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 map from a totally ordered set into $A$. (In order to have words in the usual sense, the ordered set should be finite but, as the definition presented here does not require this 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. Then we 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 be map 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 $(f\circ g)(q(x))=g(x)$.

Title	alternative treatment of concatenation
Canonical name	AlternativeTreatmentOfConcatenation
Date of creation	2013-03-22 17:24:10
Last modified on	2013-03-22 17:24:10
Owner	rspuzio (6075)
Last modified by	rspuzio (6075)
Numerical id	6
Author	rspuzio (6075)
Entry type	Definition
Classification	msc 68Q70
Classification	msc 20M35
Related topic	Word