Presentations of monoids

Question

Is it true that every f.g. monoid is a quotient of a free one by some relations ? Given a monoid $M$ how can I construct the data in $$\Sigma^*/\sim~$$ ?

Answer 1

Let $A$ be an algebraic structure, and $(a_i)_{i\in I}$ a set of generators for $A$. Then $A$ can be presented as the quotient of the free algebra on generators $(x_i)_{i\in I}$ by all the relations which hold between the generators of $A$: $t(x_{i_1},\dots,x_{i_n})\sim t'(x_{j_1},\dots,x_{j_m})$ whenever $t$ and $t'$ are terms such that $t(a_{i_1},\dots,a_{i_n})= t'(a_{j_1},\dots,a_{j_m})$ holds in $A$.

Now for every algebraic structure $A$, we can take the entire underlying set of $A$ as a set of generators. This shows that every $A$ is a quotient of a free algebra, as Zhen Lin noted in the comments. When $A$ is infinite, this construction presents $A$ as a quotient of an infinitely generated free algebra. On the other hand, if $A$ is finitely generated, we can pick a finite generating set for $A$ - then the construction above presents $A$ as a quotient of a finitely generated free algebra.