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I have 2 mathematical objects $F_a$ and $F_b$ and I am interested in the set that contains $(), (F_a), (F_b), (F_a,F_b), (F_a,F_a), (F_b,F_b), (F_b,F_a), (F_a,F_a,F_a)...$ (all the ordered finite sequences composed of any amount of $F_a$s and $F_b$s)

I guess I could describe it as the union of $\{F_a, F_b\}^i$ for $i$ in $\mathbb{N}$ but there is probably a shortest way to write it.

Vinyl_cape_jawa
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Anne Aunyme
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  • Not an answer but perhaps a useful observation. These tuples are (almost) naturally described by the nonnegative integers in binary notation. Just think all finite strings of $0$s and $1$s. You do have to worry though about leading zeros. Start each sequence with a decimal point and a $1$. Then you have the dyadic rationals in the interval [1/2, 1). – Ethan Bolker Oct 10 '19 at 15:30
  • @EthanBolker actually you can also have a bijection with non-zero positive integers by adding a leading 1. – Anne Aunyme Oct 10 '19 at 15:38
  • Indeed. I suggested the decimals because I first thought they would solve the problem all by themselves. Then added the $1$ without realizing that it made the decimals unnecessary. I don't think there's a well known name for the structure you want, so invent one for your reader. – Ethan Bolker Oct 10 '19 at 15:43

2 Answers2

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For a set $A$, a usual way to denote the set of finite sequences of elements of $A$ is $$A^{< \mathbb N}.$$

mathcounterexamples.net
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In addition to this answer, the notation $A^*$ comes to mind. I've seen this in a few contexts:

  • In symbolic dynamics, an alphabet $A$ is a (finite) set. A word is a finite sequence in which each term comes from $A$. You will often see the notations $$ A^0 := \varnothing,\qquad A^n := \{ (a_1, a_2, \dotsc, a_n) : a_j \in A \}, \qquad A^* := \bigcup_{j=0}^{\infty} A^j. $$ The set $A^n$ is the set of all words of length $n$, while $A^*$ is the set of all words of finite length. In the context of the original question, the set $$ \{ F_a, F_b \}^* $$ would denote the set of all sequences of finite length where each term is either $F_a$ or $F_b$.

  • In algebra, the free monoid on a set $A$ is the collection of all finite sequences in $A$ (really, this is exactly the same thing as above, only phrased in fancier language). The notation is the same as above, so we might say that $\{F_a, F_b\}^*$ is the free monoid on (or generated by) the set $\{ F_a, F_b\}$.

  • As pointed out by Mark S. in a comment, in the fields of mathematical logic and computer science, the set $A^*$ may be called the Kleene closure of a set. The star in the notations above is, in this context, the Kleene star, which is a unary operator which acts on sets. Note that the Wikipedia page gives a slightly different construction: given a set $A$, define $$A^0 := \varnothing, \qquad A^1 := A, \qquad A^{n+1} := \{ \alpha \beta : \alpha \in A^n, \beta \in A \}, $$ where $\alpha\beta$ denotes the concatenation of the sequences $\alpha$ and $\beta$. Then $$ A^* := \bigcup_{j=0}^{\infty} A^j. $$ The set $A^*$ here is precisely the same set constructed above, however the intermediate sets $A^n$ are the collections of words of length less than or equal to $n$ (rather than words of precisely length $n$).

Xander Henderson
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  • Very interesting! I won't use this notation since I already use the star as "deprived of zero" for other sets and it would get confusing but it is nice to know it exists. – Anne Aunyme Oct 13 '19 at 15:46