Quotient Monoid

The quotient of a Monoid $(M,\cdot,1)$ by a Monoid Congruence $\sim\;\subseteq M\times M$ is the universal monoid $\iota:M\to M/\sim$ with the property that $x\sim y\Rightarrow\iota(x)=\iota(y)$; EG, for any other such monoid $f:M\to N$, there exists a unique Monoid Homomorphism $[f]:M/\sim\;\to N$ making the following diagram commute:

Constructions

As our notation suggests, the quotient $M/\sim$ of the underlying set of $M$ serves as a generic way of producing quotient monoids.

When $M$ is a Commutative Monoid, we can quotient by an arbitrary Submonoid $N\subseteq M$, as we can construct a congruence