AI 中文总结
本文用有限树编码零关系代数的树模，构造识别树砖的确定性树自动机，并证明砖性与无自重叠等价，同时通过局部着色压缩箭向字母表，最优颜色数为入度与出度最大值。
AI 中文摘要
设 $\Lambda=KQ/I$ 为有限维零关系代数。我们用由 $Q$ 的箭向及其形式逆标记的有限有根树来编码 $\Lambda$ 上的 Crawley--Boevey 树模，并构造一个确定性有限自底向上树自动机，该自动机恰好识别这些编码。当一棵被接受的树没有非平凡因子-像自重叠时，我们将其定义为自动机诱导的树砖，并利用 Crawley--Boevey 的图映射基证明这等价于关联树模的砖性。我们还引入了 $Q_1$ 的局部着色，并表明箭向字母表可以在不改变树数据、图映射或砖性质的情况下被压缩。这种压缩的最优颜色数是 $Q$ 的入度和出度的最大值。最后，我们提出一个开放问题：仅由砖构成的树语言是否为正则语言。
英文摘要
Let $Λ=KQ/I$ be a finite-dimensional zero-relation algebra. We encode Crawley--Boevey tree modules over $Λ$ by finite rooted trees labelled by arrows of $Q$ and their formal inverses, and construct a deterministic finite bottom-up tree automaton recognizing exactly these encodings. We define an accepted tree to be an automata-induced tree brick when it has no non-trivial factor--image self-overlap, and use Crawley--Boevey's graph-map basis to prove that this is equivalent to brickness of the associated tree module. We also introduce local colourings of $Q_1$ and show that the arrow alphabet can be compressed without changing the tree data, graph maps, or brick property. The optimal number of colours for such a compression is the maximum of the in-degree and out-degree of $Q$. We conclude by asking whether the tree language consisting only of bricks is regular.
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