Is there an analog to Grothendieck group for general monoids?
I imagine such thing could be constructed but I have not found this construction in standard text so I am not sure if it makes sense. (Or maybe my searching ability is not good enough.)
My idea is as follows. Suppose $M$ is a monoid. Let $M^{-1}$ be another monoid isomorphic to $M$ and let $\varphi:M \to M^{-1}$ be the isomorphism between them. Form the free product $M * M^{-1}$ and define a congruence relation $R$ on $M * M^{-1}$ by equations $m\varphi(m) = 1 = \varphi(m)m$ for $m \in M$, where $1$ is the identity of $M * M^{-1}$, $M$ and $M^{-1}$. $N = (M * M^{-1})/R$ is the desired result.
For $m \in M \cup M^{-1}$ (considering $1$ as the only element in $M \cap M^{-1}$), let $m^{-1} = \varphi(m)$ if $m \in M$ and $m^{-1} = \varphi^{-1}(m)$ if $m \in M^{-1}$. Every element $m$ of $M \oplus M^{-1}$ can be written as a finite product $m = m_1 \ldots m_n$ with $m_i \in M \cup M^{-1}$. Define $m^{-1} = m_n^{-1} \ldots m_1^{-1}$. Then $mm^{-1} \equiv m^{-1}m \equiv 1 \pmod R$, so $N$ is a group.
Suppose $\theta: M \to N$ is the homomorphism $M \to M * M^{-1} \xrightarrow{\text{mod } R} N$. I would expect that this is universal in the sense that every monoid morphism $\psi: M \to P$ with $P$ a group factors through $\theta$. Is this the case? Are there any problems with this construction?
This works just fine.
The fact that a monoid morphism $\psi: M\to P$ to a group factors uniquely through $\theta$ (the uniqueness is very important!) is fairly easy to prove from first principles, but it should be abstractly intuitive as well. The point is that we've just taken $M$, freely adjoined a symbol $m_-$ for each $m\in M$, and taken the quotient by the relations $mm_-=m_- m=1$.