Involutions of (twisted) diagram monoids

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.