自由逆幺半群中的共轭语言

中文摘要 AI 辅助

我们开启对自由逆幺半群中共轭语言的研究。受群中共轭语言概念以及半群中共轭研究的启发，我们引入共轭类最短代表元的语言，并针对秩至少为2的自由逆幺半群，在自然的共轭概念下对其展开研究。我们证明，与自由群中该语言为正则语言的情况相反，自由逆幺半群中的该语言既非上下文无关也非余上下文无关；在单生成元情形下，该语言是上下文无关的。我们表明，这种非上下文无关性源自具有非平凡共轭类的元素。我们定义如下等价关系：所有具有非平凡共轭类的元素彼此相关，具有平凡共轭类的元素仅与自身相关，称该关系为UConj。我们通过提供显式的上下文无关文法，证明由代表共轭类平凡的元素的测地线构成的语言是上下文无关的。对于群，我们证明：若群是双曲的，则UConj类的最小代表元语言是正则的；对于具有标准生成元的直角Artin群，该语言是分段可测试的；对于virtually abelian群，我们证明存在一个生成元集使得该语言是分段排除的，并给出一个virtually abelian群的例子，其存在一个生成元集使得该语言不是正则的。

英文摘要

We initiate the study of conjugacy languages in free inverse monoids. Motivated by the notion of conjugacy languages in groups and the study of conjugacy in semigroups, we introduce the language of shortest representatives of conjugacy classes and study it for free inverse monoids of rank at least $2$ under the natural notion of conjugacy. We show that, contrary to the free group case, where this language is regular, in a free inverse monoid it is neither context-free nor co-context-free. In the monogenic case, this language is context-free. We show that this non-context-freeness comes from elements with nontrivial conjugacy classes. We define an equivalence relation as follows: all elements with a nontrivial conjugacy class are related and elements with trivial conjugacy class are only related to themselves. We call this relation $\mathrm{UConj}$. We show that the language consisting of geodesics representing elements whose conjugacy class is trivial is context-free by providing an explicit context-free grammar generating it. For groups, we show that the language of minimal representatives of $\mathrm{UConj}$ classes is regular if the group is hyperbolic and for right-angled Artin groups with the standard generating set, it is piecewise testable. For virtually abelian groups, we show that there is a generating set for which this language is piecewise excluding and exhibit an example of a virtually abelian group admitting a generating set for which this language is not regular.

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preliminary version, 24 pages