Involutions of (twisted) diagram monoids

Technical Summary: Involutions of (Twisted) Diagram Monoids

Problem Statement
The paper addresses the classification of involutions (self-inverse anti-automorphisms) for a broad class of diagram monoids and their associated twisted products. While the canonical involution (vertical reflection) is well-understood in diagram algebras and monoids, the landscape of all possible involutions, and their specific algebraic properties regarding regularity, remains largely unexplored for both un-twisted and twisted structures. Specifically, the authors aim to:

1. Classify all involutions of the partition ($P_n$), planar partition ($PP_n$), partial Brauer ($PB_n$), Motzkin ($M_n$), Brauer ($B_n$), and Temperley–Lieb ($TL_n$) monoids.
2. Characterize which of these involutions render the monoids star-regular ($\ast$-regular) or regular star-monoids.
3. Extend this classification and characterization to "twisted diagram monoids," which are semigroup algebras of underlying diagram monoids augmented with information about floating components (via the canonical float-counting twisting $\Phi$) or rank-based twisting ($\Psi$).

A central motivation is the observation that while diagram monoids are typically regular $\ast$-monoids (where the involution maps elements to their von Neumann inverses), their twisted counterparts often fail to be regular. The authors investigate whether these twisted structures retain weaker regularity properties, specifically star-regularity.

Methodology
The authors employ a two-pronged approach, developing general theory for twisted products before applying it to specific diagram monoids.

• General Theory of Twisted Products: The paper establishes a framework for the twisted product $T = M \times_q^\Phi S$, where $S$ is a monoid, $M$ is a commutative monoid, and $\Phi: S \times S \to \mathbb{N}$ is a twisting map. The authors introduce and utilize properties such as tightness, unitality, and Green-invariance for the twisting $\Phi$. They develop a general classification of involutions for such products, distinguishing between "pure" involutions (where the second coordinate depends only on the second coordinate of the input) and "general" involutions.
• Classification of Automorphisms: To classify involutions, the authors first determine the automorphism groups of the underlying diagram monoids. While automorphisms for $P_n$, $PB_n$, and $B_n$ were known (Mazorchuk), the authors provide new classifications for the planar monoids $PP_n$, $M_n$, and $TL_n$.
• Center Analysis: The authors prove that the centers of these diagram monoids are trivial for sufficiently large $n$. This result is crucial for the general theory, as a trivial center in $S$ implies that all involutions of the twisted product $T$ are pure.
• Regularity Characterization: Using the general classification of involutions, the authors derive necessary and sufficient conditions for $T$ to be star-regular or a regular star-monoid. This involves analyzing the interaction between the involution, the twisting map $\Phi$ (or $\Psi$), and the Moore-Penrose inverses.

Key Contributions and Results

1. Classification of Involutions in Diagram Monoids:

   • For $P_n$, $PB_n$, and $B_n$, involutions are in one-to-one correspondence with self-inverse permutations of the underlying set (conjugates of the canonical involution).
   • For the planar monoids $PP_n$, $M_n$, and $TL_n$, there are generally only two involutions: the canonical one and a 180-degree rotation (denoted $\rho$).
   • The authors characterize which involutions yield star-regular or regular star-monoid structures. Notably, $B_n$ is star-regular for many involutions but a regular star-monoid only for the canonical one.

2. Triviality of the Center:

   • The paper proves that $P_n, PP_n, PB_n, M_n$ (for $n \ge 2$) and $B_n, TL_n$ (for $n \ge 3$) have trivial centers. This is a novel result stated as being of independent interest.

3. General Theory of Twisted Products:

   • The authors classify involutions of $M \times_q^\Phi S$ under the assumption of unitality. They show that if $S$ has a trivial center, all involutions of the twisted product are pure.
   • They provide explicit formulas for Moore-Penrose inverses in star-regular twisted products.

4. Twisted Diagram Monoids (Float-Counting $\Phi$):

   • For twisted monoids over the integers ($M = \mathbb{Z}$), the authors classify all involutions.
   • Unexpected Result: While most twisted diagram monoids are not regular $\ast$-monoids, they are often star-regular. Specifically, twisted diagram monoids over the integers are always star-regular with respect to involutions extending star-regular involutions of the base monoid.
   • Exception: The twisted Brauer monoid $Z \times_1^\Phi B_2$ is unique in that it admits non-pure involutions, and interestingly, it is star-regular with respect to both its pure and non-pure involutions.
   • Regular-starity (the stronger condition where the involution maps to the inverse) is shown to be very rare, holding only in specific low-degree cases.

5. Twisted Diagram Monoids (Rank-Based $\Psi$):

   • The authors apply the same program to the rank-based twisting $\Psi$.
   • Unlike the float-counting case, the rank-based twisting yields a richer structure: there are generally more involutions, and star-regularity and regular-starity are more common.
   • Significant Finding: Over the integers, every rank-based twisted diagram monoid $Z \times_1^\Psi S$ is a regular star-monoid with respect to a specific "negative" involution. This provides new, natural infinite families of regular star-monoids.

Significance and Claims
The paper claims to provide a complete classification of involutions for the most well-studied diagram monoids and their twisted variants. The authors highlight several unexpected findings:

• The distinction between star-regularity and regular-starity is sharp; many twisted monoids are star-regular but not regular star-monoids.
• Twisted diagram monoids over the integers serve as new, natural examples of star-regular monoids, a class of structures introduced by Drazin.
• The rank-based twisting $\Psi$ offers a more robust framework for regularity than the canonical float-counting twisting $\Phi$, yielding infinite families of regular star-monoids.
• The development of a general theory for involutions of twisted products, particularly the identification of "unitality" as a key property weaker than tightness, allows for strong general results even when the twisting is not tight (as is the case for Motzkin and partial Brauer monoids).

The work is presented as a foundational step in understanding the symmetry and regularity properties of diagram algebras and their twisted counterparts, with implications for the study of cellular algebras and the structure of semigroup algebras.