University of Southampton Research Repository ePrints Soton Copyright © and Moral Rights for this thesis are retained by the author and/or other copyright owners. A copy can be downloaded for personal non-commercial research or study, without prior permission or charge. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the copyright holder/s. The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the copyright holders. When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given e.g. AUTHOR (year of submission) "Full thesis title", University of Southampton, name of the University School or Department, PhD Thesis, pagination http://eprints.soton.ac.uk University of Southampton Faculty of Social and Human Sciences School of Mathematics Covers of Acts over Monoids Alexander Bailey Thesis for the degree of Doctor of Philosophy July 2013 UNIVERSITY OF SOUTHAMPTON ABSTRACT FACULTY OF SOCIAL AND HUMAN SCIENCES SCHOOL OF MATHEMATICS Doctor of Philosophy COVERS OF ACTS OVER MONOIDS by Alexander Bailey Since they were first defined in the 1950's, projective covers (the dual of injective envelopes) have proved to be an important tool in module theory, and indeed in many other areas of abstract algebra. An attempt to generalise the concept led to the introduction of covers with respect to other classes of modules, for example, injective covers, torsion-free covers and flat covers. The flat cover conjecture (now a Theorem) is of particular importance, it says that every module over every ring has a flat cover. This has led to surprising results in cohomological studies of certain categories. Given a general class of objects X , an X -cover of an object A can be thought of a the `best approximation' of A by an object from X . In a certain sense, it behaves like an adjoint to the inclusion functor. In this thesis we attempt to initiate the study of different types of covers for the category of acts over a monoid. We give some necessary and sufficient conditions for the existence of X -covers for a general class X of acts, and apply these results to specific classes. Some results include, every S-act has a strongly flat cover if S satisfies Condition (A), every S-act has a torsion free cover if S is cancellative, and every S-act has a divisible cover if and only if S has a divisible ideal. We also consider the important concept of purity for the category of acts. Giving some new characterisations and results for pure monomorphisms and pure epimorphisms. i Contents Abstracti Contents iii Author's declaration vii Acknowledgements ix Introduction1 1 Preliminaries5 1.1 Set theory.............................5 1.1.1 Zorn's Lemma......................5 1.1.2 Ordinal Numbers.....................7 1.1.3 Cardinal numbers....................8 1.2 Category theory.........................9 1.2.1 Types of morphism.................... 10 1.2.2 Terminal and initial objects............... 12 1.2.3 Slice and coslice categories............... 13 1.2.4 Functors and adjoints.................. 13 1.2.5 Covers and envelopes................... 16 1.3 Semigroup theory......................... 18 2 Acts over monoids 21 2.1 The category of S-acts...................... 21 2.2 Congruences............................ 23 2.3 Colimits and limits of acts.................... 25 iii iv CONTENTS 2.3.1 Coproducts and products................ 26 2.3.2 Pushouts and pullbacks................. 27 2.3.3 Coequalizers and equalizers............... 28 2.3.4 Directed colimits..................... 29 2.4 Structure of acts......................... 33 2.5 Classes of acts.......................... 36 2.5.1 Free acts......................... 36 2.5.2 Finitely presented acts.................. 36 2.5.3 Projective acts...................... 38 2.5.4 Flat acts.......................... 38 2.5.5 Torsion free acts..................... 42 2.5.6 Injective acts....................... 42 2.5.7 Divisible acts....................... 44 2.5.8 Summary......................... 45 2.5.9 Directed colimits of classes of acts........... 45 3 Coessential covers 53 3.1 Projective coessential covers................... 54 3.1.1 Projective coessential covers of modules........ 54 3.1.2 Projective coessential covers of acts.......... 54 3.2 Flat coessential covers...................... 56 3.2.1 Flat coessential covers of modules........... 56 3.2.2 Flat coessential covers of acts.............. 56 4 Purity 59 4.1 Pure epimorphisms........................ 61 4.1.1 n-pure epimorphisms................... 65 4.1.2 X -pure congruences................... 67 4.2 Pure monomorphisms...................... 68 5 Covers of acts 77 5.1 Preliminary results on X -precovers............... 78 5.2 Examples of SF-covers...................... 82 5.2.1 The one element act over (N; +)............ 83 5.2.2 The one element act over (N; ·)............. 87 CONTENTS v 5.3 Precover implies cover...................... 91 5.4 Weak solution set condition................... 96 5.5 Weakly congruence pure..................... 98 5.6 Covers with the unique mapping property........... 101 6 Applications to specific classes 105 6.1 Free covers............................ 105 6.2 Projective covers......................... 106 6.3 Strongly flat covers........................ 108 6.4 Condition (P ) covers....................... 116 6.5 Condition (E) covers....................... 117 6.6 Flat covers............................ 117 6.7 Torsion free covers........................ 118 6.8 Principally weakly flat covers.................. 119 6.9 Injective covers.......................... 119 6.10 Divisible covers.......................... 121 7 Open Problems and Further Work 125 A Normak's Theorem 127 B Govorov-Lazard Theorem 131 Index 143 Author's declaration I, Alexander Bailey, declare that the thesis entitled Covers of Acts over Monoids and the work presented in the thesis are both my own, and have been generated by me as the result of my own original research. I confirm that: • this work was done wholly or mainly while in candidature for a research degree at this University; • where any part of this thesis has previously been submitted for a degree or any other qualification at this University or any other institution, this has been clearly stated; • where I have consulted the published work of others, this is always clearly attributed; • where I have quoted from the work of others, the source is always given. With the exception of such quotations, this thesis is entirely my own work; • I have acknowledged all main sources of help; • where the thesis is based on work done by myself jointly with others, I have made clear exactly what was done by others and what I have contributed myself; • parts of this work have been published as: Covers of acts over monoids and pure epimorphisms, A. Bailey and J. Renshaw, to appear in Proc. Edinburgh Math. Soc, 2013; Covers of acts over monoids II, A. Bailey and J. Renshaw, Semigroup Forum Vol. 87, pp 257{274, 2013. Signed ............................................................ Date ............................................................... vii Acknowledgements Firstly I would like to thank my parents for always encouraging me to pursue my dreams even when they have no idea what my thesis is about. Secondly, I must thank my supervisor Jim Renshaw. He has been so easy to work with and has consistently given me more time than is ever expected of a supervisor. I am most grateful for the excellent North British Semigroups and Appli- cations Network (NBSAN) and its organisers. Special thanks goes to Mark Kambites and Victoria Gould for all their support. Their termly meetings have provided opportunities for me and many other postgraduate students working in Semigroup Theory to meet other researchers, give seminars, help organise meetings and make lasting friendships. Among these new friend- ships I must especially thank Alistair Wallis for his hospitality in graciously housing me whenever I have visited Edinburgh. I have enjoyed our con- versations and have benefitted greatly from all of his freely given business advice. Thank you to all my University friends, for giving me many good mem- ories. In particular, Martin Fluch who made me feel welcome from day one, Martin Finn-Sell for being my fellow `semigroup theorist', Tom Harris for all our categorical conversations, Ana Khukhro for our calclulus classes, Raffaele Rainone for our many blackboard sessions, as well as Chris Cave, Joe Tait, Rob Snocken, Simon St. John-Green and Bana Al Subaiei. From my Southampton friends, special mention goes to Tristan Aubrey- Jones, David and Katie Coles, Michael and Alice Whitear-Housman and Paul Collet and his family all of whom have shown unending friendship and help and many of whom gave me a roof over my head during my `home- less' period! Above all, I must mention Ann Hutchinson without whose encouragement and support I never would have got this far. Thank you! I also acknowledge the help of the Mathematics Research Fellowship from the School of Mathematics in supporting me this last year. ix Introduction Given a category C, and a subcategory X ⊆ C, an X -cover can be thought of as the `best approximation' of an object in C by an object from X . In particular, covers (and their categorical dual, envelopes), have proved to be an important tool in module theory. This is explained succinctly in the in- troduction to G¨obel and Trlifaj's