The planar modular partition monoid

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The Planar Modular Partition Monoid

by

Nicholas Charles Ham

BEc, BSc (Hons)

Supervised by Des FitzGerald & Peter Jarvis

A thesis submitted in fulfilment of the requirements

for the Degree of Doctor of Philosophy

School of Physical Sciences

University of Tasmania

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Abstract

The primary contribution of this thesis is to introduce and examine the planar modular partition monoid for parameters m, k∈Z>0, which has simultaneously and independently generated interest from other

researchers as outlined within.

Our collective understanding of related monoids, in particular the Jones, Brauer and partition monoids, along with the algebras they generate, has heavily influenced the direction of research by a significant number of mathematicians and physicists. Examples include Schur-Weyl type dualities in representation theory along with Potts, ice-type and Andrew-Baxter-Forrester models from statistical mechanics, giving strong motivation for the planar modular partition monoid to be examined.

The original results contained within this thesis relating to the planar modular partition monoid are: the establishment of generators; recurrence relations for the cardinality of the monoid; recurrence relations for the cardinality of Green’sR,LandDrelations; and a conjecture on relations that appear to present the planar modular partition monoid whenm= 2. For diagram semigroups that are closed under vertical reflections, characterisations of Green’sR,LandHrelations have previously been established using the upper and lower patterns of bipartitions. We give a characterisation of Green’sDrelation with a similar flavour for diagram semigroups that are closed under vertical reflections.

We also give a number of analogous results for the modular partition monoid, the monoid generated by replacing diapses with m-apses in the generators of the Jones monoid, later referred to as the m-apsis monoid, and the join of the m-apsis monoid with the symmetric group.

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Acknowledgements

First and foremost I wish to express my deepest gratitude towards my supervisors, Des FitzGerald and Peter Jarvis, who have provided insightful and encouraging feedback on my progress throughout my candidacy. Furthermore, following an idea put forward to Peter by Bertfried Fauser, they originally suggested that examining the monoid generated when replacing diapses with triapses in the generators of the Jones and Brauer monoids may be a suitable topic for a doctoral thesis, and indeed it revealed itself to be both a fruitful and enjoyable topic to get my hands dirty with. It also cannot be overstated how dramatically my ability to expose my own mathematical thoughts in a clear, fluent, unambiguous and grammatically correct manner has improved under the guidance of Des, of which I am especially thankful.

I am immensely grateful to James East who provided a number of insightful suggestions, references and counterexamples at various stages, and who has been beyond generous with his own time during the final year of my candidacy. I was further included and felt like I was treated as a respected colleague within some wider research circles throughout my candidacy - in regards to which I would particularly like to mention Igor Dolinka, James Mitchell, Attila Egri-Nagy and Nicholas Loughlin.

My time at the University of Tasmania, throughout undergrad and candidacy, was made all the more enjoyable and constructive both by opportunities afforded to me and by interactions with a number of people. In that regard I give thanks to: Jet Holloway and JJ Harrison for their help catching up to speed with mathematics classes after having turned up to university somewhat behind and only having decided to pick mathematics up in my second year; Mardi Dungey, Sarah Jennings and Simon Wotherspoon who first gave me the opportunity to obtain some research experience during my undergraduate years; Barry Gardner who very kindly took three units on top of his usual workload during our honours year; Karen Bradford, Michael Brideson, Kumudini Dharmadasa, Tracy Kostiuk and Jing Tian for the opportunity to gain teaching experience along with their assistance with doing so; Arwin Kahlon for putting up with my friendly banter about physicists; and Jeremy Sumner who took the time to provide a number of useful comments on my honours research despite it being somewhat outside his area of expertise.

Identifying a number of the original results contained within this thesis was made substantially easier by the online encyclopaedia of integer sequences [53] and the free software package GAP [27] for com-putational discrete algebra. I would especially like to mention James Mitchell for the GAP semigroups package [19] which was not only useful for obtaining computational results, but was also used to generate the majority of the figures.

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with financial assistance for travelling to the annual Victorian Algebra Conference. On that note, both Marcel Jackson and Brian Davey have been notably helpful towards ensuring all Australian post graduate algebra students feel both included and valued within the Australian algebra community.

Finally, I would like to express my eternal gratitude towards my family, especially my parents and my brother, in particular for both their support and willingness to tolerate my own stubbornness, which would be almost impossible to understate at times though has also contributed significantly to many of my most cherished achievements.

To the best knowledge and belief of the author this thesis contains no material which has been accepted for a

degree or diploma by the University of Tasmania or any other institution, no material previously published or

written by another person, except by way of background information where duly acknowledged, and no material

that infringes copyright.

This thesis may be made available for loan, copying and communication in accordance with the Australian

Copyright Act 1968.

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Chapter 1 Introduction

As with any increasingly vast body of human knowledge, to give an exhaustive historical account would quickly turn into its own body of work. Nevertheless the author has attempted to provide the reader with a historical account of where, how and why the diagram semigroups relevant to this thesis arose and have since blossomed into a thriving area of research in their own right.

In 1896, Eliakim Moore [49] established a presentation by way of generators and relations for the symmet-ric groupSk. Later in 1927, Issai Schur [55] identified that a duality, now famously known as Schur-Weyl duality, holds between the symmetric group algebra C[Sk] and the general linear group GLn(C)

(in-vertible n×n matrices over the complex fieldC with a fixed ordered basis). Ten years later Richard

Brauer [8] identified that an analogous duality holds between what is now-known as the Brauer algebra

Cξ[Bk] and the orthogonal group On(C), it would be almost impossible to overstate the influence that

the identification of these dualities has since had on the direction of research by a significant number of mathematicians and physicists.

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The Partition monoid was then introduced independently by Jones [37] and Martin [47] after both considered generalisations of the Temperley-Lieb algebra and Potts models from statistical mechanics. Jones [37] did so when considering the centraliser of the tensor representation of the symmetric group Sn (when treated as the group of alln×npermutation matrices) in the endomorphism ring End(V⊗k), whereV is ann-dimensional vector space on whichSn acts diagonally on V⊗k.

Graham and Lehrer [28] introduced and provided a unified framework for obtaining a great deal of information about the representation theory of what are referred to ascellular algebras. East [14] then showed that under certain compatibility assumptions, the semigroup algebra of an inverse semigroup is cellular if the group algebras of its maximal subgroups are cellular, while Guo and Xi [31] and Wilcox [63] later examined cellularity for twisted semigroup algebras, allowing the cellularity of such algebras to be established using the theory on cellular algebras introduced in [28].

Motivated to generalise the aforementioned duality results from Jones [37], Tanabe [59] examined the centraliser algebra in the endomorphism ring of V⊗k _{on which the unitary reflection groups of type} G(m, p, n) act diagonally, whereG(m, p, n) is an index-psubgroup ofG(m,1, n), andG(m,1, n) is a group ofn×nmatrices whose non-zero entries aremth roots of unity. Note that the unitary reflection groups were introduced by Shephard and Todd [56]. A year later, FitzGerald and Leech [26] independently examined the structure of both the monoid of block bijections and monoid of uniform block bijectionsFk, which has previously been described as the largest factorisable inverse submonoid of the dual symmetric inverse semigroupI∗

k (see [24]). Kosuda [38] then identified that in the case wherem > k, the centraliser algebra considered by Tanabe corresponds to the monoid of uniform block bijectionsFk.

Fitzgerald [24] and Kosuda [38] both independently gave the same presentation by way of generators and relations for the monoid of uniform block bijectionsFk. While the aforementioned presentation ofFk was economical in terms of the number of generators used, the inherent symmetry possessed by the monoid of uniform block bijections Fk was not reflected in the defining relations. With the addition of more generators, Kosuda [38] and Kudryavtseva and Mazorchuk [45] also independently gave an equivalent set of relations that better reflected the symmetry possessed, and using the same set of generators East [15] also arrived at an equivalent set of defining relations. Kosuda [41] further constructed a complete set of representatives of the irreducible representations of the algebra of uniform block bijections_Cξ_[F

k].

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presentation by way of generators and relations for the modular partition monoidMm_k for all 1≤m≤k.

A number of presentations by way of generators and relations have recently been given for diagram semigroups: Halverson and Ram [33] gave presentations for the planar partition monoid PPk and the partition monoidPk, with their exposition since becoming reasonably famous as a survey-style treatment of the partition algebras; Kudryavtseva and Mazorchuk [45] gave presentations for the Brauer monoid Bk, the partial Brauer monoid and, as previously mentioned, the monoid of uniform block bijections

Fk; Posner, Hatch and Ly [52] have suggested a presentation of the Motzkin monoid, though the paper appears to remain in preprint; and Easdown, East and FitzGerald [13] gave a presentation for the dual symmetric inverse monoid, which may also be described as the monoid of block bijections.

Motivated by Fauser, Jarvis and King’s work on symmetric functions and generalised universal characters [20, 21, 22, 23], Fauser put the idea forward to Jarvis that it may be fruitful to consider the consequences of replacing the diapses in the generators of the Jones and Brauer monoids with triapses. When the author sought suggestions from potential supervisors on possibly suitable doctoral research topics, Jarvis shared this idea from Fauser. The author felt like the topic had more than enough potential to be fun, which it most certainly has been, while also appearing to have the potential for applications which seems increasingly promising.

During a visit to Leeds in July of 2015, James East had a discussion with Chwas Ahmed, who is currently one of Paul Martin’s doctoral students. It turned out Ahmed has also been examining the planar modular partition monoid during his candidacy. In a preprint on the arXiv, Ahmed, Martin and Mazorchuk [1] study the number of principal ideals, along with the number of principal ideals generated by an element of fixed rank, of the (planar) modular partition monoids.

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relation for diagram semigroups closed under the vertical flip involution ∗, then count the number of Green’s D, R and L relations for the modular and planar modular partition monoids. In Chapter 6 we conjecture a presentation of the planar modular-2 partition monoid, establish a bound on reduced

PM2k-words and conjecture a number of further results on the quest to identify candidates forPM

2

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Chapter 2 Background

2.1 Preliminary notation and terminology

The reader may find it of use to have precise definitions for the terminology and notation that will be used consistently throughout this thesis. Whenever it has been possible to do so the author has attempted to select terminology and notation that, at least in the author’s opinion, is preferably descriptive, succinct, unambiguous and grammatically correct. Furthermore, the author has also attempted to select termi-nology and notation that is commonly used, at least in the author’s experience, however the termitermi-nology and notation that the author has selected comes from a range of different sources rather than being identical to any particular reference on related literature.

§

2.1.1 Sets

2.1.1 Definition: We denote by:

(i) Rthe set of real numbers; and

(ii) Zthe set of integers, that isZ={. . . ,−2,−1,0,1,2, . . .}.

Furthermore for each integerz∈_Zand subset of the integersA⊆_Z, we denote by:

(i) _Z≥z the set of integers greater than or equal to z, for example Z≥0 = {0,1,2, . . .} is the set of

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(ii) Z>zthe set of integers greater thanz, for exampleZ>0={1,2,3, . . .}is the set of positive integers;

and

(iii) zAthe set {za:a∈A}, for examplez_Z={. . . ,−2z,−z,0, z,2z, . . .}, z_Z>0={z,2z,3z, . . .} and

zZ≥0={0, z,2z, . . .}.

2.1.2 Definition: For each real numberx∈R, we denote bybxcthe greatest integer less thanx, which

is typically referred to asthe floor ofx.

2.1.3 Definition: Given setsAandB, we denote by:

(i) A−B the set containing elements ofAthat are not elements ofBas A−B, that isA−B={a∈ A:a /∈B};

(ii) A×B the Cartesian product of AandB, that is the set{(a, b) :a∈A, b∈B}.

Furthermore givenn∈_Z≥2 setsA1, . . . , An, we denote by:

(i) Sn

i=1Ai the union ofA1, . . . , An; and

(ii) Tn

i=1Ai the intersection ofA1, . . . , An.

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2.1.2 Families of subsets

2.1.4 Definition: LetX be a set. Afamily of subsets ofX, which we will refer to more succinctly as afamily of X when we may do so without any contextual ambiguity, is a set Asuch that each element ofAis a subset ofX, that isA⊆X for allA∈ A. Thepower set of X, which we denote asP(X), is the family of all subsets ofX.

The reader should note it follows by definition that given a setX,P(P(X)) is the family of families of subsets ofX.

2.1.5 Definition: LetX be a set andAbe a family of subsets ofX. We denote by:

(i) S

A∈AAthe union of the elements from A; and

(ii) T

A∈AAthe intersection of the elements from A.

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Sec 2.1:Preliminary notation and terminology

(i) S

A∈Af(A) the union of the elements fromAmapped underf; and

(ii) T

A∈Af(A) the intersection of the elements fromAmapped underf.

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2.1.3 Binary relations

2.1.6 Definition: LetX be a set. Abinary relation onX is a subsetRofX×X. 2.1.7 Definition: A binary relationRon a setX is said to be:

(i) reflexive if (x, x)∈ Rfor allx∈X;

(ii) symmetric if (x, y)∈ Rimplies (y, x)∈ Rfor allx, y∈X;

(iii) antisymmetric if (x, y)∈ Rimplies (y, x)

∈ Rfor all distinctx, y∈X;

(iv) transitiveif (x, y),(y, z)∈ Rimplies (x, z)∈ Rfor allx, y, z∈X; and

(v) total if (x, y)∈ Ror (y, x)∈ Rfor allx, y∈X.

We refer to reflexivity, symmetry, antisymmetry, transitivity and totality asproperties of binary relations.

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2.1.4 Order relations

2.1.8 Definition: A binary relationRon a setX is referred to as:

(i) apre-order if it is reflexive and transitive;

(ii) apartial order if it is an antisymmetric pre-order; and

(iii) atotal order if it is a total partial order.

2.1.9 Definition: LetX be set partially ordered by≤and A be a subset ofX. An element a∈A is referred to as:

(i) the greatest element ofAifais greater than or equal to every element ofA;

(ii) the least element of Aifais less than or equal to every element ofA;

(iii) a minimal element of Aif there does not exist an element ofAthat is greater thana; and

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Furthermore an elementx∈X is referred to as:

(i) an upper bound ofAif every element of Ais less than or equal tox;

(ii) a lower bound of Aif every element ofAis greater than or equal tox;

(iii) the least upper bound of Aif it is the least element of the upper bounds ofA; and

(iv) the greatest lower bound ofAif it is the greatest element of the lower bounds ofA.

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2.1.5 Lattices

2.1.10 Definition: A partial order≤on a setX is referred to as a:

(i) join-semilattice if each two-element subset{x, x0_{} ⊆}_X _{has a greatest lower bound, which is}

typi-cally referred to asthe join ofxandx0, and denoted asx∨x0;

(ii) meet-semilattice if each two-element subset{x, x0_{} ⊆}_X _{has a least upper bound, which is typically}

referred to asthe meet ofxandx0, and denoted asx∧x0; and

(iii) lattice if it is both a join-semilattice and a meet-semilattice.

For example, for each k ∈ Z>0, the subsets of {1, . . . , k} are partially ordered by ⊆, set union gives

us a lattice join operation and set intersection gives us a lattice meet operation. Hence the subsets of {1, . . . , k}form a lattice under⊆.

LetX be a set. The reader should note it follows inductively that:

(i) given a join-semilattice≤on a setX, every finite subsetA⊆X has a least upper bound which is referred to asthe join of the elements of A; and

(ii) given a meet-semilattice≤on a setX, every finite subsetA⊆X has a greatest lower bound which is referred to asthe meet of the elements of A.

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For example transitivity is a closable property of binary relations. The transitive closure of a binary relationRis stated in Proposition 2.1.12.

2.1.12 Proposition: If R is a binary relation on a set X then the transitive closure of R is the set of all (x, x0) ∈ X ×X such that there exist a1, . . . , an ∈ X satisfying x = a1, x0 = an, and (a1, a2), . . . ,(an−1, an)∈ R.

Proof. See page 337 of [46].

Given a closable property of binary relations P:

(i) a lattice-meet operation is defined by taking theP relation generated by the intersection of two binary relations with propertyP;

(ii) a lattice-join operation is defined by taking the P relation generated by the union of two binary relations with propertyP.

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2.1.6 Equivalence relations

2.1.13 Definition: A binary relation Ron a set X is referred to as an equivalence relation if it is a symmetric pre-order.

2.1.14 Definition: Let∼be an equivalence relation on a setX. The equivalence class ofx∈X is the set {y ∈X :y ∼x}, which we denote as [x]. The set of equivalence classes {[x] :x∈X}is denoted by X/∼.

Note that reflexivity, symmetry and transitivity are all preserved under intersections of relations, hence equivalence relations are also preserved under intersections, which forms a lattice meet operation on equivalence relations.

However while reflexivity and symmetry are preserved under unions of binary relations, transitivity may not be, consequently the union of two equivalence relations may not be an equivalence relation. Never-theless we already noted that transitivity is a closable property of relations, hence giving us Proposition 2.1.15.

2.1.15 Proposition: Let ∼1 and ∼2 be two equivalence relations. The smallest equivalence relation

under set inclusion containing the union∼1∪ ∼2 is equal to the transitive closure of the union∼1∪ ∼2.

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We shall adopt the style of indicating that there will be no explicit proof (as in the case above) by terminating the statement itself with.

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2.1.7 The finer and coarser than pre-orders

2.1.16 Definition: LetX be a set andA,Bbe families of subsets ofX. We say thatAis finer thanB and thatBis coarser than A, which we denote asA B, if for eachA∈ Athere existB∈ Bsuch that A⊆B. We denote the finer than relation as.

Note that the finer than relationis trivially reflexive and transitive, and hence pre-orders families of subsets. Further note that is not a partial order, for example the families{{1,2}}and {{1,2},{1}} are finer than and coarser than each other.

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2.1.8 Set partitions

2.1.17 Definition: LetX be a set and Abe a collection of non-empty subsets ofX. We say thatX:

(i) ispairwise disjoint if for eachA, A0 ∈ A,A∩A0 is the empty set;

(ii) coversX, oris a cover of X, ifS

A∈AAis equal toX; and

(iii) partitionsX, oris a partition ofX, ifAis pairwise disjoint and coversX.

2.1.18 Definition: Letk∈Z≥0. Each partitionαof{1, . . . , k}may begraphically depicted as follows:

(i) for eachj ∈ {1, . . . , k}, the vertexj is depicted as the point (j,0)∈_R2_{; and}

(ii) lines connecting points in{1, . . . , k} × {0}are drawn non-linearly below the horizontal line{(x,0) : x∈R} and between the two vertical lines{(1, y),(k, y) :y ∈R} such that the connected

compo-nents form the blocks ofα.

For example, Figure 2.1 illustrates a graphical depiction of the partition{{1,5},{2,3},{4},{6,7,8},{9}, {10,11}}.

Figure 2.1: Given k = 11, as outlined in Definition 2.1.18, the partition {{1,5},{2,3},{4},{6,7,8},{9},

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2.1.19 Definition: The number of partitions of a setX such that|X|=kis referred to asthek-th Bell number, which we denote as Bk.

Note the Bell numbersBk are listed on the OEIS [53] as sequenceA000110.

When restricted to set partitions, the finer-than relation is additionally antisymmetric, hence set parti-tions are partially ordered by the finer-than relation.

2.1.20 Proposition: LetX be a set. IfAis a family of subsets ofX that coversX then there exists a finest partition of X that is coarser thanA.

Proof. Let ∼ be the relation on A where for each A, A0 ∈ A, A ∼ A0 if there exist k ∈ Z≥2 and

A1, . . . , Ak ∈ Asuch thatA1 =A, Ak =A0 andAi∩Ai+16=∅for alli∈ {1, . . . , k−1}. It is trivially

the case that∼is an equivalence relation and hence partitions the elements ofA.

LetB={S

A0_∈_[_A_]A0:A∈ A}. It follows from how we defined∼that

S

B∈BB =

S

A∈AA=X and for

eachB, B0 ∈ B, B∩B0 =_∅, henceB partitionsX. Furthermore for eachA∈ A, A⊆S

A0_∈_[_A_]A0 ∈ B,

henceAis finer thanB.

LetC be a partition of X that is coarser than A. LetA, A0 ∈ Asuch thatA∼A0, therefore there exist k∈Z≥2,A1, . . . , Ak∈ A and such thatA1 =A,Ak=A0 and Ai∩Ai+1 6=∅for alli∈ {1, . . . , k−1}.

It follows fromCbeing coarser thanAthat for eachi∈ {1, . . . , k}, there existCi∈ Csuch thatAi ⊆Ci. Now for each i∈ {1, . . . , k−1}, Ai∩Ai+1 6=∅impliesCi∩Ci+1 6=∅, which requiresCi=Ci+1 since

C is a partition. Hence we must haveS

A0_∈_[_A_]A0 ⊆Ci∈ C, giving thatBis finer thanC. It follows from finer than partially ordering partitions that ifC is also finer thanAthen C=A.

For example the finest partition coarser than{{1,2},{2,3}}is{{1,2,3}}.

Given a setX, the intersection of two partitions ofX is also a partition ofX, consequently the operation of taking the intersection of two partitions of X forms a lattice meet operation. The union of two partitions of X may not be a partition ofX. However, the operation of taking the finest partition of X coarser than the union of two partitions ofX, which is well-defined as was established in Proposition 2.1.20, forms a lattice join operation.

2.1.21 Proposition: Let X be a set. With the above operations the set of partitions ofX forms a lattice.

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X. Distinct equivalence relations form distinct partitions, each partition is formed by an equivalence relation, and the lattice operations match. Hence the lattice of partitions of X and the lattice of equivalence relations onX are isomorphic to each other.

2.1.22 Proposition: Let X be a set. The lattice of partitions of X and the lattice of equivalence relations onX are isomorphic to each other.

2.1.23 Proposition: IfX is a set,Ais a partition ofX andY ⊆X then{A∩Y :A∈ A}is a partition ofY.

2.1.24 Definition: Let X be a set, A be a partition of X and Y ⊆ X. We refer to the partition {A∩Y :A∈ A}as the partitionArestricted toY orthe restriction toY of the partitionA.

2.1.25 Definition: Letk∈Z≥0andm∈Z>0. A partitionαof{1, . . . , k} is referred to as:

(i) non-crossingorplanar if it may be graphically depicted, as in Definition 2.1.18, without connected components crossing; and

(ii) m-divisibleif for each blockb∈α, the number of vertices inbis divisible bym, that is|b| ∈m_Z>0.

The lattice of non-crossing partitions of{1, . . . , k}:

(i) was first identified by Kreweras [44];

(ii) is self-dual and admits a symmetric chain decomposition [58], with further structural and enumer-ative properties of chains in [18]; and

(iii) arise in the context of algebraic and geometric combinatorics, topological problems, questions in probability theory and mathematical biology [57].

The poset ofm-divisible non-crossing partitions of{1, . . . , k}:

(i) was first considered by Edelman [18]; and

(ii) has cardinality (

(m+1)k

k )

mk+1 [3, 4], [18], which is the Pfaff-Fuss-Catalan sequence [29, page 347], [62].

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2.1.9 Modular arithmetic

2.1.26 Definition: Givenm∈Z≥0 anda, b∈Z, it is said thata is congruent tobmodulo m, which is

typically denoted as a≡b(mod m), if the differenceb−abetweena andbis equal to a product of an integer andm, that isb−a∈m_Z.

2.1.27 Proposition: For each a1, a2, b1, b2∈Zsuch that a1≡b1(mod m) anda2≡b2(mod m):

(i) a1+a2≡b1+b2 (modm); and

(ii) a1−a2≡b1−b2 (modm).

Proof. Since a1 ≡b1 (modm) and a2 ≡b2 (mod m), there existsx, y∈Zsuch that b1−a1 =mxand

b2−a2=my. It trivially follows that (b1+b2)−(a1+a2) =m(x+y)∈mZand (b1−b2)−(a1−a2) =

m(x−y)∈m_Z.

2.1.28 Proposition: For each m ∈Z>0, congruence modulo m is an equivalence relation on integers

withm equivalence classes{mZ+ 1, . . . , mZ+m}.

Proof. Leta, b, c∈_Zsuch thata≡b(modm) andb≡c(mod m), hence there existsx, y∈_Zsuch that b−a=mxandc−b=my. Now:

(i) a−a= 0∈mZgiving reflexivity;

(ii) a−b=m(−x)∈mZgiving symmetry; and

(iii) c−a=c−b+b−a=m(y+x)∈m_Zgiving transitivity.

Hence congruence modulo mis an equivalence relation. Furthermore:

(i) the integers congruent toamodulomare trivially equal tom_Z+a; and

(ii) there trivially exists a distinctd∈ {1, . . . , m}such thata∈m_Z+d.

2.1.29 Proposition: For each m, n ∈ Z≥0 and a1, . . . , an, b1, . . . , bn ∈ Z, ifai ≡ bi (mod m) for all i∈ {1, . . . , n}then Σn

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Proof. For each i ∈ {1, . . . , n}, sinceai ≡bi (mod m) there existsxi ∈ Zsuch that bi−ai =mxi. It trivially follows that Σn_i₌₁bi−Σin=1ai= Σni=1(bi−ai) = Σni=1mxi=mΣni=1xi∈mZ.

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2.1.10 Integer partitions

2.1.30 Definition: For eachn∈Z>0 andp1, . . . , pn∈Z>0:

(i) {p1, . . . , pn}is aninteger partition ofΣni=1pi; and

(ii) (p1, . . . , pn) is anordered integer partition of Σni=1pi.

For example the five integer partitions of 4 are{1,1,1,1},{2,1,1},{2,2},{3,1}and{4}, while the eight ordered integer partitions of 4 are (1,1,1,1), (2,1,1), (1,2,1), (1,1,2), (2,2), (3,1), (1,3), (4).

2.1.31 Definition: For eachk∈_Z>0, we shall denote:

(i) the number of integer partitions ofkasp(k);

(ii) the number of integer partitions ofkinto parts of size less than or equal tomasp(m, k) (see Table 2.1 for example values);

(iii) the number of ordered integer partitions of kasop(k); and

(iv) the number of ordered integer partitions ofkinto parts of size less than or equal tomasop(m, k) (see Table 2.2 for example values).

Table 2.1: Example values forp(m, k).

m k

1 2 3 4 5 6 7 8 9 10

1 1 1 1 1 1 1 1 1 1 1

2 1 2 2 3 3 4 4 5 5 6

3 1 2 3 4 5 7 8 10 12 14

4 1 2 3 5 6 9 11 15 18 23

5 1 2 3 5 7 10 13 18 23 30

2.1.32 Proposition: For eachm, k∈Z>0:

(i) p(m, k) =

           

          

1 m= 1,

p(k, k) m > k,

1 +p(m−1, k) m=k,

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Sec 2.2:Semigroups and monoids

Table 2.2: Example values forop(m, k).

m k

1 2 3 4 5 6 7 8 9 10

1 1 1 1 1 1 1 1 1 1 1

2 1 2 3 5 8 13 21 34 55 89

3 1 2 4 7 13 24 44 81 149 274

4 1 2 4 8 15 29 56 108 208 401

5 1 2 4 8 16 31 61 120 236 464

(ii) p(k) =p(k, k);

(iii) op(k) = 1 + Σk_i₌₁−1op(i) = 2max{0,k−1}; and

(iv) op(m, k) =

   

  

op(k) m≤k,

Σm_i₌₁op(m, k−i) m > k.

2.2 Semigroups and monoids

In this section we outline a number of elementary definitions and results from the theory of semigroups and monoids that will pop up at various times throughout this thesis. Vastly more comprehensive expositions of the fundamentals of semigroups may be found in [9, 10], [34] and [35]

2.2.1 Definition: A setSequipped with an associative binary operation, commonly denoted by (s, s0)7→ ss0 for alls, s0∈S, is typically referred to as asemigroup.

2.2.2 Definition: Given a semigroupS, for eacha∈S andA, B⊆S we denote by:

(i) aB the set{ab:b∈B};

(ii) Abthe set{ab:a∈A}; and

(iii) ABthe set{ab:a∈A, b∈B}.

2.2.3 Definition: Let S be a semigroup. A subset T ⊆ S is referred to as a subsemigroup of S if T T ⊆T. Note that, when convenient to do so, we denote such a relationship asT ≤S.

2.2.4 Definition: Given a semigroup S and a subset G⊆S, the semigroup generated by G, which is typically denoted by hGi, is the smallest subsemigroup ofS for whichGis a subset, or equivalently the set of all finite combinations of elements fromGunder the operation ofS.

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2.2.6 Definition: Given a semigroupS, an element e∈S is referred to asthe identity of S if for each s∈S,se=s=es.

Given a semigroupSwith identity elemente∈S, Proposition 2.2.7 justifies referring toeasthe identity ofS rather than asan identity of S.

2.2.7 Proposition: Given a semigroupS, ife, e0∈S are identity elements then e=e0.

Proof. e=ee0 =e0.

2.2.8 Definition: A semigroupS containing an identity elemente∈S, that is so thates=s=sefor alls∈S, is typically referred to as amonoid.

2.2.9 Definition: Given a semigroup S,S1 typically denotes:

(i) S whenS is already a monoid; and

(ii) S∪ {e}such that se=s=esfor alls∈S whenS is not already a monoid.

2.2.10 Proposition: IfS is a semigroup thenS1_{is a monoid.}

2.2.11 Definition: Given a monoid M with identity e ∈ M, a subset N ⊆ M is referred to as a submonoid ofM ifN is a subsemigroup ofM ande∈N.

Note it is possible for a monoidM to contain a distinct monoidN that is not a submonoid ofM. For example given a monoidM with identitye, formM∪ {f}such thatf m=m=mf for allm∈M, then M andM ∪ {f} are both monoids, andM is a subsemigroup of M∪ {f}, while M is not a submonoid ofM∪ {f}.

§

2.2.1 Idempotents

2.2.12 Definition: Anidempotent of a semigroupS is an elements∈S satisfyings2₌_s.

We denote the idempotents of a semigroupS asE(S).

§

2.2.2 Ideals

2.2.13 Definition: A subsetI⊆S is referred to as:

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(ii) aright ideal ifI is closed when multiplying on the right by elements ofS, that is ifIS⊆I; and

(iii) an ideal if it is both a left and a right ideal, that is ifSI, IS⊆I.

2.2.14 Proposition: For each a∈S:

(i) Sa∪ {a}is a left ideal;

(ii) aS∪ {a}is a right ideal; and

(iii) SaS∪Sa∪aS∪ {a}is an ideal.

Proof. For each a∈S we have:

(i) S(Sa∪ {a}) =SSa∪Sa⊆Sa∪ {a};

(ii) (aS∪ {a})S=a(SS)∪aS⊆Sa∪ {a}; and

(iii) S(SaS∪Sa∪aS∪ {a}) =SSaS∪SSa∪SaS∪Sa⊆SaS∪Sa∪aS∪ {a} and (SaS∪Sa∪aS∪ {a})S=SaSS∪SaS∪aSS∪aS ⊆SaS∪Sa∪aS∪ {a}.

2.2.15 Definition: LetS be a semigroup. For eacha∈S:

(i) Sa∪ {a}is referred to asthe principal left ideal generated bya;

(ii) aS∪ {a}is referred to asthe principal right ideal generated by a; and

(iii) SaS∪Sa∪aS∪ {a}is referred to asthe principal ideal generated by a.

§

2.2.3 Regular

∗

-semigroups

2.2.16 Definition: LetS be a semigroup. Aninvolution is a unary operation∗:S →S such that∗ is:

(i) its own inverse, that is (s∗)∗=sfor alls∈S; and

(ii) an anti-automorphism, that is (st)∗=t∗s∗ for alls, t∈S.

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Regular∗-semigroups were introduced by Nordahl and Scheiblich [50].

2.2.18 Proposition: If a submonoidT of a regular∗-semigroupSis closed under the involution∗, that isT∗=T, thenT is also a regular∗-semigroup.

Proof. It trivially follows by definition that:

(i) for eacha∈T we have a∗∈T,a∗∗=aandaa∗a=a; and

(ii) for eacha, b∈T we have a∗_{, b}∗_{, b}∗_a∗_∈_T _{and (ab)}∗₌_b∗_a∗_.

2.2.19 Definition: Let S be a regular ∗-semigroup. An elementp∈S is referred to as aprojection if it satisfiesp2=p=p∗.

Note that all projections are idempotents, which trivially follows from the condition for a projection containing the condition for idempotency.

2.2.20 Proposition: LetS be a regular∗-semigroup. The projections, and more inclusively the idem-potents, ofS are partially ordered byp≤q if and only ifpq=p=qpfor allp, q∈S.

Proof. Clearly≤is reflexive and antisymmetric leaving transitivity. Letp, q, r∈S such thatp≤q≤r. Thenpr= (pq)r=p(qr) =pq=p=qp=rqp=rp.

2.2.21 Proposition: Given a regular∗-semigroupS:

(i) for eachs∈S,ss∗ ands∗sare projections;

(ii) every projection can be written asss∗ ands∗sfor somes∈S;

(iii) the set of idempotents E(S) is closed under involution; and

(iv) every idempotent is the product of two projections.

Proof. (i) For each s ∈ S, (ss∗)(ss∗) = (ss∗s)s∗ = ss∗ = (s∗)∗s∗ = (ss∗)∗, similarly (s∗s)(s∗s) = s∗s= (s∗s)∗;

(ii) For eachp∈S such thatp2₌_p₌_p∗_,_p₌_p2₌_pp∗₌_p∗_p;

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(iv) For each idempotente∈E(S), (ee∗)∗(e∗e) =ee∗e∗e=ee∗e=e.

§

2.2.4 Green’s relations

Green’s relations, introduced by James Alexander Green in [30], are five equivalence relations that partition the elements of a semigroup with respect to the principal ideals that may be generated.

2.2.22 Definition: Given a semigroupS, for eacha, b∈S:

(i) (a, b)∈≤R if and only ifaS∪ {a} ⊆bS∪ {b}; and

(ii) (a, b)∈≤L if and only ifSa∪ {a} ⊆Sb∪ {b}.

2.2.23 Proposition: ≤R and≤L are pre-orders onS.

Proof. Reflexivity is obvious and transitivity follows from⊆being transitive on sets.

2.2.24 Definition: Given a semigroupS,Green’s relations on S are defined as follows:

(i) R=≤R∩ ≤−R1={(a, b)∈S×S:aS∪ {a}=bS∪ {b}};

(ii) L=≤L ∩ ≤−L1={(a, b)∈S×S:Sa∪ {a}=Sb∪ {b}};

(iii) H=L ∩ R;

(iv) D={(a, b)∈S×S: there existsc∈S such thataLc andcRb}; and

(v) J ={(a, b)∈S×S:SaS∪ {a}=SbS∪ {b}}.

2.2.25 Proposition: Given a semigroup S,

(i) Green’sR,L,H,DandJ relations are equivalence relations;

(ii) D=R ◦ L=L ◦ R=R ∨ L; and

(iii) ifS is finite then Green’s DandJ relations are equal.

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(i) RT ₌_RS_∩_(T_×_T_);

(ii) LT ₌_LS_∩_(T_×_T_{); and}

(iii) HT ₌_HS_∩_(T_×_T).

Proof. Proposition 2.4.2 in [35] (also in [32]).

§

2.2.5 Green’s relations on regular

∗

-semigroups

2.2.27 Proposition: Given a regular∗-semigroupS, for eacha, b∈S:

(i) (a, b)∈≤R if and only if aa∗≤bb∗;

(ii) (a, b)∈≤L if and only if a∗a≤b∗b;

(iii) (a, b)∈ Rif and only if aa∗ =bb∗;

(iv) (a, b)∈ Lif and only ifa∗a=b∗b; and

(v) (a, b)∈ Dif and only if there existsc∈S such thata∗a=c∗c andcc∗=bb∗.

Proof. Suppose (a, b) ∈≤R, and hence that there exist s ∈ S1 such that as = b. Then aa∗(b)b∗ =

(aa∗a)sb∗ = (as)b∗=bb∗ =b(s∗a∗) = bs∗(a∗aa∗) =b(b∗)aa∗, giving us aa∗≤bb∗. Conversely suppose aa∗ ≤bb∗. Then a(a∗bb∗b) = (aa∗bb∗)b =bb∗b =b, hence (a, b) ∈≤R, establishing Part (i). Part (ii)

follows dually to Part (i). Parts (iii), (iv) and (v) follow from applying Parts (i) and (ii) to how Green’s R,L andDrelations are defined.

2.2.28 Proposition: Given a regular∗-semigroupS:

(i) eachRclass contains precisely one projection; and

(ii) eachLclass contains precisely one projection.

Proof. Let a, b ∈ S and suppose (aa∗, bb∗) ∈ R, and hence that (aa∗_)(aa∗₎∗ _{= (bb}∗_)(bb∗₎∗_. _Then

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§

2.2.6 Congruence relations

2.2.29 Definition: Given a semigroup S, acongruence relation is an equivalence relation ∼ ⊆S×S that is compatible with the operation forS. The equivalence classes of a congruence relation are often also referred to ascongruence classes.

The congruence classes of a congruence relation form a monoid where for each s, s0∈S, the congruence class containing s multiplied by the congruence class containing s0 is equal to the congruence class containingss0, that is [s][s0] = [ss0].

Note that congruence relations are closed under intersections, hence given a relation Ron a semigroup S, the intersection of all congruence relations containing Ris itself a congruence relation, and hence is the smallest congruence relation containingR.

2.2.30 Definition: Given n ∈ Z>0 binary relations R1, . . . , Rn on a semigroup S, the congruence relation generated byR1, . . . ,Rn, which we denote byhR1, . . . ,Rni†, is the intersection of all congruence relations that contain∪n

i=1Ri.

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2.2.7 Free semigroups and free monoids

2.2.31 Definition: Given a setX:

(i) thefree semigroup ofX, which is often denoted byX+_{, is the set of non-zero finite strings from}_X

together with string concatenation, that isX+₌_{Πn

i=1xi:n∈Z>0 andx1, . . . , xn∈S}; and

(ii) the free monoid of X, which is often denoted by X?_{, adjoins the empty string, which acts as the} identity under string concatenation, to the free semigroupX+_.

§

2.2.8 Presentations

2.2.32 Definition: Apresentation of a semigroupS consists of:

(i) a set of generators Σ; and

(ii) a set of binary relations{R1, . . . ,Rn} on the free semigroup Σ+ generated by Σ,

such thatS ∼= Σ+/hR1, . . . ,Rni

†

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Perhaps more descriptively, given a semigroupS, a generating set Σ of S along with n∈ Z>0 binary

relations R1, . . . , Rn form a presentation of S when any two words in the free semigroup Σ+ whose products form the same element ofS are able to be shown as equivalent in an abstract manner using the binary relationsR1, . . . ,Rn.

2.3 Fundamentals of diagram semigroups

There are fewer times more appropriate than a doctoral thesis to give a complete coverage of the foun-dations for diagram semigroups, consequently the author has taken the opportunity to do just that.

§

2.3.1 Diagrams

D

k

2.3.1 Definition: Letk∈Z≥0. Ak-diagram is a reflexive and symmetric binary relation on{1, . . . , k}∪

{10_{, . . . , k}0_{}. We denote by}_Dk _{the set of all}_{k-diagrams, and will refer to}_{k-diagrams more succinctly as}

diagrams either whenkmay be any positive integer or when the value ofkis contextually unambiguous.

2.3.2 Definition: Letk∈Z≥0 andδ∈Dk. We refer to:

(i) elements of{1, . . . , k} as upper vertices;

(ii) elements of{10_{, . . . , k}0_}_{as lower vertices; and}

(iii) elements ofδas edgesor lines.

Note providedk∈Z≥0 is contextually unambiguous, we may specify a diagram using an edge set that

is neither reflexive nor symmetric with the implicit understanding that we are actually referring to the reflexive and symmetric closure. For example givenk= 2, if a diagram is specified as{(1,2),(2,20)}, we will implicitly mean the diagram{(1,1),(2,2),(10,10),(20,20),(1,2),(2,1),(2,20), (20,2)} ∈D2.

2.3.3 Definition: Letk∈Z≥0. Each diagramδ∈Dk is typicallydepicted as follows (see Figure 2.2 for

an example):

(i) for eachj∈ {1, . . . , k}, the upper vertexj is depicted as the point (j,1) while the lower vertexj0 is depicted as the point (j,0); and

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[image:33.595.217.383.124.207.2]

Sec 2.3:Fundamentals of diagram semigroups

Figure 2.2: {(2,10),(3,4),(4,7),(7,70),(70,80),(5,6),(8,60),(60,50),(20,30),(30,40)} ∈ D8 may be depicted, as

outlined in Definition 2.3.3, as follows:

1 2

10

3 4 7

70 80

5 6 8

50 60 20 30 40

Note that vertex labels are trivially recoverable without their inclusion, consequently vertex labels are omitted from all figures proceeding Figure 2.2.

We will adopt the convention that lines between two upper vertices along with lines between two lower vertices will be non-linear, and that lines between an upper vertex and a lower vertex will be, inclusive of vertical sections, piecewise linear.

§

2.3.2 Diagram equivalence

∼

_D

Let k ∈ Z≥0 and δ ∈ Dk. If we take the transitive closure of δ we get an equivalence relation on

{1, . . . , k} ∪ {10, . . . , k0}, the equivalence classes of which form a partition of{1, . . . , k} ∪ {10, . . . , k0}.

2.3.4 Definition: Let k∈Z≥0. Theconnected components of ak-diagramδ∈Dk are the equivalence

classes of the transitive closure ofδ.

For example the connected components of the diagram depicted in Figure 2.2 are:

{{1},{2,10},{3,4,7,70,80},{5,6},{8,50,60},{20,30,40}}.

Givenk∈Z≥0, the connected components of eachk-diagram form a partition of{1, . . . , k} ∪ {10, . . . , k0}.

Note however that there exist distinct diagrams whose connected components form precisely the same partition of {1, . . . , k} ∪ {10_{, . . . , k}0_{}. For example Figure 2.3 depicts two diagrams whose connected}

components form the same partition.

2.3.5 Definition: Let k ∈Z≥0. We say that two diagrams δ, δ0 ∈Dk are equivalent if they have the

same connected components. Furthermore we denote by ∼_D the binary relation that relates equivalent diagrams.

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Figure 2.3: Given k = 6, the diagrams δ = {(1,2),(2,3),(3,4),(4,5),(4,40),(30,4),(40,50)}, δ0 = {(1,2),

(1,4),(3,4),(5,40),(5,50),(6,60),(50,60)} ∈D6 are depicted below.

δ= δ0 =

While δ and δ0 do not contain the same edges, the connected components of both δ and δ0 are {{1,2,3,4},{5,6,40,50,60},{10_,₂0_,₃0_}}_{which is a partition of}_{{1, . . . ,}_{6} ∪ {1}0_{, . . . ,}₆0_}.

equivalence relation onk-diagrams.

§

2.3.3 Bipartitions

2.3.6 Definition: Letk∈Z≥0. Ak-bipartitionis a set partition of{1, . . . , k} ∪ {10, . . . , k0}. We denote

byPk the set of all k-bipartitions, and will refer tok-bipartitions more succinctly as bipartitions either whenk may be any positive integer or when the value ofk is contextually unambiguous. Furthermore we refer to elements of bipartitions asblocks.

TypicallyP0 is treated as the singleton set containing the empty bipartition.

2.3.7 Definition: For each k ∈ Z≥0 and α ∈ Pk, we denote by Dα the distinct equivalence class of diagrams inDk/∼_D such that for eachδ∈Dα, the connected components ofδare the bipartition α.

To depict the bipartitionα∈ Pkwe select any diagramδ∈Dα, and treat blocks of bipartitions as being synonymous with connected components of diagrams.

No consistent convention has been followed by the author for which diagram will be selected to depict a given bipartition, though the author has attempted to select diagrams that are aesthetically pleasing whilst ensuring a visible distance between distinct connected components.

§

2.3.4 Block types

It is often convenient to establish and use various bits of notation and terminology for blocks of bipar-titions. Terminology tends to differ between authors, though definitions are typically either identical or equivalent.

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(i) U(b) the subset of upper vertices contained in b, that isb∩ {1, . . . , k};

(ii) u(b) the number of upper vertices contained inb, that is|U(b)|;

(iii) L(b) the subset of lower vertices contained inb, that isb∩ {10_{, . . . , k}0_{}; and}

(iv) l(b) the number of lower vertices contained inb, that is|L(b)|.

2.3.9 Definition: Given a bipartition α∈ Pk, the type of a blockb ∈αis the pair u(b), l(b)

, where u(b), l(b)∈ {0, . . . , k} such thatu(b) +l(b)>0 are the number of upper vertices and number of lower vertices in blockb respectively.

It will be convenient for us to use some further terminology for blocks.

2.3.10 Definition: Givenm, n∈Z>0 we refer to:

(i) a block of type (m, n) as atransversal;

(ii) a block of type (1,1) as atransversal line;

(iii) a block of type (m, m) asuniform;

(iv) a transversal line{i, i0}where i∈ {1, . . . , k} as avertical line;

(v) a block of type (m,0) or (0, m) as anon-transversal;

(vi) a block of type (2,0) or (0,2) as anon-transversal line;

(vii) a block of type (1,1), (2,0) or (0,2) as aline;

(viii) a non-transversal containing mconsecutive vertices as anm-apsis;

(ix) 1-apses asmonapses, 2-apses asdiapses and 3-apses astriapses; and

(x) a transversal of type (2,2) containing the same two adjacent upper and lower vertices as a (2, 2)-transapsis.

§

2.3.5 Product graphs

Γ

δ

1

, . . . , δ

p

2.3.11 Definition: For eachk∈_Z≥0,p∈Z>0 andδ1, . . . , δp∈Dk:

(i) byδ1 we denote the reflexive and symmetric relation on {1, . . . , k} ∪ {12, . . . , k2} where for each

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(ii) byδp we denote the reflexive and symmetric relation on{1p, . . . , kp} ∪ {10, . . . , k0}where for each j∈ {1, . . . , k}, every instance of the upper vertexj inδp has been relabelled tojp inδp; and

(iii) for each i∈ {2, . . . , p−1}, byδi we denote the reflexive and symmetric relation on{1i, . . . , ki} ∪ {1i+1_{, . . . , k}i+1_}_{where for each}_j_{∈ {1, . . . , k}, every instance of the upper vertex}_j _in _δ

i has been relabelled to ji in δi and every instance of the lower vertexj0 inδi has been relabelled toji+1 in δi.

The product graph of δ1, . . . , δp is the union Sp_i₌₁δi, which is a reflexive and symmetric relation on {1, . . . , k} ∪ {10_{, . . . , k}0_{} ∪}₍Sp

i=2{1i, . . . , ki}) that is often denoted as Γ δ1, . . . , δp

.

For each k ∈ Z≥0 and δ ∈ Dk, it trivially follows by definition that Γ δ = δ, consequently we will

implicitly interchange between the two whenever it is convenient to do so.

2.3.12 Definition: For eachk∈_Z≥0, p∈Z>0 and δ1, . . . , δp∈Dk, the product graph Γ δ1, . . . , δp

is typically depicted by vertically stackingδ1, . . . , δp such that for eachj∈ {1, . . . , p},δj is depicted as the j-th upper most diagram. Being more precise (see Figure 2.4 for an example):

(i) for each j ∈ {1, . . . , k}, the upper vertex j is depicted as the point (j, p), the lower vertex j0 is depicted as the point (j,0), and for each i ∈ {2, . . . , p}, the vertex ji−1 _{is depicted as the point}

(j, i−1); and

(ii) for eachi∈ {1, . . . , p}, each ofδi’s edges are drawn within the convex hull of the points{1, . . . , k}× {p−i, p−i+ 1}.

We will often label the vertical layers of a product graph by whichever is more convenient out of the associated diagram or the underlying bipartition for each layer.

2.3.13 Definition: For each k ∈ Z≥0, p, q ∈ Z>0 and δ1, . . . , δp, η1, . . . , ηq ∈ Dk, the product graph of Γ δ1, . . . , δp

and Γ η1, . . . , ηq

, which we denote by Γ Γ δ1, . . . , δp

,Γ η1, . . . , ηq

, is formed by vertically stacking Γ δ1, . . . , δp

above Γ η1, . . . , ηq

in an analogous way to how product graphs of diagrams are formed.

2.3.14 Proposition: For eachk∈Z≥0,p, q∈Z>0andδ1, . . . , δp, η1, . . . , ηq ∈Dk,

Γ Γ δ1, . . . , δp

,Γ η1, . . . , ηq

= Γ δ1, . . . , δp, η1, . . . , ηq

.

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[image:37.595.208.398.138.265.2]

Figure 2.4: Let k = 5, δα = {(1,10),(2,20),(3,4),(5,30),(6,40),(7,90),(8,9),(50,80),(60,70)} ∈ D5 and δβ = {(1,4),(1,10),(10,20),(2,3),(5,60),(30,60),(6,7),(8,9),(8,70),(40,50),(80,90)} ∈D5. The product graph Γ δα, δβ

, as outlined in Definition 2.3.12, is depicted as:

Γ δα, δβ

=

δα

δβ

stack of δ1, . . . , δp with the vertical stack ofη1, . . . , ηq.

2.3.15 Corollary: Forming product graphs of product graphs is associative.

Proof. For each k∈Z≥0,p, q, r∈Z>0 andδ1, . . . , δp, η1, . . . , ηq, ξ1, . . . , ξr∈Dk,

Γ Γ Γ δ1, . . . , δp,Γ η1, . . . , ηq,Γ ξ1, . . . , ξr= Γ Γ δ1, . . . , δp, η1, . . . , ηq,Γ ξ1, . . . , ξr = Γ δ1, . . . , δp, η1, . . . , ηq, ξ1, . . . , ξr

= Γ Γ δ1, . . . , δp

,Γ η1, . . . , ηq, ξ1, . . . , ξr

= Γ Γ δ1, . . . , δp,Γ Γ η1, . . . , ηq,Γ ξ1, . . . , ξr.

2.3.16 Definition: Let k ∈ Z≥0, p, q ∈ Z>0 and δ1, . . . , δp, η1, . . . , ηq ∈ Dk. We say that the two product graphs Γ δ1, . . . , δp

and Γ η1, . . . , ηq

are equivalent, which we denote by Γ δ1, . . . , δp

∼_D

Γ η1, . . . , ηq

, if their connected components restricted to {1, . . . , k} ∪ {10_{, . . . , k}0_} _{are equal, with the}

implicit understanding that we forget about any then-empty components following our restriction.

2.3.17 Proposition: For eachk∈Z≥0, p, q, r, s∈Z>0 andδ1, . . . , δp, η1, . . . , ηq, ξ1, . . . , ξr,ω1, . . . , ωs ∈Dk, if:

(i) Γ δ1, . . . , δp∼_DΓ η1, . . . , ηq; and

(ii) Γ ξ1, . . . , ξr

∼_DΓ ω1, . . . , ωs

,

then Γ δ1, . . . , δp, ξ1, . . . , ξr

∼_DΓ η1, . . . , ηq, ω1, . . . , ωs

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Proof. First note that       

Γ δ1, . . . , δp, ξ1, . . . , ξr

= Γ Γ δ1, . . . , δp

,Γ ξ1, . . . , ξr

; and

Γ η1, . . . , ηq, ω1, . . . , ωs

= Γ Γ η1, . . . , ηq

,Γ ω1, . . . , ωs

.

For eachj ∈ {1, . . . , k}, we relabel the vertexjp+1toj00in the (p+ 1)-th vertex row of the product graph Γ δ1, . . . , δp, ξ1, . . . , ξr

, and relabel the vertex jr+1 _to _j00 _{in the (r}_{+ 1)-th vertex row of the product}

graph Γ η1, . . . , ηq, ω1, . . . , ωs

.

Let i, j ∈ {1, . . . , k} ∪ {10_{, . . . , k}0_} _{such that there exists a path} _p

i,j from i to j in the product graph Γ δ1, . . . , δp, ξ1, . . . , ξr. Let t ∈ Z≥2 such that the minimum number of times pi,j crosses between Γ δ1, . . . , δp

and Γ ξ1, . . . , ξr

is t−2. Therefore there exists t vertices v1, . . . , vt ∈ {1, . . . , k} ∪ {100_{, . . . , k}00_{} ∪ {1}0_{, . . . , k}0_} _{and paths} _p

v1,v2, . . . , pvt−1,vt such that v1 = i, vt = j and for each l ∈

{1, . . . , t−1}, pvl,vl+1 is a path from vl to vl+1 that is either a subset of Γ δ1, . . . , δp

or a subset of Γ ξ1, . . . , ξr

.

For each l ∈ {1, . . . , t−1}, it follows from Γ δ1, . . . , δp ∼_D Γ η1, . . . , ηq and Γ ξ1, . . . , ξr ∼_D Γ ω1, . . . , ωs that there exists a path from vl to vl+1, which we denote as p0vl,vl+1, that is either a

subset of Γ η1, . . . , ηqor a subset of Γ ω1, . . . , ωsdepending on whether Γ δ1, . . . , δpor Γ ξ1, . . . , ξr contains the pathpvl,vl+1.

FinallySt−1

l=1p0vl,vl+1 forms a path fromitoj and for eachl∈ {1, . . . , t−1},p

0

vl,vl+1 is either a subset of

Γ η1, . . . , ηq

or a subset of Γ ω1, . . . , ωs

. Hence i andj, which share the same connected component of Γ δ1, . . . , δp, ξ1, . . . , ξr

, also share the same connected component of Γ η1, . . . , ηq, ω1, . . . , ωs

.

The same argument in the opposite direction establishes that for any two vertices i, j ∈ {1, . . . , k} ∪ {10_{, . . . , k}0_}_{that share the same connected component of the product graph Γ} _η

1, . . . , ηq, ω1, . . . , ωs

,iand jalso share the same connected component of Γ δ1, . . . , δp, ξ1, . . . , ξr

. Hence Γ δ1, . . . , δp, ξ1, . . . , ξr

∼_D

Γ η1, . . . , ηq, ω1, . . . , ωs

.

§

2.3.6 The partition monoid

P

k

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to depict products of bipartitions in the process. The second approach multiplies bipartitions directly. Since both products are equivalent, which we will later establish, we will only make a distinction between the two when it is relevant to do so.

2.3.18 Definition: Let k∈_Z≥0 andα, β ∈ Pk. Theproduct of αand β, denoted asαβ, is formed as follows:

(i) Letδα∈Dαandδβ∈Dβ;

(ii) two verticesi, j∈ {1, . . . , k} ∪ {10_{, . . . , k}0_}_{share the same block in the product}_αβ_{if they share the}

same connected component of the product graph Γ δα, δβ

, that is the blocks of the productαβare the connected components of Γ δα, δβ

restricted to{1, . . . , k} ∪ {10_{, . . . , k}0_} _{with any then-empty}

connected components implicitly removed.

For example considerδα, δβ∈D8from Figure 2.4 where the product graph Γ δα, δβis depicted. It is eas-ily verified that the underlying bipartition ofδαisα={{1,10},{2,20},{3,4},{5,30},{6,40},{7,90},{8,9}, {50_,₈0_},_{70_,₈0_{}} ∈ P}

8 and that the underlying bipartition of δβ is β = {{1,4,10,20},{2,3},{5,30,60}, {6,7},{8,9,70},{40_,₅0_},_{80_,₉0_{}} ∈ P}

8, that isδα∈Dαandδβ∈Dβ. Hence two verticesi, j∈ {1, . . . , k}∪ {10_{, . . . , k}0_}_{share the same block in}_αβ_{if there is a path from}_i_to_j_{in the product graph Γ} _δ

α, δβ

that is depicted in Figure 2.4. Giving usαβ={{1,6,10,20},{2,5},{3,4},{7,30,60,70},{8,9},{40_,₅0_},_{80_,₉0_{}} ∈}

P8.

Note that in order for the bipartition product to be well-defined, the choice of representative diagrams from DαandDβ must be arbitrary, which is conveniently a restricted case of Proposition 2.3.17.

2.3.19 Corollary: For eachk∈Z≥0,α, β∈ Pk,δα, δα0 ∈Dαandδβ, δβ0 ∈Dβ, Γ δα, δβ

∼_DΓ δ0_α, δ_β0

.

Proof. A restricted case of Proposition 2.3.17.

2.3.20 Proposition: For each k∈Z≥0andα, β, γ∈ Pk, (αβ)γ=α(βγ).

Proof. Let δα ∈ Dα, δβ ∈ Dβ, δαβ ∈ Dαβ and δγ ∈ Dγ. Note that the connected components of δαβ and the connected components of Γ δα, δβrestricted to {1, . . . , k} ∪ {10, . . . , k0} are the blocks of αβ, and hence that the connected components of δαβ are equal to the connected components of Γ δα, δβ

restricted to{1, . . . , k} ∪ {10_{, . . . , k}0_{}, hence}_δ

αβ∼_DΓ δα, δβ

.

Note it follows from Proposition 2.3.17 that the connected components of Γ δαβ, δγ

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components of Γ Γ δα, δβ, δγ= Γ δα, δβ, δγ are equal. Hence the blocks of (αβ)γ are equal to the connected components of Γ δα, δβ, δγ

restricted to{1, . . . , k} ∪ {10_{, . . . , k}0_}.

By an analogous argument the blocks ofα(βγ) are also equal to the connected components of Γ δα, δβ, δγ

restricted to{1, . . . , k} ∪ {10, . . . , k0}, giving us (αβ)γ=α(βγ).

2.3.21 Definition: For eachk∈_Z>0, we denote by idk the bipartition

n

{j, j0_}_:_j _{∈ {1, . . . , k}}o_{∈ P}

k

(see Figure 2.5 for a depiction of id4).

Figure 2.5: Givenk= 4,

id4= ∈ P4

Note idk trivially acts as both a left and right identity, and hence is the identity of the bipartition product.

2.3.22 Definition: Given k∈Z≥0,the partition monoid, which is often also denoted asPk for conve-nience, consists of allk-bipartitionsPk along with the bipartition product.

2.3.23 Corollary: For eachk∈Z≥0,p∈Z≥2α1, . . . , αp∈ Pk andδαj ∈Dαj wherej∈ {1, . . . , p}, the

blocks ofα1. . . αp are the connected components of Γ δα1, . . . , δαp

.

Proof. Follows inductively on pin an analogous way to the proof of Proposition 2.3.20, where we estab-lished that the bipartition product is associative.

Letp∈Z≥2andα1, . . . , αp∈ Pk. When depicting the formation of the productα1. . . αp, we will equate the product graph Γ(α1, . . . , αp) to any representative diagram δ ∈ Dα1...αp, with the implicit

under-standing that we really mean the connected components on the product graph Γ(α1, . . . , αp) restricted to{1, . . . , k} ∪ {10_{, . . . , k}0_}_{are equal to the connected components of the diagram depicting the product}

α1. . . αp.

Next we review how the product of two bipartitions may equivalently be formed directly without using representative diagrams.

2.3.24 Proposition: For eachk∈Z≥0 andα, β∈Dk,αβ is the finest partition coarser thanα∨∪β∧

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(i) α∨ denotes the partition of {1, . . . , k} ∪ {100, . . . , k00} such that for each j ∈ {1, . . . , k}, the lower

vertexj0 in αhas been relabelled toj00 inα∨; and

(ii) β∧ _{denotes the partition of} _{100_{, . . . , k}00_{} ∪ {1}0_{, . . . , k}0_} _{where for each} _j _{∈ {1, . . . , k}, the upper}

vertexj inβ has been relabelled toj00in β∧.

Proof. Letδα∈Dα andδβ ∈Dβ. The product graph Γ δα, δβ

is formed in an equivalent way to how α∨∪β∧ is formed, and the transitive closure of Γ δα, δβ

is the finest equivalence relation containing Γ δα, δβ. Hence the associated partition must be the finest partition coarser thanα∨∪β∧.

For example, consider α={{1,5,40,50},{2,3,4},{10_},_{20_,₃0_}}, _β₌_{{1,_4,_5,₁0_,₂0_,₃0_},_{2,_3},_{40_,₅0_{}} ∈}

P5. The formation of the product αβ as outlined in Definition 2.3.18 is depicted in Figure 2.6,

alterna-tively it may be computed as follows:

(i) α∨={{1,5,400,500},{2,3,4},{100},{200,300}};

(ii) β∧={{100_,₄00_,₅00_,₁0_,₂0_,₃0_},_{200_,₃00_},_{40_,₅0_}};

(iii) {{1,5,100,400,500,10,20,30},{2,3,4},{200_,₃00_},_{40_,₅0_}}_{is the finest partition coarser than}_α

∨∪β∧;

(iv) {{1,5,10,20,30},{2,3,4},{},{40_,₅0_}} _{are the blocks of the finest partition coarser than} _α

∨ ∪β∧

restricted to{1, . . . , k} ∪ {10_{, . . . , k}0_{}; and}

(v) {{1,5,10,20,30},{2,3,4},{40,50}} are the non-empty blocks of the finest partition coarser than α∨∪β∧ restricted to{1, . . . , k} ∪ {10, . . . , k0}, which is the bipartitionαβ and depicted in Figure

2.6.

Figure 2.7 depicts the formation of a more intricate product of bipartitions than we have considered up to now, which we proceed to discuss. Notice that when forming the productαβ:

(i) the upper non-transversal blocks ofαand lower non-transversal blocks ofβ are preserved;

(ii) the upper 2-apsis block {5,6} ∈ β joins to the transversal lines{4,50}, {5,60} ∈ α, forming the upper non-transversal{4,5} ∈αβ;

(iii) the lower 3-apsis block{10_,₂0_,₃0_{} ∈}_α_{and upper 3-apsis block} _{1,_2,_{3} ∈}_β _{cap each other off;}

(iv) the lower non-transversals {90_,₁₀0_},_{110_{} ∈} _α _{and upper non-transversals} _{9},_{10,_{11} ∈} _β _cap

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[image:42.595.236.355.120.284.2]

Figure 2.6: Givenk= 5 andα={{1,5,40,50},{2,3,4},{10},{20,30}}, β={{1,4,5,10,20,30},{2,3},{40,50}} ∈ P5,

α

β

αβ

=

(v) the transversal lines{3,40},{7,70} ∈αjoin to the transversal{4,7,30} ∈β, forming the transversal {3,7,30} ∈αβ; and

(vi) the blocks{8,130},{80_,₁₂0_{} ∈}_α_{join to the transversal blocks}_{8,₄0_,₇0_,₈0_},_{12,₁₁0_{} ∈}_{β, forming}

the transversal block{8,40,70,80,110} ∈αβ.

Figure 2.7: Givenα= n

{1,2},{3,40},{4,50},{5,60},{6},{7,70},{8,130},{9,10,11,12,13},{10,20,30},{80,120},

{90,100},{110}o,β =n{1,2,3},{4,7,30},{5,6},{8,40,70,80},{9},{10,11},{12,13,110},{10,20},{50,60},{90,100},

{120,130}o∈ P13,

α

β

αβ

=

§

2.3.7 Horizontal sum

⊕

:

P

k1

× P

k2

→ P

k1+k2

2.3.25 Definition: For eachk1, k2∈Z≥0,α∈ Pk1 andβ ∈ Pk2,the horizontal sum of αandβ, which

[image:42.595.157.436.475.638.2]

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(i) let βB _{denote the partition of} _{k₁_{+ 1, . . . , k}₁₊_k₂_{} ∪ {(k}₁_{+ 1)}0_{, . . . ,}_(k

1+k2)0} where for each

j∈ {1, . . . , k2}, the upper vertexj in β has been relabelled to k1+j and the lower vertexj0 in β

has been relabelled to (k1+j)0 in βB; and

(ii) the horizontal sumα⊕β is the union α∪βB_{∈ P} k1+k2.

The horizontal sum of diagrams is defined analogously as follows.

2.3.26 Definition: For eachk1, k2∈Z≥0,δ∈Dk1 andη ∈Dk2, the horizontal sum ofδ andη, which

we denote asδ⊕η, is formed as follows:

(i) letηB denote the translation of ηwhere for eachj∈ {1, . . . , k2}, the upper vertexj inη has been

relabelled tok1+j and the lower vertex j0 inη has been relabelled to (k1+j)0 inηB; and

(ii) the horizontal sumδ⊕η is the unionδ∪ηB_∈_D k1+k2.

For each k1, k2 ∈Z≥0, α∈ Pk1 and β ∈ Pk2, it trivially follows by definition that Dα⊕Dβ =Dα⊕β.

Hence if we select representative diagrams δα ∈ Dα and δβ ∈ Dβ, we may depict α⊕β by δα⊕δβ without having to explicitly determineα⊕β directly to either verify thatδα⊕δβ∈Dα⊕β or to select a representative diagram fromDα⊕β.

Figure 2.8 depicts the horizontal sum of α = {{1,2},{3,6,50,60},{4,5}, {10_,₄0_},_{20_,₃0_{}} ∈ P}

6 and

[image:43.595.138.470.514.707.2]

β ={{1,2,3,30_,₄0_},_{4,_8,₅0_},_{10_,₂0_},_{60_,₇0_,₈0_}}.

Figure 2.8: Givenk1= 6,k2= 8,

∈ P6and

α=

∈ P8,

β=

∈ P14.

α⊕β =

The planar modular partition monoid