Very little is known about the structure of the intersection of two $k$-generated monoids of words, even for $k=3$. Here we investigate the case of $k$-maximal monoids, that is, monoids whose basis of cardinality $k$ cannot be non-trivially decomposed into at most $k$ words. We characterize the intersection in the case of two $3$-maximal monoids.

中文翻译：

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$3$-Maximal Submonids 的交集

关于两个由 $k$ 生成的词的幺半群的交集的结构知之甚少，即使是 $k=3$。在这里，我们研究了 $k$-maximal monoids 的情况，即基数 $k$ 的基础不能被非平凡地分解为最多 $k$ 个词的幺半群。我们在两个 $3$-maximal 幺半群的情况下描述了交集。