Two-generator minimal common left multiple

PDF p. 19: Lemma 4.15; (4.16)

Let $I$ be an associate of $J$ and let distinct $a,b\in I^c$ satisfy $\mathrm{\mid }{\mathrm{\Phi }}_{I+a+b}\setminus {\mathrm{\Phi }}_I\mathrm{\mid }<\mathrm{\infty }$. There exist an associate $K$ and $v_{a,b,I}\in {\mathrm{H}\mathrm{o}\mathrm{m}}_{\mathcal{B}\mathcal{H}}(I,K)$ such that $v_{a,I},v_{b,I}$ are left divisors of $v_{a,b,I}$ in $W$. It has exactly two standard expressions, of the same length, one starting with $v_{a,I}$ and one with $v_{b,I}$. Any Brink–Howlett morphism from $I$ left-divisible by both $v_{a,I},v_{b,I}$ is left-divisible by $v_{a,b,I}$. Thus $v_{a,b,I}$ is minimal both for left divisibility and for standard-expression length among such morphisms.

More explicitly, put $T=I\cup \{a,b\}$, and let $D$ be the union of the connected components of the diagram induced on $T$ that meet $\{a,b\}$. Then $W_D$ is finite, and we choose the morphism above to be $v_{a,b,I}=(w_D{w}_{D\cap I}{)}^{-1}$, with $K$ the associate target supplied by the finite-relative-longest construction. Here $w_L$ denotes the longest element of the finite parabolic subgroup $W_L$.

The article cites Brink–Howlett (1999), p. 326 Type R2, Proposition 2.4 and Lemma 4.1. That source was not read; an earlier scaffold reference to “Lemma 7” did not identify the article’s actual citation and is not used. The five intermediate items instead use independently inspected Dyer–Gimenez Lemma 1.25, Lemma 3.12(a), Theorem 1.22(b) and the exactly-two assertion in §1.27, with explicit reading limits. Prefix minimality and the initial factors are derived independently of the malformed indexing in the latter paragraph. No defect in the original article is asserted.