In the paper "Schreier split epimorphisms between monoids" by Bourn, Nelson, Martins-Ferreira, Montoli and Sobral, Semigroup Forum June 2014, the authors prove a characterization of groups among monoids. I am hoping for a simpler proof of one direction.

**Definitions.** A *splitting* is a monoid homomorphism $f:A\to B$ with a specified section $s:B\to A$. A splitting is *left homogeneous* if for every $b\in B$, multiplication on the left by $s(b)$ is bijective $\mathrm{Ker}f\to f^{-1}(b)$. Analogously for *right homogeneous*. A splitting is *homogeneous* if it's both left and right homogeneous.

**Remark.** For the characterization, I think it suffices to assume the multiplication maps are surjective, not necessarily bijective.

**Theorem.** For a monoid $B$, TFAE

- $B$ is a group;
- Every splitting $A\overset{s}{\leftrightarrows}B$ over $B$ is homogeneous.

That 1$\implies$2 is straightforward and appears in proposition 3.4. The converse implication 2$\implies$1 is part of corollary 5.7 and involves an ordeal with internal relations.

Is there a simple(r) proof that 2$\implies$1?