发表机构
Université du Luxembourg; Centro de Investigação em Matemática e Aplicações (CIMA) Departamento de Matemática, Escola de Ciências e Tecnologia da Universidade de Évora; Center for Mathematics and Applications (NOVA Math), NOVA School of Science and Technology (NOVA FCT)（卢森堡大学; 埃武拉大学理学院数学系数学研究与应用中心; 新里斯本科技大学数学与应用中心）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文引入代表语言框架，定义格林语言，用芒树刻画自由逆幺半群的格林语言，证明其形式语言复杂性及增长紧性。
AI 中文摘要
受近期关于群中共轭语言研究的启发，我们引入了一个通用框架，用于研究与有限生成半群上等价关系相关的“代表语言”。在充分发展这一概念后，我们将其具体化到格林关系上，并得到相应的绿色语言，这些语言构成了本文的核心研究对象。对于自由逆幺半群，我们通过芒树描述这些语言，并建立它们之间的若干结构关系，包括对$\mathcal{D}$-、$\mathcal{R}$-、$\mathcal{L}$-和$\mathcal{H}$-语言的特征刻画。随后，我们研究了这些绿色语言在形式语言理论中的复杂性。对于秩至少为2的自由逆幺半群，我们证明$\mathcal{H}$-Geo是上下文无关且共上下文无关的，但不是正则的，而$\mathcal{R}$-Geo、$\mathcal{L}$-Geo和$\mathcal{D}$-Geo既不是上下文无关的也不是共上下文无关的。相反，在单生成情形下，$\mathcal{D}$-Geo是正则的，其余绿色语言是确定型上下文无关的。我们进一步将这些结果与ShortLex语言及自由逆幺半群的增长联系起来，特别证明了$(\text{FIM}_X,X)$是增长紧的。文章最后收集了一些开放问题及未来研究方向。
英文摘要
Motivated by recent work on conjugacy languages in groups, we introduce a general framework for studying \emph{languages of representatives} associated to equivalence relations on finitely generated semigroups. After developing this notion in full generality, we particularize it to Green's relations and obtain the corresponding Green languages, which form the central objects of this article. For free inverse monoids, we describe these languages via Munn trees and establish several structural relations among them, including characterizations of the $\mathcal{D}$-, $\mathcal{R}$-, $\mathcal{L}$-, and $\mathcal{H}$-languages. We then investigate the formal language-theoretic complexity of these Green languages. For free inverse monoids of rank at least two, we show that $\mathcal{H}$-Geo is context-free and co-context-free but not regular, while $\mathcal{R}$-Geo, $\mathcal{L}$-Geo, and $\mathcal{D}$-Geo are neither context-free nor co-context-free. In contrast, in the monogenic case, $\mathcal{D}$-Geo is regular and the remaining Green languages are deterministic context-free. We further relate these results to the ShortLex language and the growth of free inverse monoids, proving in particular that $(\text{FIM}_X,X)$ is growth tight. The article concludes with a collection of open problems and directions for future research.
