Independent arbitrary-rank factorization and simple-class inputs

Every arrow $(B,x,A)$ in $\mathcal{B}$ admits a factorization into Brink–Howlett generators $g_n\cdots g_1$ satisfying $\mathrm{\ell }(x)={\sum }_i\mathrm{\ell }(g_i)$. Every Brink–Howlett generator $g$ satisfies $\mathrm{\mid }C(g)\mathrm{\mid }=1$. Both assertions hold for arbitrary rank.

Import the first assertion from Dyer–Gimenez, Proposition 3.13, with the notation and component condition of Lemma 1.25 and §1.27. Import the second assertion from Theorem 1.22(b): in the real compression of the standard root-action groupoid-set, the Brink–Howlett generators are exactly the simple morphisms. Section 1.19 defines simple by the inversion set having cardinal one. By the preceding lemma, this inversion set is precisely $C(g)$. Section 1.4 explicitly excludes a standing finite-rank assumption. These two statements are external inputs. Their proofs are not claimed to have been reconstructed locally.