2 – 3 October 2014Monash University, Clayton

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participants 2

abstracts 3

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information

Welcome to the 32nd Victorian Algebra Conference.This year’s conference features 22 talks, including 7 by students.

venue

All talks are in Bld 29, theatre S13; see the map on Page 8. The theatre has blackboards, aswell as a computer and data projector. Registration and morning & afternoon teas will benext to the conference theatre. Restrooms and a water dispenser can be found on the groundfloor of Bld 28: if you enter the building, turn right, and you’ll find them behind the lifts.

There is free-parking available in the blue permit areas in the N1 multi-level car park and theSE4 multi-level car park, see the map on Page 8. (You must display a permit for all otherblue parking areas at all times.)

dinner

The conference dinner is on Thursday, 7.00pm, at Taste Baguette, on Clayton Campus (Bld11, see the map on Page 8). The location will be open from 6:30pm, serving drinks.

lunch

There are many options for lunch in the Campus Centre (Bld 10); additional coffee shops arelocated in the S.T.R.I.P. (Bld 75) or New Horizon (Bld 82).

coffee/tea

During the morning and afternoon breaks, the conference will provide coffee, tea, and aselection of sweets in front of the conference theatre. The morning break on Friday will alsofeature some additional snacks. If you want plain water, there is a water dispenser on theground floor of Bld 28.

internet

The University-wide wireless network is available via ‘eduroam’ (for those with accounts atparticipating institutions).

sponsors

Thanks to the Australian Mathematical Society for its support of the Victorian AlgebraGroup over the past 32 years. Thanks also to the School of Mathematical Sciences for itssupport of this year’s conference.

questions?

Feel free to ask us, or one of the other locals, if you need assistance during the conference.

Enrico Carlini, Heiko Dietrich, Ian Wanless– Conference Organisers

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participantsas of 29 September 2014

Dr Iain Aitchison University of MelbourneMiss Gair Asha La Trobe UniversityMr Darcy Best Monash UniversityDr Jose Burillo Universitat Politecnica de CatalunyaDr Graham Clarke RMIT UniversityDr Michael Coons The University of NewcastleDr Zajj Daugherty University of MelbourneProf Brian Davey La Trobe UniversityDr Heiko Dietrich Monash UniversityDr Norman Do Monash UniversityDr James East University of Western SydneyDr Murray Elder The University of NewcastleDr Enrico Carlini Monash UniversityDr Graham Farr Monash UniversityDr Barry Gardner University of TasmaniaDr Alex Ghitza Melbourne UniversityDr Michael Giudici The University Of Western AustraliaMiss Amelia Gontar University of SydneyMr Josh Grant Monash UniversityMrs Charles Gray La Trobe UniversityMr Cheng Guo University of Technology, SydneyMs Lucy Ham La Trobe UniversityMr Nick Ham University of TasmaniaMiss Sophie Ham Monash UniversityMr John Harrison The University of NewcastleDr Roozbeh Hazrat University of Western SydneyDr Nhan Bao Ho La Trobe UniversityMr Joshua Howie University of MelbourneDr Marcel Jackson La Trobe UniversityDr Tomasz Kowalski La Trobe UniversityMrs Jayama Mahamendige Monash UniversityDr Daniel Mathews Monash UniversityMrs Stacey Mendan La Trobe UniversityProf Chuck Miller University of MelbourneMr Pol Naranjo Barnet Universitat Politecnica de CatalunyaDr Padraig O Cathain Monash UniversityDr Jane Pitkethly La Trobe UniversityDr Arun Ram University of MelbourneDr Lawrence Reeves University of MelbourneDr Amin Sakzad Monash UniversityMiss Tian Sang University of MelbourneMr Murray Smith La Trobe UniversityMr Christopher Taylor La Trobe UniversityMs Na Wang Monash UniversityDr Ian Wanless Monash UniversityMr Timothy Wilson Monash UniversityDr David Wood Monash UniversityDr Wenting Zhang La Trobe University

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abstracts

1. Artin Conjectures: links with geometry,and an elliptic curve

Iain Aitchison(with Lawrence Reeves, Ryuji Abe)

A simple graph derived from the parity of inte-gers leads to a well-studied class of knots andlinks. The simple geometry of these in turn raisesan interesting connection with Artin’s Conjecturerelated to generators of finite fields. In turn,this leads to problems associated with another ofArtin’s Conjectures, that which led to the Rie-mann Hypothesis over finite fields. In our contextthis relates to counting points on a specific andsimple elliptic curve. This talk presents these rela-tionships, culminating in a tantalizing elementaryconjecture concerning quadratic residues.

2. Parity of transversals in latin squares

Darcy Best

A transversal of a latin square of order n is a setof n entries which has exactly one representativefrom each row, column and symbol. A long stand-ing conjecture of Ryser states that the number oftransversals in a latin square is congruent to theorder of the latin square modulo 2. In 1990, Bala-subramanian confirmed this conjecture to be truein the even orders. In this talk, we extend thisproof to show that in squares of order 2 mod 4,the number of transversals is necessarily a mul-tiple of 4. Moreover, the set of transversals maybe further broken down into smaller classes whichalso contain a special property modulo 2.

3. Metric Properties of Baumslag-Solitargroups

Jose Burillo(with Murray Elder)

We use Britton’s normal forms to find estimatesfor the word metric in the Baumslag-Solitar groups.With these estimates, we find lower bounds for thegrowth rate for the groups.

4. From rational to regular

Michael Coons

I will provide an introduction to regular sequencesby surveying some classical examples and resultsas well as presenting a few new ones.

5. Topological recursion and the quantumcurve for monotone Hurwitz numbers

Norman Do(with Alastair Dyer, Daniel Mathews)

Take a permutation and count the number of waysto express it as a product of a fixed number oftranspositions — you have calculated a Hurwitznumber. By adding a mild constraint on such fac-torisations, one obtains the notion of a monotoneHurwitz number. We have recently shown thatthe monotone Hurwitz problem fits into the so-called topological recursion/quantum curve para-digm. This talk will attempt to give the flavourof what exactly the previous sentence means.

6. There are 132 069 776 distincttransformation semigroups on 4 points

James East(with Attila Egri-Nagy, James Mitchell)

We enumerated and classified transformation semi-groups acting on up to 4 points. The degree 4 caserequires extensive computation and in this talk wedescribe how the ideal structure of the full trans-formation semigroups allows us to parallelize theenumeration. We also give a short summary of theclassification results and the new open problems.

7. Equations in free groups and freemonoids

Murray Elder(with Laura Ciobanu, Volker Diekert)

An equation in a free group/monoid is an expres-sion like aX2 = XY , where a is an element of thegroup/monoid and X,Y are variables. A solutionis an assignment of elements to X and Y so thatthe equation is true in the group/monoid.

Using some clever new results by Diekert, Jez andPlandowski, we are able to describe the set of allsolutions to an equation in a free group or freemonoid with involution, as a formal language ofreasonably low complexity. In my talk I will de-scribe the problem, the formal language class, andbriefly describe how the result works.

8. Minors and Tutte invariants foralternating dimaps

Graham Farr

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abstracts

A graph H is a minor of a graph G if it can beobtained from G by a sequence of deletions and/orcontractions of edges. Minors play a central rolein graph theory, in characterising graph propertiesand in counting various structures associated withgraphs. The theory of minors is intimately relatedto duality in graphs and matroids.

This talk will describe a theory of minors for alter-nating dimaps (i.e., orientably embedded digraphsin which the edges incident at a vertex are directedalternately into, and out of, the vertex). There arenow three fundamental reductions, rather thantwo (as for deletion and contraction), and theyare related by a transform due to Tutte (1948)called triality or trinity, which extends duality.

The theory has many analogies with standard mi-nor theory. However, these minor operations arenon-commutative. This makes their theory moredifficult, but we are still able to establish some ofthe kinds of results one would hope for in a the-ory of minors, such as excluded minor characteri-sations and enumerative invariants that obey lin-ear recurrence relations analogous to the deletion-contraction relation for the Tutte polynomial.

9. On Skew Polynomial Rings and SomeRelated Rings

Barry Gardner(with Elena Cojuhari)

Skew monoid rings (which include skew polyno-mial rings)form a rather restricted subclass of acertain class of modified monoid rings. We shallexamine their relationship with the larger classand their connection with finite normalizing ex-tensions. We shall also present a characterizationof rings with a grading by the cyclic group of order2.

10. Commuting graphs

Michael Giudici

Given a group G, the commuting graph of G is thegraph with vertices the noncentral elements of G,and two vertices are adjacent if and only if theycommute. Commuting graphs of other algebraicstructures can be defined similarly. Commutinggraphs of groups have received a lot of attentionin recent years. I will discuss some recent resultsabout the structure of such graphs, and in partic-ular their diameter and which graphs can be thecommuting graphs of groups and other algebraicstructures.

11. Polynomial Rank and TransformationBetween Symmetric Pure States

Cheng Guo

Multipartite entanglement has been widely stud-ied since it is a proven asset to information pro-cessing and computational tasks. In order to makequantitative comparison between different typesof quantum information resources via StochasticLocal Operations and Classical Communication(SLOCC), we want to find the structure of theseprotocol mathematically. We notice that the ten-sor rank of any state cannot increase by SLOCC.When these pure states are symmetric, polyno-mial rank show the upper bound of the tensor rankand the lower bound of the transformation radio.We will also introduce some recent result.

12. Preservation Theorems in FiniteStructures

Lucy Ham

It is widely known that most classical results inmodel theory fail or become meaningless when re-stricted to the realm of finite structures. A famousexample is the Los-Tarski Preservation Theorem,which says that a first-order sentence is preservedby taking extensions if and only if it is equivalentto an existential sentence: this fails at the finitelevel for both relational and algebraic signatures.

The one notable exception is Rossman’s FiniteHomomorphism-Preservation Theorem: a first-order sentence is preserved under homomorphismson finite relational structures if and only if it isequivalent, on the class of finite relational struc-tures, to an existential positive sentence.

Relativising such theorems can change their truth:the Los-Tarski Theorem for example is recoveredat the finite level when restricted to classes ofacyclic graphs closed under substructures and dis-joint unions. We show that some broad cases ofRossmans result continue to hold under relativisa-tions. But in contrast, we show the correspondingrelativised theorem fails in algebraic signatures,giving perhaps the first case of a classical preser-vation theorem holding for relational structuresat the finite level, but not for algebraic structuresat the finite level. It remains unknown if Ross-mans result (unrelativised) holds in the algebraicsetting.

13. The Fauser Monoid

Nick Ham

The Jones monoid Jn is generated by hook dia-grams, by including generators of the symmetricgroup we get the Brauer monoid Bn. We will belooking at the monoids generated when we replacethe hooks in the generators of Jn and Bn withtriapses, the latter of which we refer to as theFauser monoid Fn. I will begin by discussing var-ious bits of information about these monoids, such

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abstracts

as: characterising and counting the elements; dif-ficulties with terminology; Green’s relations; andcomplications with my so far hopeless search fora presentation. Then end by discussing what wehope to achieve with the semigroup algebra of Fn.

14. Graded Algebras, classifications viaGraded K-theory

Roozbeh Hazrat

From a graph (e.g., cities and flights betweenthem) one can generate an algebra which capturesthe movements along the graph.

This talk is about one type of such correspon-dences, i.e., Leavitt path algebras.

Despite being introduced only 10 years ago, Leav-itt path algebras have arisen in a variety of dif-ferent contexts as diverse as analysis, symbolicdynamics, noncommutative geometry and repre-sentation theory. In fact, Leavitt path algebrasare algebraic counterpart to graph C*-algebras,a theory originated and nourished in Australianuniversities which has become an area of inten-sive research globally. There are strikingly paral-lel similarities between these two theories. Evenmore surprisingly, one cannot (yet) obtain the re-sults in one theory as a consequence of the other;the statements look the same, however the tech-niques to prove them are quite different (as thenames suggest, one uses Algebra and other Anal-ysis). These all suggest that there might be abridge between Algebra and Analysis yet to beuncovered.

In this talk, we introduce Leavitt path algebrasand then try to understand the behaviour and toclassify them by means of (graded) K-theory. Wewill ask nice questions!

15. On the closureness of impartialcombinatorial games

Nhan Bao Ho

An impartial combinatorial game is a two-playergame in which the players move alternately, fol-lowing some given set of rules of moves. We as-sume the game has only a finite number of posi-tions each of which can be visited at most once andso the game must terminate after some numberof moves. The game has no hidden informationand no element of luck. The player who makesthe last move wins. In sum of two games, a playercan choose any game and then makes a legal movein that game. The next player can choose eitherof game to move. We discuss the closureness oftwo families of impartial games called ”tame” and”miserable” under the sum operator.

16. The relative 1-line property for knotexteriors

Joshua Howie

The relative 1-line property is a condition on afamily of intersections of surface subgroups of a 3-manifold group constructed from the knot group.

The relative 1-line property allows us to give analgebraic characterisation of what it means for aknot to be alternating, answering an old questionof Ralph Fox. We can also use this to characterisea class of knots which have alternating projectionsonto higher genus surfaces.

We will also consider the problem of decidingwhether two surface subgroups have the relative1-line property.

17. Ghosts on a plane. And nilpotence.

Marcel Jackson

A Natural Duality is a dual equivalence betweena class of algebraic objects and a class of topologi-cal structures, arising over some generating objectby way a particular canonical construction. Manyclassical dualities fall under this umbrella, includ-ing vector space duality (over the field of scalars)and Stone’s duality for Boolean algebras (over thetwo element Boolean algebra).

Not every algebraic object can carry a natural du-ality (as a generating object), and so it emerges asof obvious interest to understand which can andwhich cannot.

In this talk we return to the currently incom-plete classification of finite dualisable semigroups.We introduce a geometric argument for showingnondualisability of semigroups (and other alge-bras), and find it sufficiently powerful to showthat almost all finite semigroups are not dualis-able. Amongst other broad applications of thetechnique is a fresh proof that a finite group isnondualisable when it has a nonabelian Sylow sub-group.

18. Quasivarieties of directed paths

Tomasz Kowalski(with Marcel Jackson, Michal Stronkowski)

A quasivariety is a class of relational structuresdefined by a set of quasi-identities, that is, univer-sally quantified formulas ϕ1& . . .&ϕn → ψ, whereϕi and ψ are atomic. In the language of directedgraphs atomic formulas are: x = y, and E(x, y)(stating that there is a directed edge between xand y). A quasivariety Q is finitely based, if Q isdefined by a finite set of quasi-identities. A qua-sivariety Q is finitely generated, if it is preciselythe class of isomorphic copies of substructures ofdirect products of a finite set S of structures.

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abstracts

It is well known that in the class of (undirected)graphs there are only 4 finitely based quasivari-eties, but for directed graphs there are infinitelymany. We investigate quasivarieties generated bydirected paths and show that: (1) If Q is a qua-sivariety generated by ”mostly upward” directedpaths, then Q is finitely based if and only if it isfinitely generated. (2) There are continuum manydistinct quasivarieties generated by ”mostly up-ward” paths.

Time permitting, we mention some results aim-ing at a complete characterisation of finitely basedquasivarieties of directed paths.

19. Near-autoparatopisms of Latin squares

Jayama Mahamendige(with Ian Wanless)

A Latin square L of order n is an n × n ar-ray containing n symbols from [n] = {1, 2, . . . , n}such that each element of [n] appears exactly oncein each row and each column of L. Rows andcolumns of L are indexed by elements of [n]. Theelement in the ith row and jth column is denotedby L(i, j).The set of n2 ordered triples,

O(L) = {(i, j, L(i, j)); i, j ∈ [n]}is called the orthogonal array representation of L.

The Hamming distance between two Latin squaresL and L′ is defined by

dist(L,L′) = #{d ∈ O(L); d 6∈ O(L′)}.

The elements of the form σ = (α, β, γ;λ) ∈ Sn oS3

are known as paratopisms. When λ = ε, σ issaid to be an isotopism. In other words, θ =(α, β, γ) ∈ S3

n is said to be an isotopism. Whenθ = (α, α, α) ∈ S3

n, then θ is known as an isomor-phism.

If dist(L,Lσ) = 0, that is, Lσ = L, then σ isknown as an autoparatopism. If dist(L,Lσ) =4, then σ is called a near-autoparatopism of L.If dist(L,Lθ) = 0, that is, Lθ = L, then θ isknown as autotopism. If dist(L,Lθ) = 4, thenθ is called a near-autotopism of L. The similardefinitions for isomorphisms are automorphismsand near-automorphisms.

If a submatrix M of a Latin square L is also aLatin square then M is called a subsquare of L.A subsquare of order 2 is an intercalate.

Cavenagh and Stones gave a proof to the followingresult.

For all n ≥ 2 except n ∈ [3, 4] there exists aLatin square L of order n that admits a near-automorphism. In this process, they found anecessary and sufficient condition for α to be a

near-automorphism when α has the cycle struc-ture (2, n− 2), where n ≥ 0 and n ≡ 2 mod 4.

We determine a family of Latin squares withunique subsquare of order two(intercalate) whichadmits a near-autoparatopism. In this process,we find a necessary and sufficient condition forσ = (ε, β, γ : (12)) ∈ Sn o S3 to be a near-autoparatopism when the cycle structure of β andγ are (n− 2)112 and (n− 2)121 respectively.

20. Unitary Precoding for Integer-ForcingMIMO Linear Receivers

Amin Sakzad(with Emanuele Viterbo)

A flat fading point-to-point multiple-antenna chan-nel is considered where the channel state infor-mation is known at both transmitter and re-ceiver. At the transmitter side, we use a lat-tice encoder to map information symbols to lat-tice codewords. The lattice coded layers are thenprecoded using unitary matrices satisfying non-vanishing minimum product distance. At the re-ceiver side, an integer-forcing linear receiver is em-ployed. This scheme is called ”unitary precodedinteger-forcing”. We show that by applying theproposed precoding technique full-diversity can beachieved. We then verify this result by conductingcomputer simulations in a 2×2 and 4×4 multiple-input multiple-output (MIMO) channel using full-diversity algebraic rotation precoder matrices.

21. The Boundary Between Decidabilityand Undecidability in the Tarski Signature

Murray Smith

Algebras of relations can be viewed as a natu-ral extension of Boole’s algebra of logic: as wellas the usual set theoretic operations, one hascomposition, converse and the identity relation.Tarski proposed a succinct equational axiomati-sation and showed it covered much of the basictheory. But these axioms turned out to be incom-plete, and moreover it is now known that no finitesystem of laws will suffice. Worse still: recognis-ing when a finite abstract algebra is isomorphic toan algebra of relations is undecidable!

The purely compositional fragment of relation al-gebra is easily captured: it is semigroup the-ory. So where in the signature do these complica-tions arise? We investigate identify reducts of theTarski signature lying at the boundary betweendecidability and undecidability.

22. Discriminator Varieties ofDouble-Heyting Algebras

Christopher Taylor

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Just as Boolean algebras provide an algebraiccounterpart to classical logic, Heyting algebras arethe algebraic counterpart of intuitionistic logic,where the law of the excluded middle (i.e. “p or¬p” is always true) is rejected. The rejection ofthis law means that implication p → q cannot bedefined as ¬p or q in an intuitionistic setting.

A Heyting algebra is a bounded distributive lat-tice endowed with an additional binary operation→ (corresponding to intuitionistic implication).Including the dual of implication as a further bi-nary operation gives us the class of double-Heytingalgebras.

A discriminator variety is an equational class Vthat has a term t(x, y, z) such that t is a discrimi-nator term on every subdirectly irreducible mem-ber of V, i.e. t(x, y, z) = z if x = y and x other-wise. Discriminator algebras are perhaps the mostsuccessful generalisation of Boolean algebras. Itis well-known that any discriminator variety V issemisimple, that is, every subdirectly irreduciblemember of V is simple. However the converse doesnot hold in general. We show that in the case ofdouble-Heyting algebras, the converse is true.

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· All talks are in Bld 29, S13.· Registration, tea, and coffee are next to S13.· Conference dinner is at Taste Baguette, at the back of Bld 11.

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Thursday 2 October Friday 3 October 9.00-9.20 9.20-9.30

Registration Conference opening

9.00-9.55 Cheng Guo: Polynomial rank and transformation between symmetric pure states.

9.30-10.25 Michael Giudici: Commuting graphs.

10.00-10.25 Graham Farr: Minors and Tutte invariants for alternating dimaps.

10.30-10.55 Coffee break 10.30-10.55 Coffee break

11.00-11.25 Darcy Best: Parity of transversals in latin squares.

11.00-11.25 Murray Elder: Equations in free groups and free monoids.

11.30-11.55 Lucy Ham: Preservation theorems in finite structures.

11.30-11.55 Barry Gardner: On skew polynomial rings and some related rings.

12.00-12.25 Nick Ham: The Fauser monoid. 12.00-12.25 Roozbeh Hazrat: Graded algebras, classifications via graded K-theory.

12.30-1.55 Lunch 12.30-1.55 Lunch & VAG AGM

2.00-2.25 Joshua Howie: The relative 1-line property for knot exteriors.

2.00-2.25 Nhan Bao Ho: On the closureness of impartial combinatorial games.

2.30-2.55 Jayama Mahamendige: Near-autoparatopisms of latin squares.

2.30-2.55 Marcel Jackson: Ghosts on a plane. And nilpotence.

3.00-3.25 Murray Smith: The boundary between decidability and undecidability in the Tarski signature.

3.00-3.25 Tomasz Kowalski: Quasivarieties of directed paths.

3.30-3.55 Coffee break 3.30-3.55 Coffee break

4.00-4.25 Christopher Taylor: Discriminator varieties of double-Heyting algebras.

4.00-4.25 Amin Sakzad: Unitary precoding for integer-forcing MIMO linear receivers.

4.30-4.55 Norman Do: Topological recursion and the quantum curve for monotone Hurwitz numbers.

4.30-4.55 José Burillo: Metric properties of Baumslag-solitar groups.

5.00-5.25 James East: There are 132069776 distinct transformation semigroups on 4 points.

5.00-5.25 Michael Coons: From rational to regular.

5.30-5.55 Iain Aitchison: Artin conjectures: links with geometry, and an elliptic curve.

7.00 Dinner at “Taste Baguette”

(drinks from 6.30)

All talks are in S13 (Bld 29).

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