Unique function whose product is a unique monoid element?

Question

Let $P$ be the free monoid on a set $X$. Given $p \in P$, this induces a unique function $f : [n] \to X$ with $\prod_{i\in[n]} f(i) = p$. Does this function have a name?

Answer 1

A bit of notation and terminology. The free monoid on the set $X$ is usually denoted by $X^*$. The set $X$ is the alphabet, its elements are letters. The elements of $X^*$ are called words.

Let $p = a_1 \dotsm a_n$ be a word, where $a_1, \ldots, a_n$ are letters. Then $n$ is the length of $p$ (also denoted $|p|$). The letter $a_i$ is the letter of $p$ at position $i$. Thus, if you insist to give a name to your function $f: \{1, \ldots, |p|\} \to X$, I would suggest to call it the position map.