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The word problem for finitely presented inverse monoids (GRAYR_U21SCIOM)

School of Mathematics

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Dr R Gray , Dr S Lyle No more applications being accepted Competition Funded PhD Project (Students Worldwide)
Norwich United Kingdom Pure Mathematics

About the Project

Combinatorial algebra is the study of infinite algebraic objects like groups, and more generally monoids, defined in terms of a generating set, and a set of defining relations holding among those generators. The set of generators and relations is called a presentation. When studying algebraic structures defined by presentations in generators and relations certain fundamental algorithmic problems naturally arise. The most important of these is the word problem, which asks whether there is an algorithm which takes two expressions over the generators and decides whether they represent the same element. When such an algorithm exists the word problem is said to be decidable.  

The study of the word problem involves many interesting ideas from algebra, logic, geometry, and theoretical computer science. In general there is no algorithm to solve the word problem. This has led researchers to identify and study interesting classes of finitely presented groups and monoids all of whose members have decidable word problem including: hyperbolic groups (in the sense of Gromov), one-relator groups, and groups and monoids that admit presentations by finite complete rewriting systems; see [4]. 

The main objects of study in this PhD project are algebraic structures called inverse monoids. While groups are an algebraic abstraction of permutations, and monoids of arbitrary mappings, inverse monoids correspond to partial bijections and provide an algebraic framework for studying partial symmetries of structures. In this project you will investigate the word problem for finitely presented inverse monoids. Your project will build on recent results [1,2] about the word problem for inverse monoids with a single defining relator. You will develop a theory which relates algorithmic properties of inverse monoids to properties of their subgroups. You will also study closely related algorithmic problems for groups including the prefix membership problem [3], and the rational subset membership problem.  

For more information on the supervisor for this project, please go here

This is a PhD programme. The start date is 1st October 2021. The mode of study is full time. The studentship length is 3 years. 

Entry requirements: 2:1 in Mathematics.

Funding Notes

Successful Home candidates who meet UKRI’s eligibility criteria will be awarded an EPSRC studentship in Mathematical Sciences covering fees, stipend (£15,285 pa) for 4 years. The eligibility requirements are detailed in UKRI Training Grant Guidance: (see Annex B for criteria for classification as a Home student).

Applicants to this project will also be considered for a 3 year UEA funded studentship covering stipend (£15,285 pa), tuition fees (Home only). International applicants (EU/non-EU) are eligible for UEA funded studentships but required to fund the difference between Home and International tuition fees (which are detailed on the University’s fees pages


[1] I. Dolinka, R. D. Gray. New results on the prefix membership problem for one-relator groups. arXiv:1911.06571.
[2] R. D. Gray. Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups. Invent. Math., 219:987–1008, 2020.
[3] S. V. Ivanov, S. W. Margolis, and J. C. Meakin. On one-relator inverse monoids and one-relator groups. J. Pure Appl. Algebra, 159(1):83–111, 2001.
[4] R. C. Lyndon and P. E. Schupp. Combinatorial group theory. Classics in Mathematics. Springer- Verlag, Berlin, 2001.
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