I am confused about the following structure, and would be very thankful if somebody could give me a hint.
Let $\mathbb{S}$ be a set with n elements $\mathbb{S}=\{a_1, a_2, ..., a_n\}$, and $(x,y) \in \mathbb{S}$. We have a transformation $T: x \mapsto y$, such that the transformation $T$ applied to any element in $\mathbb{S}$ gives a (potentially different) element in $\mathbb{S}$ again.
- Is $(\mathbb{S}, T)$ a special algebraic structure?
- Are there other properties that are not obvious?
Added later: What I am describing is $T: \mathbb{S} \to \mathbb{S}$, where $T$ is bijective and $\mathbb{S}$ is a finite discrete set. Is there something special about such structures? For example, it seems like $T$ always has to be a permutation-matrix. Anything more?
What you have is a permutation action of the group $\Bbb Z$ on the finite set $S$, through $(n,a)\mapsto T^n(a)$. The information is equivalent to giving the permutation $T$ of $S$, and leads to the usual stuff associated to a single permutation, such a decomposition of $S$ into cycles.