
A: The set $S$ is a torsor for the cyclic group of order $4$.
Given a group $G$, a $G$-torsor is a nonempty $G$-set $T$ with the property that
$$\forall x,y \in T \, \exists! g \in G \, gx=y.$$
For
$$G=C_4=\{e,g,g^2,g^3\},$$
one can define an action of $G$ on $S$ by letting $g^nf$ be $f^{(n)}$ (the $n$th derivative of $f$, where the $0$th derivative of a function is simply the function itself) for $n \in \{0,1,2,3\}$ and $f \in \{\sin,\cos,-\sin,-\cos\}$.
One can easily check that this action does in fact make $S$ into a $G$-torsor.
