Does a monoid $M$ freely generated by a subset $A$ mean the same thing as the free monoid $M(A)$ on $A$? For example define $A = \{a, b, ab, ba\}$ and monoid $A^*=\{\text{words over }A\}$. Then we see that $A^*$ satisfies the definition of free monoid (having universal mapping property) but it's not a monoid freely generated by $A$ as the word $aba$ can be expressed in two ways. If they don't mean the same thing, what's the relation between them then? Since they seem share a similar name. Thank you very much!
Relation between free monoid and freely generated monoid
abstract-algebracategory-theorymonoid
Best Answer
In your example, you start with the alphabet $X = \{a,b\}$. The free monoid over $X$ is indeed $X^*$, the set of words on the alphabet $X$. Next, you consider the set $A = \{a, b, ab, ba\}$. As you observed, the monoid generated by $A$ is no longer free. Actually, a set $C$ which is the basis of a free monoid is called a code. If you want to learn everything on codes, I recommend you the book:
J. Berstel, D. Perrin and Ch. Reutenauer, Codes and Automata, volume 129 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, 2009. 634 pages.