Congruence lattices of finite diagram monoids

Abstract

We give a complete description of the congruence lattices of the following finite diagram monoids: the partition monoid, the planar partition monoid, the Brauer monoid, the Jones monoid (also known as the Temperley-Lieb monoid), the Motzkin monoid, and the partial Brauer monoid. All the congruences under discussion arise as special instances of a new construction, involving an ideal I, a retraction I → M onto the minimal ideal, a congruence on M , and a normal subgroup of a maximal subgroup outside I.Roughly speaking, if α ∈ P n , then α * is obtained by reflecting (a graph representing) α in the horizontal axis midway between the two rows of vertices. It is easy to see that α * * = α, (αβ) * = β * α * , αα * α = α, for all α, β ∈ P n . It follows that P n is a regular * -semigroup, in the sense of Nordahl and Scheiblich [35], with respect to this operation. We have several obvious identities, such as dom(α * ) = codom(α) and ker(α * ) = coker(α). This symmetry/duality will allow us to shorten several proofs. Other diagram monoidsIn this subsection, we introduce a number of important submonoids of the partition monoid P n . Following [33], the Brauer and partial Brauer monoid are defined by B n = {α ∈ P n : all blocks of α have size 2} and PB n = {α ∈ P n : all blocks of α have size at most 2}, Green's equivalences R, L , J , H and D reflect the ideal structure of a semigroup S, and are the fundamental structural tool in semigroup theory. They are defined as follows. We write S 1 = S if S is a monoid; otherwise S 1 is the monoid obtained from S by adjoining an identity element to S. Then, for a, b ∈ S,further, H = R ∩ L , and D is the join R ∨ L : i.e., the least equivalence containing R and L . It is well known that D = R • L = L • R for any semigroup S, and that D = J when S is finite (as is the case for all semigroups considered in this article). If K is any of Green's relations, and if a ∈ S, we write K a = {b ∈ S : a K b} for the K -class of a in S. The set S/J = {J a : a ∈ S} of all J -classes of S is partially ordered as follows. For a, b ∈ S, we say that J a ≤ J b if a ∈ S 1 bS 1 . If T is a subset of S that is a union of J -classes, we write T /J for the set of all J -classes of S contained in T . The reader is referred to [7, Chapter 2], [21, Chapter 2] or [36, Appendix A] for a more detailed introduction to Green's relations.Green's equivalences on all diagram monoids considered in this article are governed by (co)domains, (co)kernels and ranks, as specified in the following proposition. For P n this was first proved in [40], though the terminology there was different. For the other monoids see [10, Theorem 2.4] and also [16][17][18]40]. The proposition will be used frequently throughout the paper without explicit reference.Proposition 2.1. Let K n be any of the monoids P n , PB n , B n , PP n , M n , I n , J n , O n . If α, β ∈ K n , then (i) α R β ⇔ dom(α) = dom(β) and ker(α) = ker(β), (ii) α L β ⇔ codom(α) = codom(β) and coker(α) = coker(β),Remark 2.2. A number of consequences and simplifications ar...

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).…”

Congruences on infinite partition and partial Brauer monoids

“…While there are some intriguing parallels between the theories of diagram and transformation monoids, the results of [17] also highlighted some striking differences. For example, while the congruence lattices of finite full transformation monoids form chains under inclusion, the same is not true for any of the diagram monoids studied in [17]. For each of these, the lattice has a prism-shaped lower part, the existence of which is partly explained by structural properties of the minimal ideal.…”

“…The article [17] initiated the study of congruences on diagram monoids. A congruence on an algebraic structure S is an equivalence relation compatible with all the basic operations on S; the set Cong(S) of all such congruences forms a lattice under inclusion.…”

“…Nevertheless, congruences are the tools for constructing quotient structures, defining kernels of homomorphisms/representations, and so on. The main results of [17] are the complete descriptions of the congruence lattices of finite partition, planar partition, Brauer, partial Brauer, Temperley-Lieb and Motzkin monoids. Inspiration was drawn from Mal'cev's classic study of full transformation monoids [1,40], an excellent account of which may be found in [7,Section 10.8].…”

“…Inspiration was drawn from Mal'cev's classic study of full transformation monoids [1,40], an excellent account of which may be found in [7,Section 10.8]. While there are some intriguing parallels between the theories of diagram and transformation monoids, the results of [17] also highlighted some striking differences. For example, while the congruence lattices of finite full transformation monoids form chains under inclusion, the same is not true for any of the diagram monoids studied in [17].…”