An automata-based test for bricks over string algebras

TL;DR

This paper introduces a new automaton-based method to analyze string modules over gentle algebras, linking automata theory with algebraic structures to identify brick modules.

Contribution

It develops a decorated automaton called a multi-entry inverse automaton (MIA) to classify brick modules over string algebras, extending previous work on Sturmian words.

Findings

Automaton accepts pointed words representing string modules.

Brick modules correspond to certain equivalence classes of words.

The method generalizes prior results on Sturmian words and gentle algebras.

Abstract

Motivated by the recent work of Deaconu, Mousavand and Paquette on the connection between infinite string bricks for certain gentle algebras and Sturmian words, we develop a decorated version of a deterministic automaton, called a multi-entry inverse automaton (MIA, for short) that accepts pointed words. We then associate an MIA $\mathsf M_{\Lambda\delta}$ over $\{0,1\}$ to a string algebra $\Lambda$, and show that strings over $\Lambda$ can be viewed as certain equivalence classes of the pointed words accepted by $\mathsf M_{\Lambda\delta}$. By defining (weak) brick words over this MIA, we show that a finite/infinite string module (resp. band module) is a brick if and only if every word in the associated equivalence class of pointed binary words is a brick word (resp. a weak brick word) over $\mathsf M_{\Lambda\delta}$. The result of Deaconu et al. follows as an immediate consequence.