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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.02326 |
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Table of Contents:
- Given a special biserial algebra $Λ$ over an algebraically closed field, let $\mathrm{rad}_Λ$ denote the radical of its module category. The authors showed with Sinha that the stable rank of a special biserial algebra $Λ$, i.e., the least ordinal $γ$ satisfying $\mathrm{rad}_Λ^γ=\mathrm{rad}_Λ^{γ+1}$, is strictly bounded above by $ω^2$. 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 $ω$, 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.