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:

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

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

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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:

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

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If $y\in u$, then $(f\circ g)(q(x))=g(x)$.
Title  alternative treatment of concatenation 

Canonical name  AlternativeTreatmentOfConcatenation 
Date of creation  20130322 17:24:10 
Last modified on  20130322 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 