Definition:Associative Word on Set

Definition
Let $X$ be a set.

Let $n\geq 0$ be a natural number.

A formal word of length $n$ on $X$ is a mapping:
 * $w : \left[ 1 \,.\,.\, n\right] \to X$

from the integer interval from $1$ to $n$.

Empty formal word
The empty formal word on $X$ is the empty mapping $\varnothing \to X$.

Also defined as
A formal word of length $n$ on $X$ is also seen defined as a mapping:
 * $w : \left[ 0 \,.\,.\, n-1 \right] \to X$

Also see

 * Definition:Composition of Formal Words
 * Definition:Reduced Formal Word on Set