Standard expressions are minimum generator words

For an article arrow $x:I\to K$, Transfer to the root-action Brink–Howlett groupoid identifies the root-action arrow with $(I,x,K):K\to I$ and identifies every article elementary label with a Brink–Howlett generator, in both directions. Reverse the path order of the factors of a standard expression to obtain a composable root-action word with the same label product and factor count. The same reversal applies to arbitrary generator words and their inverse arrows, and is involutive. Hence minimum word lengths in these two conventions agree. This uses the standard-expression convention of Standard expressions and their lengths.

By Every standard expression counts the same parallel classes, a standard expression has $\mathrm{\mid }C(x)\mathrm{\mid }$ factors, where $C(x)$ is taken for the reversed root-action arrow with the same label. Dyer–Gimenez Theorem 1.22(b) makes the real compressed root-action signed groupoid-set principal with Brink–Howlett generators as simple morphisms; §1.19 defines principal by minimum generator length, allowing inverse generators, equal to inversion cardinality. By Parallel classes of roots and finite inversion sets, that cardinality is $\mathrm{\mid }C(x)\mathrm{\mid }$. The correspondence in step 1.1 therefore makes every standard expression a minimum word, with the stated minimum number of factors.

Dyer–Gimenez Lemma 1.25(d), specialized to singleton additions as in §1.27, makes inverse elementary generators elementary in the root-action groupoid. Reverse a root-action expression and invert every factor. This operation is an involution, preserves factor count and composability, and takes words for $x$ to words for $x^{-1}$. A shorter word on either side would give a shorter word on the other. Thus it restricts to the claimed bijection of minimum words.