What's the groupification of the cardinal numbers (without using AC)? The set-theoretic cardinal numbers form an abelian monoid under addition (ignoring size problems). Under the axiom of choice, given two cardinal numbers $c$ and $d$, there exists a cardinal number $e$ such that $c + e = d + e = e$. We now determine its groupification (or "Grothendieck group"): Considering formal differences, $c - d = (c + e) - (d + e) = e - e = 0$. So the groupification of the cardinal numbers is the trivial group.
What happens if we don't assume the Axiom of Choice? Things don't seem as easy because the cardinal numbers may only form a Partially Order Set (thanks to the CSB theorem). I don't know if they satisfy other properties in general.

Even without the axiom of choice, $x+x\cdot\aleph_0=x\cdot\aleph_0$. In fact, $x+x=x$ if and only if $x\cdot\aleph_0=x$.
So, given any two sets $C$ and $D$, take $E=(C\cup D)\times\omega$. Then, let's denote the cardinals with the lowercase letters, we get that $c+e=d+e=e$.
