Automating the stable rank computation for special biserial algebras

TL;DR

This paper introduces automata-based algorithms to analyze the stable rank of special biserial algebras, providing bounds and characterizations of associated linear orders with implications for algebraic and automata theory.

Contribution

It presents a novel automata-theoretic approach to compute and analyze the stable rank and related linear orders of special biserial algebras, with complexity analysis.

Findings

Stable rank of special biserial algebras is bounded by ω^2.

Certain linear orders called hammocks are finite description linear orders.

Order types of these linear orders match languages recognized by finite automata.

Abstract

Given a special biserial algebra $\Lambda$ over an algebraically closed field, let $\mathrm{rad}_\Lambda$ denote the radical of its module category. The authors showed with Sinha that the stable rank of a special biserial algebra $\Lambda$, i.e., the least ordinal $\gamma$ satisfying $\mathrm{rad}_\Lambda^\gamma=\mathrm{rad}_\Lambda^{\gamma+1}$, 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.