Enumeration of idempotents in planar diagram monoids

Abstract

We classify and enumerate the idempotents in several planar diagram monoids: namely, the Motzkin, Jones (a.k.a. Temperley-Lieb) and Kauffman monoids. The classification is in terms of certain vertexand edge-coloured graphs associated to Motzkin diagrams. The enumeration is necessarily algorithmic in nature, and is based on parameters associated to cycle components of these graphs. We compare our algorithms to existing algorithms for enumerating idempotents in arbitrary (regular * -) semigroups, and give several tables of calculated values.Section 3, that calculate the number of idempotents in the Jones, Kauffman and Motzkin monoids; C++ implementations of these algorithms can be found at [30]. Finally, Section 5 gives several tables of calculated values, including comparative run-times of the various algorithms.The reader is referred to the monographs of Higgins [19] and Howie [20] for background on semigroups in general, and to the introduction of [9] and references therein-in particular to foundational articles of Jones [23], Martin [28] and Halverson and Ram [18]-for background and relevant detail on the partition, Brauer and partial Brauer monoids. Existing algorithmsIn this section, we describe two existing approaches to counting idempotents in semigroups such as those we study in this article. The first applies to any finite semigroup, while the second applies to any finite regular * -semigroup; see below for the definitions. In Section 5, we will discuss the performance of these two approaches when applied to the diagram monoids we are concerned with, and we will compare them with the new algorithms presented in Section 4.The first approach to counting idempotents in an arbitrary finite semigroup S is simple: create the elements of S, and then check if x 2 = x for each x ∈ S; see Algorithm 1. If the semigroup S is generated by A ⊆ S, then S can be enumerated using the Froidure-Pin Algorithm [16]. If the complexity of multiplying elements in S is assumed to be constant, then the complexity of the Froidure-Pin Algorithm is O |S||A| , so Algorithm 1 has complexity O |S||A| + |S| = O |S||A| .Algorithm 1 Count idempotents in a semigroup S 1: n := 0 2: for x ∈ S do 3:if x 2 = x then 4:n ← n + 1 5: return n The approach just described requires the creation of each element of S in order to check which elements are idempotents. When S is very large, this can be impractical, in terms of both space and time. The second approach improves on the first in the case that S is a regular * -semigroup. To describe it, we must first recall some background.Recall from [34] that a semigroup S is a regular * -semigroup if there is a unary operation * : S → S such that x * * = x, (xy) * = y * x * and xx * x = x for all x, y ∈ S. For the remainder of this section, we fix a finite regular * -semigroup S. Recall that Green's relations R, L , J , H and D are defined on S as follows. Let x, y ∈ S. We say that x R y if xS = yS, that x L y if Sx = Sy, and that x J y if SxS = SyS. The relation H is defined to be the intersection of R and...

Identities of the Kauffman Monoid $\mathcal{K}_4$ and of the Jones monoid $\mathcal{J}_4$

“…Kauffman monoids play an important role in knot theory, low-dimensional topology, topological quantum field theory, quantum groups, etc. As algebraic objects, these monoids belong to the family of so-called diagram or Brauer-type monoids that originally arose in representation theory [Brauer, 1937] and have been intensively studied from various viewpoints over the last two decades; see, e.g., [Auinger, 2012[Auinger, , 2014Auinger et al, 2012Auinger et al, , 2015East, 2017, 2018;Dolinka et al, 2015Dolinka et al, , 2019East, 2011aEast, ,b, 2014aEast, ,b, 2018East, , 2019aEast and FitzGerald, 2012;East and Gray, 2017;East et al, 2018;FitzGerald and Lau, 2011;Mazorchuk, 2006, 2007;Lau and FitzGerald, 2006;Maltcev and Mazorchuk, 2007;Mazorchuk, 1998Mazorchuk, , 2002 and references therein.…”

Presentations for Temperley-Lieb algebras

“…Our method also involves replacing the algebra with a monoid, but this time with a finite one, the so-called Temperley-Lieb monoid, TL n . This monoid is sometimes called the Jones monoid in the literature, and denoted J n ; see for example [2,5,8,9], and especially [18] for a discussion of naming conventions. The main innovation in our proof is in the use of two apparently new submonoids L n and R n of TL n ; roughly speaking, these each capture half of the complexity of TL n itself, and we have a natural factorisation TL n = L n R n (Proposition 4.1).…”

Classification of congruences of twisted partition monoids

“…The above connection with semigroup algebras was formalised by Wilcox [60], but the idea has its origins in the work of Jones [35] and Kauffman [40]; see also [33]. Partition monoids, and other diagram monoids, have been studied by many authors, as for example in [2,4,6,[17][18][19]21,23,29,42,47,50,53]; see [22] for many more references. Studies of twisted diagram monoids include [5, 7, 11, 14-16, 41, 43].…”