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Generalizations of Artin and Coxeter monoids

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Ffitch, Edward Stefan (2019) Generalizations of Artin and Coxeter monoids. PhD thesis, University of Warwick.

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Official URL: http://webcat.warwick.ac.uk/record=b3423507~S15

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Abstract

This thesis is divided into three chapters.

The first chapter looks at a class of generalized Coxeter monoids, the 'CI-monoids' appearing in [23]. We extend results by S.V. Tsaranov [28] and classify all the CI-monoids that have a zero element - an element of the monoid absorbing anything on the left and right. Following this, we partially resolve the classification of the finite CI-monoids, making use of the theory of rewriting systems [20].

The second chapter is an investigation into a related class of monoids, the 'AI-monoids' appearing in [23]. In accordance with [9, 12], we conjecture that every AI-monoid A has a finite Garside family, a distinguished subfamily of A such that every element of A has a certain 'greedy' normal decomposition. We establish the conjecture for a number of cases, and resolve Conjecture 11.12 of [23].

The final chapter extends partial results to the Embedding Conjecture for the monoid of left self-distributivity MLD, as presented by P. Dehornoy [4, p. 428-436, §9.6], [5, p. 518-524, §11.3]. After outlining the theory of leftdistributivity, we consider orthogonality properties of MLD and use these to establish the Embedding Conjecture for other large subfamilies of MLD not previously considered.

Item Type: Thesis (PhD)
Subjects: Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH): Monoids -- Research, Coxeter groups, Artin's conjecture
Official Date: February 2019
Dates:
DateEvent
February 2019Submitted
Institution: University of Warwick
Theses Department: Mathematics Institute
Thesis Type: PhD
Publication Status: Unpublished
Supervisor(s)/Advisor: Krammer, Daan
Extent: ix, 164 leaves
Language: eng

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