alternative treatment of concatenation

It is possible to define words and concatenation in terms of ordered sets. Let $A$be a set, which we shall call our alphabet. Define a word on $A$ to be a mapfrom a totally ordered set into $A$. (In order to have words in the usual sense, theordered set should be finite but, as the definition presented here does not requirethis condition, we do not impose it.)

Suppose that we have totally ordered sets and $(v,\prec )$ and words $f:u\to A$and $g:v\to A$. Let $u\coprod v$ denote the disjoint union of $u$ and $v$ and let$p:u\to u\coprod v$ and $q:u\to u\coprod v$ be the canonical maps. Thenwe may define an order $\ll$ on $u\coprod v$ as follows:

• If $x\in u$ and $y\in u$, then $p(x)\ll p(y)$ if and only if .

• If $x\in u$ and $y\in v$, then $p(x)\ll q(y)$.

• If $x\in v$ and $y\in v$, then $q(x)\ll q(y)$ if and only if $x\prec y$.

We define the concatenation of $f$ and $g$, which will be denoted $f\circ g$, to bemap from $u\coprod v$ to $A$ defined by the following conditions:

• If $x\in u$, then $(f\circ g)(p(x))=f(x)$.

• If $y\in u$, then $(f\circ g)(q(x))=g(x)$.