Let A be a set of symbols. The free monoid on A is the structure (A*,.,ø), where
The free monoid M on a single letter, say A = {c}, has as elements
This monoid has the same shape as (N,0,+) via the correspondence
Multiplication in M corresponds to addition in the exponent of c and thus to addition in N.
Our next issue is how to phrase that monoids may appear in different guises but are essentially the same.