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Semigroup Congruences: Computational Techniques and Theoretical Applications SEMIGROUP CONGRUENCES: COMPUTATIONAL TECHNIQUES AND THEORETICAL APPLICATIONS Michael Torpey A Thesis Submitted for the Degree of PhD at the University of St Andrews 2019 Full metadata for this item is available in St Andrews Research Repository at: http://research-repository.st-andrews.ac.uk/ Please use this identifier to cite or link to this item: http://hdl.handle.net/10023/17350 This item is protected by original copyright 2 Declarations Candidate's declaration I, Michael Colin Torpey, do hereby certify that this thesis, submitted for the degree of PhD, which is approx- imately 59,000 words in length, has been written by me, and that it is the record of work carried out by me, or principally by myself in collaboration with others as acknowledged, and that it has not been submitted in any previous application for any degree. I was admitted as a research student at the University of St Andrews in September 2014. I received funding from an organisation or institution and have acknowledged the funder(s) in the full text of my thesis. Date: . Signature of candidate: . Supervisor's declaration I hereby certify that the candidate has fulfilled the conditions of the Resolution and Regulations appropriate for the degree of PhD in the University of St Andrews and that the candidate is qualified to submit this thesis in application for that degree. Date: . Signature of supervisor:. Permission for publication In submitting this thesis to the University of St Andrews we understand that we are giving permission for it to be made available for use in accordance with the regulations of the University Library for the time being in force, subject to any copyright vested in the work not being affected thereby. We also understand, unless exempt by an award of an embargo as requested below, that the title and the abstract will be published, and that a copy of the work may be made and supplied to any bona fide library or research worker, that this thesis will be electronically accessible for personal or research use and that the library has the right to migrate this thesis into new electronic forms as required to ensure continued access to the thesis. I, Michael Colin Torpey, confirm that my thesis does not contain any third-party material that requires copyright clearance. The following is an agreed request by candidate and supervisor regarding the publication of this thesis: no embargo on print copy; no embargo on electronic copy. Date: . Signature of candidate: . Date: . Signature of supervisor:. Underpinning Research Data or Digital Outputs I, Michael Colin Torpey, hereby certify that no requirements to deposit original research data or digital outputs apply to this thesis and that, where appropriate, secondary data used have been referenced in the full text of my thesis. Date: . Signature of candidate: . 3 4 Abstract Computational semigroup theory is an area of research that is subject to growing interest. The development of semigroup algorithms allows for new theoretical results to be discovered, which in turn informs the creation of yet more algorithms. Groups have benefitted from this cycle since before the invention of electronic computers, and the popularity of computational group theory has resulted in a rich and detailed literature. Computational semigroup theory is a less developed field, but recent work has resulted in a variety of algorithms, and some important pieces of software such as the Semigroups package for GAP. Congruences are an important part of semigroup theory. A semigroup's congruences deter- mine its homomorphic images in a manner analogous to a group's normal subgroups. Prior to the work described here, there existed few practical algorithms for computing with semigroup congruences. However, a number of results about alternative representations for congruences, as well as existing algorithms that can be borrowed from group theory, make congruences a fertile area for improvement. In this thesis, we first consider computational techniques that can be applied to the study of congruences, and then present some results that have been produced or precipitated by applying these techniques to interesting examples. After some preliminary theory, we present a new parallel approach to computing with con- gruences specified by generating pairs. We then consider alternative ways of representing a congruence, using intermediate objects such as linked triples. We also present an algorithm for computing the entire congruence lattice of a finite semigroup. In the second part of the thesis, we classify the congruences of several monoids of bipartitions, as well as the principal factors of several monoids of partial transformations. Finally, we consider how many congruences a finite semigroup can have, and examine those on semigroups with up to seven elements. 5 Contents List of figures, tables and algorithms 9 Preface 11 Acknowledgements 15 1 Preliminaries 16 1.1 Basic notation . 16 1.2 Semigroups . 17 1.3 Generators . 19 1.4 Homomorphisms . 20 1.5 Congruences . 21 1.6 Generating pairs . 24 1.7 Presentations . 28 1.8 Ideals . 30 1.9 Green's relations . 31 1.10 Regularity . 34 1.11 Element types . 34 1.11.1 Transformations . 35 1.11.2 Partial transformations . 36 1.11.3 Order-preserving elements . 37 1.11.4 Bipartitions . 38 1.12 Computation & decidability . 42 1.13 Union–find . 45 I Computational techniques 50 2 Parallel method for generating pairs 51 2.1 Reasons for parallelisation . 52 2.2 Applicable representations of a semigroup . 53 2.3 Program inputs . 54 2.4 Program outputs . 56 2.5 Finding a presentation . 56 2.6 The methods . 60 2.6.1 Pair orbit enumeration . 60 2.6.2 The Todd{Coxeter algorithm . 63 2.6.3 Rewriting systems . 78 2.7 Running in parallel . 84 2.8 Benchmarking . 85 2.8.1 Examples . 85 2.8.2 Comparison on random examples . 86 2.9 Future work . 90 2.9.1 Pre-filling the Todd{Coxeter algorithm with a left Cayley graph . 96 6 2.9.2 Interaction between the Knuth{Bendix and Todd{Coxeter algorithms . 96 2.9.3 Using concrete elements in the Todd{Coxeter algorithm . 97 2.9.4 Left and right congruences with the Knuth{Bendix algorithm . 97 3 Converting between representations 99 3.1 Ways of representing a congruence . 99 3.1.1 Generating pairs . 99 3.1.2 Groups: normal subgroups . 100 3.1.3 Completely (0-)simple semigroups: linked triples . 101 3.1.4 Inverse semigroups: kernel{trace pairs . 104 3.1.5 Rees congruences . 106 3.2 Converting between representations . 106 3.2.1 Normal subgroups and linked triples . 107 3.2.2 Generating pairs of a Rees congruence . 107 3.2.3 Linked triple from generating pairs . 108 3.2.4 Generating pairs from a linked triple . 111 3.2.5 Kernel and trace from generating pairs . 116 3.2.6 Kernel and trace of a Rees congruence . 120 3.2.7 Trivial conversions . 121 3.3 Further work . 123 3.3.1 Generating pairs from a kernel{trace pair . 123 3.3.2 Rees congruences from generating pairs . 123 3.3.3 Regular semigroups . 124 4 Calculating congruence lattices 126 4.1 The algorithm . 128 4.1.1 Principal congruences . 128 4.1.2 Adding joins . 130 4.2 Implementation . 132 4.3 Examples . 133 II Theoretical applications 137 5 Congruences of the Motzkin monoid 138 5.1 The Motzkin monoid Mn ............................... 140 5.2 Lifted congruences . 142 5.3 Congruence lattice of Mn ............................... 144 5.4 Other monoids . 156 5.4.1 IN-pairs . 156 5.4.2 Results . 157 6 Principal factors and counting congruences 164 6.1 Congruences of principal factors . 164 6.1.1 Principal factors . 164 6.1.2 Full transformation monoid Tn ........................ 165 6.1.3 Other semigroups . 169 6.1.4 Further work . 173 6.2 The number of congruences of a semigroup . 174 6.2.1 Congruence-free semigroups . 174 6.2.2 Congruence-full semigroups . 175 6.2.3 Semigroups with fewer congruences.
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