Diagram Monoids and Their Congruences
Lead Research Organisation:
University of St Andrews
Department Name: Mathematics and Statistics
Abstract
Diagram monoids have over the last few years come into sharp focus, because of their fundamental role in the formation of diagram algebras, which in turn have applications in representation theory and theoretical physics, and also because of their intriguing algebraic and combinatorial properties. Until recently, investigations were largely concerned with the basic semigroup-theoretic and combinatorial properties of these monoids, for instance determination of Green's equivalences, computations of minimal generating sets and defining relations, and characterisations and counting of idempotents. PI (NR) and James East (JE) have recently started a collaboration on another important aspect of these monoids, namely their congruences. In a ground-breaking paper (joint with J.D. Mitchell and M. Torpey) they classified all the congruences on all classical diagram monoids over a finite set. In the process they started developing what looks like a promising general theory of semigroup congruences, at least for finite semigroups. Arising from this work are a number of strands, each important in its own right, and the project proposed here is designed to enable these strands to be followed up and for their full potential to be realised. We have identified four work packages -- dealing with infinite partition monoids, infinite diagram monoids, finite and infinite twisted diagram monoids, and development of a general theory of congruences -- which will be investigated in the course of four research visits by NR to JE over a period of two years.
Planned Impact
The main impact will be on the academic areas to which the proposed project belongs: theory of semigroups, diagram algebras, algebra in general, and combinatorics. Because of strong links between semigroup theory and discrete mathematics on one side, and information technology on the other (particularly via theoretical computer science, theory of automata and languages, complexity, etc.) there will be realistic, secondary impacts on the latter. Finally, there will be a tangible impact on the UK mathematics and research overall as a consequence of a high-powered, and extremely timely research proposed here.
Organisations
People |
ORCID iD |
Nik Ruskuc (Principal Investigator) |
Publications
Brookes M
(2023)
Heights of one- and two-sided congruence lattices of semigroups
East J
(2018)
Congruence lattices of finite diagram monoids
in Advances in Mathematics
East J
(2022)
Properties of congruences of twisted partition monoids and their lattices
in Journal of the London Mathematical Society
East J
(2022)
Classification of congruences of twisted partition monoids
in Advances in Mathematics
East J
(2022)
Congruences on Infinite Partition and Partial Brauer Monoids
in Moscow Mathematical Journal
East J
(2023)
Congruence Lattices of Ideals in Categories and (Partial) Semigroups
in Memoirs of the American Mathematical Society
Description | A complete classification of all congruences on a number of finite diagram monoids. General theory for computing congruences in specific monoids, which covers all the currently known instances in literature. Determination of congruences on twisted partition monoids, and a detailed analysis of their properties. |
Exploitation Route | The results of this project are likely to be used by a wide range of researchers in different disciplines, from semigroup theory, theory of diagram algebras and general algebra. |
Sectors | Education |