LINEAR ALGEBRA OVER SEMIRINGS A thesis submitted to The University of Manchester for the degree of Doctor of Philosophy in the Faculty of Engineering and Physical Sciences DAVID WILDING School of Mathematics 2014 Contents Abstract . 5 Declaration . 6 Copyright . 6 Acknowledgements . 7 Introduction 9 1 Background and motivation . 9 2 Basic definitions, notation and conventions . 17 3 Summary of the thesis . 19 Semirings, modules and matrices 25 4 Semirings and modules . 25 5 Linear functions and homomorphisms . 29 6 Matrices and Green's relations . 35 7 Direct products and monoid semirings . 42 The three main problems 49 8 Kernels and separation . 49 9 Extensions and exactness . 54 10 Conjugations on semirings . 63 Transferring exactness 67 11 Exact matrix and product semirings . 67 12 Ideals and exact subsemirings . 69 Linear algebra over rings 77 13 Orthogonal complements and exact annihilators . 77 14 Commutative elementary divisor rings . 81 3 4 Contents Residuated structures 91 15 Preliminary order theory . 91 16 Ordered monoids and actions . 96 17 Residuation and enriched categories . 101 Linear algebra over residuated lattices 107 18 Matrix residuation . 107 19 Involutive residuated lattices . 114 20 Subsets of groups and monoids . 118 Bibliography 125 Index of terminology 133 Sections 1 to 20 contain approximately 37,000 words. 5 Abstract Linear Algebra Over Semirings was submitted by David Wilding to The University of Manchester on 10 September 2014 for the degree of Doctor of Philosophy. Motivated by results of linear algebra over fields, rings and tropical semirings, we present a systematic way to understand the behaviour of matrices with entries in an arbitrary semiring. We focus on three closely related problems concerning the row and column spaces of matrices. This allows us to isolate and extract common properties that hold for different reasons over different semirings, yet also lets us identify which features of linear algebra are specific to particular types of semiring. For instance, the row and column spaces of a matrix over a field are isomorphic to each others' duals, as well as to each other, but over a tropical semiring only the first of these properties holds in general (this in itself is a surprising fact). Instead of being isomorphic, the row space and column space of a tropical matrix are anti- isomorphic in a certain order-theoretic and algebraic sense. The first problem is to describe the kernels of the row and column spaces of a given matrix. These equivalence relations generalise the orthogonal complement of a set of vectors, and the nature of their equivalence classes is entirely dependent upon the kind of semiring in question. The second, Hahn-Banach type, problem is to decide which linear functionals on row and column spaces of matrices have a linear extension. If they all do, the underlying semiring is called exact, and in this case the row and column spaces of any matrix are isomorphic to each others' duals. The final problem is to explain the connection between the row space and column space of each matrix. Our notion of a conjugation on a semiring accounts for the different possibilities in a unified manner, as it guarantees the existence of bijections between row and column spaces and lets us focus on the peculiarities of those bijections. Our main original contribution is the systematic approach described above, but along the way we establish several new results about exactness of semirings. We give sufficient conditions for a subsemiring of an exact semiring to inherit exactness, and we apply these conditions to show that exactness transfers to finite group semirings. We also show that every Boolean ring is exact. This result is interesting because it allows us to construct a ring which is exact (also known as FP-injective) but not self-injective. Finally, we consider exactness for residuated lattices, showing that every involutive residuated lattice is exact. We end by showing that the residuated lattice of subsets of a finite monoid is exact if and only if the monoid is a group. 6 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or other institute of learning. 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DocID=487), in any relevant Thesis restriction declarations deposited in the University Library, The University Library's regulations (see http://www. manchester.ac.uk/library/aboutus/regulations) and in The University's policy on Presentation of Theses. 7 Acknowledgements The research contained in this thesis was generously supported by The University of Manchester Faculty of Engineering and Physical Sciences. My supervisor, Mark Kambites, suggested exactness of semirings as a potential research topic long before I even began my PhD study, and I would like to express my great appreciation to him for guiding me down such an interesting path. I would also like to deeply thank Marianne Johnson, as together the pair have been a constant source of inspiration. I have always found them very keen to discuss my ideas and involve me in their research. Working with Marianne and Mark has been an enormously enjoyable experience, and this thesis would be nowhere near what it is today without their input and advice. Another person who always enthusiastically engaged with my research is Amit Kuber. During our years together at Manchester we have had countless discussions about our work, and I will miss our mathematical joint ventures immensely. I am also very grateful to Philip Bridge, Anthony Chiu, Andrew Davies, Markus Pfeiffer, Laura Phillips, Joe Razavi, Harold Simmons, Matthew Taylor, Marcus Tressl and David Ward for all the useful and interesting discussions we have had. The School of Mathematics at Manchester has a large number of research groups, and I have learnt a great deal as a result of my involvement in several of them. The Mathematical Logic Group, the Tropical Mathematics Reading Group and the Centre for Interdisciplinary Computational and Dynamical Analysis made me feel extremely welcome. I also owe a large part of my mathematical development to the Mathematical Foundations Group (School of Computer Science) and the North British Semigroups and Applications Network. I would finally like to thank my family and friends. It would not have been pos- sible to complete this thesis without their welcome distractions and encouragement. Introduction 1 Background and motivation A semiring is an algebraic structure in which we can add and multiply elements, where multiplication distributes over addition, but in which neither subtraction nor division are necessarily possible. These assumptions might appear prohibitively weak; what interesting facts about semirings can we hope to establish if we cannot even rely on subtraction? But, of course, we do not usually attempt to prove grand results about all structures of a particular kind (sets, groups, rings, and so on). Rather, we focus our energy on structures that are specialised enough to allow meaningful, non-trivial, things to be said about them (well-ordered sets, simple groups, Noetherian rings). With this precedent in mind, the definition of a semiring is entirely fit for purpose. It is weak enough to encompass an extraordinary variety of mathematical objects, yet just strong enough to provide a general framework for matrices and linear algebra that does not need to be rebuilt time and time again as the class of semirings is charted. Semirings were first explicitly defined and deemed worthy of study by Vandiver [82] in 1934, but the notion is so simple and pervasive that several authors before and since independently came very close to formulating a modern abstract definition. For instance, in 1847 Boole [11] axiomatised what we would now call an idempotent semiring (see page 15), and later Hilbert [39] and Huntington [42, 43] formulated axioms for the semiring of non-negative integers. Golan [34] gives a brief history of semiring theory, along with a very useful guide to the various alternative names under which semirings continue to appear. For the brave reader, G lazek[31] provides a comprehensive catalogue of the extensive and disparate literature on semirings and their applications. We cannot possibly describe all the areas of mathematics (and beyond) that have found a use for semirings, so in this section we only discuss a few instances when matrices over semirings are of particular importance.