发表机构
Bar-Ilan University; University of California-Berkeley（巴伊兰大学; 加州大学伯克利分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究有限简单图的Margolis-Rhodes幺半群的结构与复杂性，给出路径和圈的组合枚举，并证明圈长至少4或路径长至少13时复杂性为2。
AI 中文摘要
我们研究了有限简单图G的Margolis-Rhodes幺半群MR(G)的结构、组合、理想理论和Krohn-Rhodes复杂性，其中G在拓扑上被视为一维单纯复形。除了完整的幺半群，我们还考察了一些子半群，包括St(G)（由完全逆像是边或空集的条件定义）和Inj(G)（所有部分一一连续函数的幺半群）。我们给出了路径和圈上的显式组合枚举和结构。我们计算了Green关系，特别表明正则J-类的偏序与G的诱导子图的偏序集同构。最后，我们将这些结构不变量应用于Krohn-Rhodes复杂性理论。已知Margolis-Rhodes幺半群的复杂性至多为2，而St(G)和Inj(G)的复杂性至多为1。我们证明，对于圈，其Margolis-Rhodes幺半群的复杂性为2当且仅当圈的长度至少为4。对于路径，我们证明如果路径长度至少为13，则其Margolis-Rhodes幺半群的复杂性为2。
英文摘要
We investigate the structural, combinatorial, ideal-theoretic and Krohn-Rhodes complexity of the Margolis-Rhodes monoid, MR(G), of a finite simple graph G, viewed topologically as a 1-dimensional simplicial complex. Alongside the full monoid, we examine some subsemigroups including St(G), defined by the condition that the full inverse image is an edge or the empty set and Inj(G), the monoid of all partial 1-1 continuous functions. We provide explicit combinatorial enumerations and struture for paths and cycles. We compute Green's relations showing in particular that the partial order of regular J-classes is isomorphic to the poset of induced subgraphs of G. Finally, we apply these structural invariants to Krohn-Rhodes complexity theory. It is known that the Margolis-Rhodes monoid has complexity at most 2 and 1 for St(G) and Inj(G). We show that for cycles the complexity of its Margolis-Rhodes monoid is 2 if and only if the cycle is of length at least 4. For paths, we prove that the complexity of its Margolis-Rhodes monoid is 2 if the path length is at least 13.
