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James East

Publications and source records attributed to James East.

At least 37 records · Page 2Linked to original sources
arXiv2021

On the enumeration of integer tetrahedra

We consider the problem of enumerating integer tetrahedra of fixed perimeter (sum of side-lengths) and/or diameter (maximum side-length), up to congruence. As we will see, this problem is considerably more difficult than the corresponding problem for triangles, which has long been solved. We expect there are no closed-form solutions to the tetrahedron enumeration problems, but we explore the extent to which they can be approached via classical methods, such as orbit enumeration. We also discuss algorithms for computing the numbers, and present several tables and figures that can be used to visualise the data. Several intriguing patterns seem to emerge, leading to a number of natural conjectures. The central conjecture is that the number of integer tetrahedra of perimeter $n$, up to congruence, is asymptotic to $n^5/C$ for some constant $C\approx 229000$.

math.CO↗
arXiv2021

Properties of congruences of twisted partition monoids and their lattices

We build on the recent characterisation of congruences on the infinite twisted partition monoids $\mathcal{P}_{n}^Φ$ and their finite $d$-twisted homomorphic images $\mathcal{P}_{n,d}^Φ$, and investigate their algebraic and order-theoretic properties. We prove that each congruence of $\mathcal{P}_{n}^Φ$ is (finitely) generated by at most $\lceil\frac{5n}2\rceil$ pairs, and we characterise the principal ones. We also prove that the congruence lattice $\textsf{Cong}(\mathcal{P}_{n}^Φ)$ is not modular (or distributive); it has no infinite ascending chains, but it does have infinite descending chains and infinite antichains. By way of contrast, the lattice $\textsf{Cong}(\mathcal{P}_{n,d}^Φ)$ is modular but still not distributive for $d>0$, while $\textsf{Cong}(\mathcal{P}_{n,0}^Φ)$ is distributive. We also calculate the number of congruences of $\mathcal{P}_{n,d}^Φ$, showing that the array $\big(|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|\big)_{n,d\geq 0}$ has a rational generating function, and that for a fixed $n$ or $d$, $|\textsf{Cong}(\mathcal{P}_{n,d}^Φ)|$ is a polynomial in $d$ or $n\geq 4$, respectively.

math.RA↗
arXiv2021

Classification of congruences of twisted partition monoids

The twisted partition monoid $\mathcal{P}_n^Φ$ is an infinite monoid obtained from the classical finite partition monoid $\mathcal{P}_n$ by taking into account the number of floating components when multiplying partitions. The main result of this paper is a complete description of the congruences on $\mathcal{P}_n^Φ$. The succinct encoding of a congruence, which we call a C-pair, consists of a sequence of $n+1$ congruences on the additive monoid $\mathbb{N}$ of natural numbers and a certain $(n+1)\times\mathbb{N}$ matrix. We also give a description of the inclusion ordering of congruences in terms of a lexicographic-like ordering on C-pairs. This is then used to classify congruences on the finite $d$-twisted partition monoids $\mathcal{P}_{n,d}^Φ$, which are obtained by factoring out from $\mathcal{P}_n^Φ$ the ideal of all partitions with more than $d$ floating components. Further applications of our results, elucidating the structure and properties of the congruence lattices of the ($d$-)twisted partition monoids, will be the subject of a future article.

math.RA↗
arXiv2021

Lattice paths and submonoids of $\mathbb Z^2$

We study a number of combinatorial and algebraic structures arising from walks on the two-dimensional integer lattice. To a given step set $X\subseteq\mathbb Z^2$, there are two naturally associated monoids: $\mathscr F_X$, the monoid of all $X$-walks/paths; and $\mathscr A_X$, the monoid of all endpoints of $X$-walks starting from the origin $O$. For each $A\in\mathscr A_X$, write $π_X(A)$ for the number of $X$-walks from $O$ to $A$. Calculating the numbers $π_X(A)$ is a classical problem, leading to Fibonacci, Catalan, Motzkin, Delannoy and Schroder numbers, among many other well-studied sequences and arrays. Our main results give relationships between finiteness properties of the numbers $π_X(A)$, geometrical properties of the step set $X$, algebraic properties of the monoid $\mathscr A_X$, and combinatorial properties of a certain bi-labelled digraph naturally associated to $X$. There is an intriguing divergence between the cases of finite and infinite step sets, and some constructions rely on highly non-trivial properties of real numbers. We also consider the case of walks constrained to stay within a given region of the plane. Several examples are considered throughout to highlight the sometimes-subtle nature of the theoretical results.

math.CO↗
arXiv2021

Presentations for P^K

It is a classical result that the direct product AxB of two groups is finitely generated (finitely presented) if and only if A and B are both finitely generated (finitely presented). This is also true for direct products of monoids, but not for semigroups. The typical (counter)example is when A and B are both the additive semigroup P = {1,2,3,...} of positive integers. Here P is freely generated by a single element, but P^2 is not finitely generated, and hence not finitely presented. In this note we give an explicit presentation for P^2 in terms of the unique minimal generating set; in fact, we do this more generally for P^K, the direct product of arbitrarily many copies of P.

math.RA↗
arXiv2021

Congruences on infinite partition and partial Brauer monoids

We give a complete description of the congruences on the partition monoid $P_X$ and the partial Brauer monoid $PB_X$, where $X$ is an arbitrary infinite set, and also of the lattices formed by all such congruences. Our results complement those from a recent article of East, Mitchell, Ruskuc and Torpey, which deals with the finite case. As a consequence of our classification result, we show that the congruence lattices of $P_X$ and $PB_X$ are isomorphic to each other, and are distributive and well quasi-ordered. We also calculate the smallest number of pairs of partitions required to generate any congruence; when this number is infinite, it depends on the cofinality of certain limit cardinals.

math.GR↗
arXiv2021

Ehresmann theory and partition monoids

This article concerns Ehresmann structures in the partition monoid $P_X$. Since $P_X$ contains the symmetric and dual symmetric inverse monoids on the same base set $X$, it naturally contains the semilattices of idempotents of both submonoids. We show that one of these semilattices leads to an Ehresmann structure on $P_X$ while the other does not. We explore some consequences of this (structural/combinatorial and representation theoretic), and in particular characterise the largest left-, right- and two-sided restriction submonoids. The new results are contrasted with known results concerning relation monoids, and a number of interesting dualities arise, stemming from the traditional philosophies of inverse semigroups as models of partial symmetries (Vagner and Preston) or block symmetries (FitzGerald and Leech): "surjections between subsets" for relations become "injections between quotients" for partitions. We also consider some related diagram monoids, including rook partition monoids, and state several open problems.

math.RA↗
arXiv2021

Deep learning for detection and segmentation of artefact and disease instances in gastrointestinal endoscopy

The Endoscopy Computer Vision Challenge (EndoCV) is a crowd-sourcing initiative to address eminent problems in developing reliable computer aided detection and diagnosis endoscopy systems and suggest a pathway for clinical translation of technologies. Whilst endoscopy is a widely used diagnostic and treatment tool for hollow-organs, there are several core challenges often faced by endoscopists, mainly: 1) presence of multi-class artefacts that hinder their visual interpretation, and 2) difficulty in identifying subtle precancerous precursors and cancer abnormalities. Artefacts often affect the robustness of deep learning methods applied to the gastrointestinal tract organs as they can be confused with tissue of interest. EndoCV2020 challenges are designed to address research questions in these remits. In this paper, we present a summary of methods developed by the top 17 teams and provide an objective comparison of state-of-the-art methods and methods designed by the participants for two sub-challenges: i) artefact detection and segmentation (EAD2020), and ii) disease detection and segmentation (EDD2020). Multi-center, multi-organ, multi-class, and multi-modal clinical endoscopy datasets were compiled for both EAD2020 and EDD2020 sub-challenges. The out-of-sample generalization ability of detection algorithms was also evaluated. Whilst most teams focused on accuracy improvements, only a few methods hold credibility for clinical usability. The best performing teams provided solutions to tackle class imbalance, and variabilities in size, origin, modality and occurrences by exploring data augmentation, data fusion, and optimal class thresholding techniques.

cs.CV↗
arXiv2021

Presentations for Temperley-Lieb algebras

We give a new and conceptually straightforward proof of the well-known presentation for the Temperley-Lieb algebra, via an alternative new presentation. Our method involves twisted semigroup algebras, and we make use of two apparently new submonoids of the Temperley-Lieb monoid.

math.RA↗
arXiv2020

Idempotents and one-sided units: Lattice invariants and a semigroup of functors on the category of monoids

For a monoid $M$, we denote by $\mathbb G(M)$ the group of units, $\mathbb E(M)$ the submonoid generated by the idempotents, and $\mathbb G_L(M)$ and $\mathbb G_R(M)$ the submonoids consisting of all left or right units. Writing $\mathcal M$ for the (monoidal) category of monoids, $\mathbb G$, $\mathbb E$, $\mathbb G_L$ and $\mathbb G_R$ are all (monoidal) functors $\mathcal M\to\mathcal M$. There are other natural functors associated to submonoids generated by combinations of idempotents and one- or two-sided units. The above functors generate a monoid with composition as its operation. We show that this monoid has size $15$, and describe its algebraic structure. We also show how to associate certain lattice invariants to a monoid, and classify the lattices that arise in this fashion. A number of examples are discussed throughout, some of which are essential for the proofs of the main theoretical results.

math.GR↗
arXiv2020

Congruence lattices of ideals in categories and (partial) semigroups

This paper presents a unified framework for determining the congruences on a number of monoids and categories of transformations, diagrams, matrices and braids, and on all their ideals. The key theoretical advances present an iterative process of stacking certain normal subgroup lattices on top of each other to successively build congruence lattices of a chain of ideals. This is applied to several specific categories of: transformations; order/orientation preserving/reversing transformations; partitions; planar/annular partitions; Brauer, Temperley--Lieb and Jones partitions; linear and projective linear transformations; and partial braids. Special considerations are needed for certain small ideals, and technically more intricate theoretical underpinnings for the linear and partial braid categories.

math.GR↗
arXiv2020

Endoscopy disease detection challenge 2020

Whilst many technologies are built around endoscopy, there is a need to have a comprehensive dataset collected from multiple centers to address the generalization issues with most deep learning frameworks. What could be more important than disease detection and localization? Through our extensive network of clinical and computational experts, we have collected, curated and annotated gastrointestinal endoscopy video frames. We have released this dataset and have launched disease detection and segmentation challenge EDD2020 https://edd2020.grand-challenge.org to address the limitations and explore new directions. EDD2020 is a crowd sourcing initiative to test the feasibility of recent deep learning methods and to promote research for building robust technologies. In this paper, we provide an overview of the EDD2020 dataset, challenge tasks, evaluation strategies and a short summary of results on test data. A detailed paper will be drafted after the challenge workshop with more detailed analysis of the results.

eess.IV↗
arXiv2019

Structure of principal one-sided ideals

We give a thorough structural analysis of the principal one-sided ideals of arbitrary semigroups, and then apply this to full transformation semigroups and symmetric inverse monoids. One-sided ideals of these semigroups naturally occur as semigroups of transformations with restricted range or kernel.

math.GR↗
arXiv2019

Sandwich semigroups in diagram categories

This paper concerns a number of diagram categories, namely the partition, planar partition, Brauer, partial Brauer, Motzkin and Temperley-Lieb categories. If $\mathcal K$ denotes any of these categories, and if $σ\in\mathcal K_{nm}$ is a fixed morphism, then an associative operation $\star_σ$ may be defined on $\mathcal K_{mn}$ by $α\star_σβ=ασβ$. The resulting semigroup $\mathcal K_{mn}^σ=(\mathcal K_{mn},\star_σ)$ is called a sandwich semigroup. We conduct a thorough investigation of these sandwich semigroups, with an emphasis on structural and combinatorial properties such as Green's relations and preorders, regularity, stability, mid-identities, ideal structure, (products of) idempotents, and minimal generation. It turns out that the Brauer category has many remarkable properties not shared by any of the other diagram categories we study. Because of these unique properties, we may completely classify isomorphism classes of sandwich semigroups in the Brauer category, calculate the rank (smallest size of a generating set) of an arbitrary sandwich semigroup, enumerate Green's classes and idempotents, and calculate ranks (and idempotent ranks, where appropriate) of the regular subsemigroup and its ideals, as well as the idempotent-generated subsemigroup. Several illustrative examples are considered throughout, partly to demonstrate the sometimes-subtle differences between the various diagram categories.

math.GR↗
arXiv2019

Green's relations and stability for subsemigroups

We prove new results on inheritance of Green's relations by subsemigroups in the presence of stability of elements. We provide counterexamples in other cases to show in particular that not all right-stable semigroups are embeddable in left-stable semigroups. This is carried out in the context of a survey of the various closely related notions of stability and minimality of Green's classes that have appeared in the literature over the last sixty years, and which have sometimes been presented in different forms.

math.GR↗
arXiv2019

Idempotents and one-sided units in infinite partial Brauer monoids

We study monoids generated by various combinations of idempotents and one- or two-sided units of an infinite partial Brauer monoid. This yields a total of eight such monoids, each with a natural characterisation in terms of relationships between parameters associated to Brauer graphs. We calculate the relative ranks of each monoid modulo any other such monoid it may contain, and then apply these results to determine the Sierpinski rank of each monoid, and ascertain which ones have the semigroup Bergman property. We also make some fundamental observations about idempotents and units in arbitrary monoids, and prove some general results about relative ranks for submonoids generated by these sets. Dedicated to Dr Des FitzGerald on the occasion of his 70th birthday.

math.GR↗