Automating the stable rank computation for special biserial algebras

Given a special biserial algebra $\mathrm{\Lambda }$ over an algebraically closed field, let ${\mathrm{r}\mathrm{a}\mathrm{d}}_{\mathrm{\Lambda }}$ denote the radical of its module category. The authors showed with Sinha that the stable rank of a special biserial algebra $\mathrm{\Lambda }$, i.e., the least ordinal $\gamma$ satisfying ${\mathrm{r}\mathrm{a}\mathrm{d}}_{\mathrm{\Lambda }}^{\gamma }={\mathrm{r}\mathrm{a}\mathrm{d}}_{\mathrm{\Lambda }}^{\gamma +1}$, is strictly bounded above by ${\omega }^2$. We use finite automata to give simple algorithmic proofs, complete with their time complexity analyses, of two key ingredients in the proof of this result–the first one states that certain linear orders called hammocks associated with such algebras are \emph{finite description linear orders}, i.e., they lie in the smallest class of linear orders that contains finite linear orders and $\omega$, and that is closed under isomorphisms, order-reversals, binary sums, co-lexicographic products and finitary shuffles. We also document a complete proof of the result that the class of order types(=order-isomorphism classes) of finite description linear orders coincides with that of languages of finite automata under inorder.