Editing Free monoid (section)

=== Natural numbers ===
The monoid ('''N<sub>0</sub>''',+) of [[natural numbers]] (including zero) under addition is a free monoid on a singleton free generator, in this case, the natural number 1. 
According to the formal definition, this monoid consists of all sequences like "1", "1+1", "1+1+1", "1+1+1+1", and so on, including the empty sequence. 
Mapping each such sequence to its evaluation result<ref>Since addition of natural numbers is associative, the result doesn't depend on the order of evaluation, thus ensuring the mapping to be well-defined.</ref>
and the empty sequence to zero establishes an isomorphism from the set of such sequences to '''N<sub>0</sub>'''.
This isomorphism is compatible with "+", that is, for any two sequences ''s'' and ''t'', if ''s'' is mapped (i.e. evaluated) to a number ''m'' and ''t'' to ''n'', then their concatenation ''s''+''t'' is mapped to the sum ''m''+''n''.