Residuation Theory aims to contribute to literature in the field of ordered algebraic structures, especially on the subject of residual mappings. The book is divided into three chapters. Chapter 1 focuses on ordered sets; directed sets; semilattices; lattices; and complete lattices. Chapter 2 tackles Baer rings; Baer semigroups; Foulis semigroups; residual mappings; the notion of involution; and Boolean algebras. Chapter 3 covers residuated groupoids and semigroups; group homomorphic and isotone homomorphic Boolean images of ordered semigroups; Dubreil-Jacotin and Brouwer semigroups; and lolimorphisms. The book is a self-contained and unified introduction to residual mappings and its related concepts. It is applicable as a textbook and reference book for mathematicians who plan to learn more about the subject.

Many branches of algebra are linked by the theory of modules. Since the notion of a module is obtained essentially by a modest generalisation of that of a vector space, it is not surprising that it plays an important role in the theory of linear algebra. Modules are also of great importance in the higher reaches of group theory and ring theory, and are fundamental to the study of advanced topics such as homological algebra, category theory, and algebraic topology. The aim of this text is to develop the basic properties of modules and to show their importance, mainly in the theory of linear algebra.

At the end of each section we have supplied a number of exercises. These provide ample opportunity to consolidate the results in the body of the text, and we include lots of hints to help the reader gain the satisfaction of solving problems.

The aim of this series of problem-solvers is to provide a selection of worked examples in algebra designed to supplement undergraduate algebra courses. We have attempted, mainly with the average student in mind, to produce a varied selection of exercises while incorporating a few of a more challenging nature. Although complete solutions are included, it is intended that these should be consulted by readers only after they have attempted the questions. In this way, it is hoped that the student will gain confidence in his or her approach to the art of problem-solving which, after all, is what mathematics is all about.

The problems, although arranged in chapters, have not been 'graded' within each chapter so that, if readers cannot do problem n this should not discourage them from attempting problem n + 1. A great many of the ideas involved in these problems have been used in examination papers of one sort or another. Some test papers (without solutions) are included at the end of each book; these contain questions based on the topics covered.

It, as it is often said, mathematics is the queen of science then algebra is surely the jewel in her crown. In the course of its vast development over the last half-century, algebra has emerged as the subject in which one can observe pure mathematical reasoning at its best. Its elegance is matched only by the ever-increasing number of its applications to an extraordinarily wide range of topics in areas other than 'pure' mathematics.

Here our objective is to present, in the form of a series of five concise volumes, the fundamentals of the subject. Broadly speaking, we have covered in all the now traditional syllabus that is found in first and second year university courses, as well as some third year material. Further study would be at the level of 'honours options'. The reasoning that lies behind this modular presentation is simple, namely to allow the student (be he a mathematician or not) to read the subject in a way that is more appropriate to the length, content, and extent, of the various courses he has to take.

Although we have taken great pains to include a wide selection of illustrative examples, we have not included any exercises. For a suitable companion collection of worked examples, we would refer the reader to our series Algebra through practice (Cambridge University Press), the first five books of which are appropriate to the material covered here.

The purpose of the present text is therefore quite simple: to provide, at an advanced undergraduate or beginning graduate level, an introductory course that can be covered in a couple of dozen lectures with enough material to provide a good idea of what categories are about and to whet the appetite of those who wish to proceed to the texts that are written by the experts in the field.

The first eleven sections can easily be used as a self-contained course for first year honours students. Here we cover all the basic material on modules and vector spaces required for embarkation on advanced courses. Concerning the prerequisite algebraic background for this, we mention that any standard course on groups, rings, and fields will suffice. Although we have kept the discussion as self-contained as possible, there are places where references to standard results are unavoidable; readers who are unfamiliar with such results should consult a standard text on abstract algebra. The remainder of the text can be used, with a few omissions to suit any particular instructor's objectives, as an advanced course. In this, we develop the foundations of multilinear and exterior algebra. In particular, we show how exterior powers lead to determinants. In this edition we include also some results of a ring-theoretic nature that are directly related to modules and linear algebra. In particular, we establish the celebrated Wedderburn-Artin Theorem that every simple ring is isomorphic to the ring of endomorphisms of a finite-dimensional module over a division ring. Finally, we discuss in detail the structure of finitely generated modules over a principal ideal domain, and apply the fundamental structure theorems to obtain, on the one hand, the structure of all finitely generated abelian groups and, on the other, important decomposition theorems for vector spaces which lead naturally to various canonical forms for matrices.

Although this second edition is algebraically larger than the first edition, it is geometrically smaller. The reason is simple: the first edition was produced at a time when rampant inflation had caused typesetting to become very expensive and, regrettably, publishers were choosing to produce texts from camera-ready material (the synonym of the day for typescript). Nowadays, texts are still produced from camera-ready material but there is an enormous difference in the quality. The intervening years have seen the march of technology: typesetting by computer has arrived and, more importantly, can be done by the authors themselves. This is the case with the present edition. It was set entirely by the author, without scissors, paste, or any cartographic assistance, using the mathematical typesetting system TEX developed by Professor Donald Knuth, and the document preparation system LATEX developed by Dr Leslie Lamport. To be more precise, it was set on a Macintosh II computer using the package MacTEX developed by FTL systems Inc. of Toronto. We record here our gratitude to Lian Zerafa, President of FTL, for making this wonderful system available.

An Ockham algebra is a natural generalisation of a well known and important notion of a Boolean algebra. Regarding the latter as a bounded distributive lattice with complementation (a dual automorphism of period 2) by a dual endomorphism that satisfies the de Morgan laws, this seemingly modest generalisation turns out to be extremely wide. The variety of Ockham algebras has infinitely many subvarieties including those of de Morgan algebras, Stone algebras, and Kleene algebras. Following pioneering work by Berman in 1977, many papers have appeared in this area of lattice theory to which several important results in the theory of universal algebra are highly applicable. This is the first unified account of some of this research. Particular emphasis is placed on Priestly's topological duality, which involves working with ordered sets and order-reversing maps, hereby involving many problems of a combinatorial nature. Written with the graduate student in mind, this book provides an ideal overview of this are of increasing interest.

The word 'basic' in the title of this text could be substituted by 'elementary' or by 'an introduction to'; such are the contents. We have chosen the word 'basic' in order to emphasise our objective, which is to provide in a reasonably compact and readable form a rigorous first course that covers all of the material on linear algebra to which every student of mathematics should be exposed at an early stage.

By developing the algebra of matrices before proceeding to the abstract notion of a vector space, we present the pedagogical progression as a smooth transition from the computational to the general, from the concrete to the abstract. In so doing we have included more than 125 illustrative and worked examples, these being presented immediately following definitions and new results. We have also included more than 300 exercises. In order to consolidate the student's understanding, many of these appear strategically placed throughout the text. They are ideal for self-tutorial purposes. Supplementary exercises are grouped at the end of each chapter. Many of these are 'cumulative' in the sense that they require a knowledge of material covered in previous chapters. Solutions to the exercises are provided at the conclusion of the text.

We are greatly indebted to Dr Jie Fang for assistance with proofreading and for checking the exercises.

The present text is a greatly enhanced version of the now unavailable Volume 2 in the series Essential Student Algebra. We acknowledge the privilege of having it included in the launch of the Springer SUMS series.

In preparing this second edition we decided to take the opportunity of including, as in our companion volume Further Linear Algebra in this series, a chapter that gives a brief introduction to the use of MAPLE in dealing with numerical and algebraic problems in linear algebra. We have also included some additional exercises at the end of each chapter. No solutions are provided for these as they are intended for assignment purposes.

Most of the introductory courses on linear algebra develop the basic theory of finite-dimensional vector spaces, and in so doing relate the notion of a linear mapping to that of a matrix. Generally speaking, such courses culminate in the diagonalisation of certain matrices and the application of this process to various situations. Such is the case, for example, in our previous SUMS volume Basic Linear Algebra. The present text is a continuation of that volume, and has the objective of introducing the reader to more advanced properties of vector spaces and linear mappings, and consequently of matrices. For readers who are not familiar with the contents of Basic Linear Algebra we provide an introductory chapter that consists of a compact summary of the prerequisites for the present volume.

In order to consolidate the student's understanding we have included a large number of illustrative and worked examples, as well as many exercises that are strategically placed throughout the text. Solutions to the exercises are also provided. Many applications of linear algebra require careful, and at times rather tedious, calculations by hand. Very often these are subject to error, so the assistance of a computer is welcome. As far as computation in algebra is concerned, there are several packages available. Here we include, in the spirit of a tutorial, a chapter that gives a brief introduction to the use of MAPLE in dealing with numerical and algebraic problems in linear algebra.

Finally, for the student who is keen on the history of mathematics, we include a chapter that contains brief biographies of those mathematicians mentioned in our two volumes on linear algebra. The biographies emphasise their contributions to linear algebra.

The notion of an order plays an important rôle not only throughout mathematics but also in adjacent disciplines such as logic and computer science. The purpose of the present text is to provide a basic introduction to the theory of ordered structures. Taken as a whole, the material is mainly designed for a postgraduate course. However, since prerequisites are minimal, selected parts of it may easily be considered suitable to broaden the horizon of the advanced undergraduate. Indeed, this has been the author's practice over many years.

A basic tool in analysis is the notion of a continuous function, namely a mapping which has the property that the inverse image of an open set is an open set. In the theory of ordered sets there is the corresponding concept of a residuated mapping, this being a mapping which has the property that the inverse image of a principal down-set is a principal down-set. It comes therefore as no surprise that residuated mappings are important as far as ordered structures are concerned. Indeed, albeit beyond the scope of the present exposition, the naturality of residuated mappings can perhaps best be exhibited using categorical concepts. If we regard an ordered set as a small category then an order-preserving mapping $f : A \rightarrow B$ becomes a functor. Then f is residuated if and only if there exists a functor $f^{+} : B \rightarrow A$ such that $(f, f^{+})$ is an adjoint pair.

Residuated mappings play a central rôle throughout this exposition, with fundamental concepts being introduced whenever possible in terms of natural properties of them. For example, an order isomorphism is precisely a bijection that is residuated; an ordered set $E$ is a meet semilattice if and only if, for every principal down-set $x^\downarrow$, the canonical embedding of $x^\downarrow$ into $E$ is residuated; and a Heyting algebra can be characterised as a lattice-based algebra in which every translation $\lambda_{x} : y \mapsto x \lor y$ is residuated. The important notion of a closure operator, which arises in many situations that concern ordered sets, is intimately related to that of a residuated mapping. Likewise, Galois connections can be described in terms of residuated mappings, and vice versa. Residuated mappings have the added advantage that they can be composed to form new residuated mappings. In particular, the set $Res E$ of residuated mappings on an ordered set $E$ forms a semigroup, and here we include descriptions of the types of semigroup that arise.

A glance at the list of contents will reveal how the material is marshalled. Roughly speaking, the text may be divided into two parts though it should be stressed that these are not mutually independent. In Chapters 1 to 8 we deal with the essentials of ordered sets and lattices, including Boolean algebras, $p$-algebras, Heyting algebras, and their subdirectly irreducible algebras. In Chapters 9 to 14 we provide an introduction to ordered algebraic structures, including ordered groups, rings, fields, and semigroups. In particular, we include a characterisation of the real numbers as, to within isomorphism, the only Dedekind complete totally ordered field, something that is rarely seen by mathematics graduates nowadays. As far as ordered groups are concerned, we develop the theory as far as proving that every Archimedean lattice-ordered group is commutative. In dealing with ordered semigroups we concentrate mainly on naturally ordered regular and inverse semigroups and provide a unified account which highlights those that admit an ordered group as an image under a residuated epimorphism, culminating in structure theorems for various types of Dubreil-Jacotin semigroups.

Throughout the text we give many examples of the structures arising, and interspersed with the theorems there are bundles of exercises to whet the reader's appetite. These are of varying degrees of difficulty, some being designed to help the student gain intuition and some serving to provide further examples to supplement the text material. Since this is primarily designed as a non-encyclopaedic introduction to the vast area of ordered structures we also include relevant references.

We are deeply indebted to Professora Doutora Maria Helena Santos and Professor Doutor Herberto Silva for their assistance in the proof-reading. The traditional free copy is small recompense for their labour. Finally, our thanks go to the editorial team at Springer for unparalleled courtesy and efficiency.