Every monoid admits a surjection from a free monoid?

Question

How can I show that every monoid $M$ admits a surjection from a free monoid $F(X) \rightarrow M$ ?

Answer 1

The set $X$ is called an alphabet. You can take $M$ itself as alphabet and then the morphism $\mu : F(M)\rightarrow M$ is just the multiplication.

To be more specific, the structure of monoid is the data of a triplet $(M,*,1_M)$ where

• $M$ is a set
• $*$ is an associative internal law in $M$,
• $1_M$ is the neutral

The elements of $F(M)$ are strings $m_1.m_2.\cdots .m_k$ where the $m_i\in M$, then $$ \mu(m_1.m_2.\cdots .m_k)=m_1*m_2*\cdots *m_k $$ the stars standing for the multiplication inside the monoid $M$.