AI 中文总结
研究对划分、平面划分等多种图幺半群及其扭曲图幺半群的对合进行分类，刻画产生星正则或正则星幺半群结构的对合，发展扭曲积对合理论，给出相关结果如布劳尔幺半群的特殊情况及整数上扭曲图幺半群的性质等。
AI 中文摘要
我们对所有最常研究的图幺半群（即划分幺半群、平面划分幺半群、部分布劳尔幺半群、莫茨金幺半群、布劳尔幺半群和坦珀利 - 利布幺半群）的对合进行分类，并刻画那些产生星正则或正则星幺半群结构的对合。然后，针对相关的扭曲图幺半群，就规范的浮点数计数扭曲和最近发现的基于秩的扭曲完成相同的程序。这需要发展扭曲积的对合的一般理论。我们的一些结果相当出乎意料。例如，布劳尔幺半群关于其许多对合是星正则的，但只有一个对合使其成为正则星幺半群。我们还将看到整数上的扭曲图幺半群总是星正则的，从而提供了星正则幺半群的新的且非常自然的例子。在此过程中，我们还（必然地）获得了一些具有独立研究价值的结果；具体来说，我们对莫茨金幺半群和坦珀利 - 利布幺半群（因此也对平面划分幺半群）的自同构进行分类，并表明我们所有的图幺半群一般都有平凡中心。
英文摘要
We classify the involutions of all of the most well-studied diagram monoids -- namely the partition, planar partition, partial Brauer, Motzkin, Brauer and Temperley--Lieb monoids -- and characterise those that give rise to star-regular or regular star-monoid structures. We then complete the same program for the associated twisted diagram monoids, with respect to both the canonical float-counting twisting, and the recently-discovered rank-based twisting. This necessitates developing a general theory of involutions of twisted products. Some of our results were quite unexpected. For example, a Brauer monoid is star-regular with respect to many of its involutions, but only a regular star-monoid for one of them. We will also see that twisted diagram monoids over the integers are always star-regular, thereby providing new and very natural examples of star-regular monoids. Along the way, we also obtain (by necessity) a number of results of independent interest; specifically, we classify the automorphisms of the Motzkin and Temperley--Lieb monoids (and hence also of the planar partition monoids), and we show that all of our diagram monoids generically have trivial centre.
Comments47 pages, 4 figures
