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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?

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  • $\begingroup$ Are describing a set $S$ with a function $T:S\to S$? If so there's not much structure apriori. If $|S|=n$ there are $n^n$ possible functions. Groups have a binary operation that satisfy identity, associativity, and inverse axioms. Those are what give the structure. $\endgroup$
    – spitespike
    Jan 16, 2015 at 7:30
  • $\begingroup$ See if 'unary operation' is what you're looking for $\endgroup$
    – spitespike
    Jan 16, 2015 at 7:33
  • $\begingroup$ In other words your searching for the name of a structure $(X,f)$ where $f:X\rightarrow X$ is a map from $X$ to itself and $X$ is finite? If yes maybe you could find this question interesting, in Keinstein's answer the term "Function Algebra" appears. $\endgroup$
    – MphLee
    Jan 16, 2015 at 7:48
  • $\begingroup$ Btw if with $T:x\rightarrow y$ you mean $T(x)=y$, you should write $T:x\mapsto y$ $\endgroup$
    – MphLee
    Jan 16, 2015 at 7:50
  • $\begingroup$ @MphLee yes, it seems like I'm searching for exactly what you said. "If yes maybe you could find this question interesting" - which question? It's very like that I would find it interesting, if you could provide me a link, that would be nice. :) $\endgroup$
    – Mario Krenn
    Jan 16, 2015 at 8:30

1 Answer 1

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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.

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  • $\begingroup$ that sounds very much like what I was searching for, thank you. Your remark on cycles, I saw/used that, that's a nice feature. Now I see it in a bigger picture, thank you! $\endgroup$
    – Mario Krenn
    Jan 16, 2015 at 9:09

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