Why in the world do they use the word "free" in "free monoid"? It driving me crazy to see where the "freedom" comes from.
Here is the Awodey's explanation of it, in terms of "baby lagebra" (sic.) but it is even more confusing:
A monoid M is freely generated by a subset A of M, if the following conditions hold
- Every element $m\in M$ can be written as a product of elements in A:
$m = a_1 \cdot_{M} ... \cdot_{M} a_n, a_i\in A$- No "nontrivial" relations hold in $M$, that is, if $a_1...a_j = a\prime_1 ... a\prime_k$, then this is required by the axioms for monoids.
to me this doesn't explain the word "free"...
Math level: novice