Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

East, James; Gray, Robert D.

doi:10.1016/j.jcta.2016.09.001

Cited by 45 publications

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References 120 publications

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“…We define the two bicomponents of ∆(S, J) as follows: one bicomponent is the collection of all orbits of L -classes of J, the other bicomponent is the collection of all orbits of R-classes of J; the bicomponents of ∆(S, J) partition its vertices into two maximal independent subsets. Note that ∆(S, J) is isomorphic to a quotient of the Graham-Houghton graph of the principal factor of J, as defined in [19,27,30] -in the case that the orbits of L -and R-classes are trivial, these graphs are isomorphic.…”

Section: Maximal Subsemigroups Arising From a Regular J -Class Coverementioning

confidence: 99%

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Maximal subsemigroups of finite transformation and diagram monoids

et al. 2018

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We describe and count the maximal subsemigroups of many well-known monoids of transformations and monoids of partitions. More precisely, we find the maximal subsemigroups of the full spectrum of monoids of order-or orientationpreserving transformations and partial permutations considered by V. H. Fernandes and co-authors (12 monoids in total); the partition, Brauer, Jones, and Motzkin monoids; and certain further monoids.Although descriptions of the maximal subsemigroups of some of the aforementioned classes of monoids appear in the literature, we present a unified framework for determining these maximal subsemigroups. This approach is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors. This allows us to concisely present the descriptions of the maximal subsemigroups, and to more clearly see their common features.

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Section: Maximal Subsemigroups Arising From a Regular J -Class Coverementioning

confidence: 99%

“…Corollary 2. 19. Let S be a finite monoid with group of units G, and suppose there exists a non-empty subset X of S \ G with the property that S = G, x if and only if x ∈ X.…”

Section: Maximal Subsemigroups Arising From Other J -Classesmentioning

confidence: 99%

Maximal subsemigroups of finite transformation and diagram monoids

et al. 2018

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“…However, the ranks of the monoids I n , T n , PT n and P n are constant and very small (all being equal to either 3 or 4, for n ≥ 3), and each monoid may be generated by elements in its top two J -classes; see [2,3,6,18]. The proper ideals of these monoids are all generated by elements in a single J -class; formulae for the ranks of the ideals of these monoids may be found in [4,5,10,19]. By contrast, as we have seen, rank(D n ) = B(n) + n grows rapidly with n, and any generating set for D n or one of its proper ideals must contain elements from all J -classes except the very bottom one.…”

Section: ··· ···mentioning

confidence: 99%

Ranks of ideals in inverse semigroups of difunctional binary relations

East

Vernitski

2017

Semigroup Forum

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The set D n of all difunctional relations on an n element set is an inverse semigroup under a variation of the usual composition operation. We solve an open problem of Kudryavtseva and Maltcev (2011), which asks: What is the rank (smallest size of a generating set) of D n ? Specifically, we show that the rank of D n is B(n) + n, where B(n) is the nth Bell number. We also give the rank of an arbitrary ideal of D n . Although D n bears many similarities with families such as the full transformation semigroups and symmetric inverse semigroups (all contain the symmetric group and have a chain of J -classes), we note that the fast growth of rank(D n ) as a function of n is a property not shared with these other families.

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“…For example, in each of the above-mentioned diagram monoids, the ideals form a chain with respect to containment. Furthermore, the idempotent-generated subsemigroup coincides with the singular part of the Brauer and partition monoids [12,28], and the proper ideals of the Jones, Brauer and partition monoids are idempotent-generated [14]. When it comes to congruences, however, it turns out that the parallels are simultaneously less tight and more subtle.…”

Section: Introductionmentioning

confidence: 99%

Congruence lattices of finite diagram monoids

East

Mitchell

Ruškuc

et al. 2018

Advances in Mathematics

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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...

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Diagram monoids and Graham–Houghton graphs: Idempotents and generating sets of ideals

Cited by 45 publications

References 120 publications

Maximal subsemigroups of finite transformation and diagram monoids

Maximal subsemigroups of finite transformation and diagram monoids

Ranks of ideals in inverse semigroups of difunctional binary relations

Congruence lattices of finite diagram monoids

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