# Notation for permutations of function composition

Given a set of functions $$F$$ of which all are functions in $$X \to X$$, where certain functions, say $$a, b \in F$$ can be composed together: $$a(b(x)) = (a \circ b)(x)$$. What is the notation for the set of all possible permutations of compositions of functions in $$F$$?

For example, say $$F = \{a, b\}$$ where $$a, b : X \to X$$, all the possible permutations of compositions of functions in $$F$$ are:

$$(a \circ a)(x)$$

$$(a \circ b)(x)$$

$$(b \circ a)(x)$$

$$(b \circ b)(x)$$

How would the set of the above be expressed in terms of $$F$$ in general? Should I not be expressing these functions as a set?

• You could write $\{ f \circ g \mid f,g \in F \}$. – Clive Newstead May 15 '19 at 16:28
• I forgot to mention that I'm looking for a general notation. My example was to simplify the example permutation set but the cardinality of $F$ is undefined and may be arbitrarily large. – Shiri May 15 '19 at 16:32
• So you don't allow $a \circ a \circ a$ in your set? – Jair Taylor May 15 '19 at 16:38
• @JairTaylor My mistake I was initially thinking of a fixed sample size of $|F|$ but yes I do allow $a \circ a \circ a$ – Shiri May 17 '19 at 10:44

Note, you haven't provided all possible compositions, since there's also $$aba$$ and $$abaaabbbaba$$. Confusingly, the mathematical literature calls these "words".
You could use the Kleene star notation (in your case, $$F^*$$ ) to describe all possible words you can make, and interpret the words as compositions.