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© 2005

Lattices and Ordered Algebraic Structures

  • A unified introduction to the theory of ordered sets and basic ordered algebraic structures; it provides an exposition of all the essentials

  • Up-to-date: it includes recent developments in the theory of residuated lattices and frames

  • Includes numerous exercises and examples

Textbook

Part of the Universitext book series (UTX)

About this book

Introduction

Lattices and Ordered Algebraic Structures provides a lucid and concise introduction to the basic results concerning the notion of an order. Although as a whole it is mainly intended for beginning postgraduates, the prerequisities are minimal and selected parts can profitably be used to broaden the horizon of the advanced undergraduate.

The treatment is modern, with a slant towards recent developments in the theory of residuated lattices and ordered regular semigroups. Topics covered include:

[bulleted list]

residuated mappings

Galois connections

modular, distributive, and complemented lattices

Boolean algebras

pseudocomplemented lattices

Stone algebras

Heyting algebras

ordered groups

lattice-ordered groups

representable groups

Archimedean ordered structures

ordered semigroups

naturally ordered regular and inverse Dubreil-Jacotin semigroups

[end od bulleted list]

Featuring material that has been hitherto available only in research articles, and an account of the range of applications of the theory, there are also many illustrative examples and numerous exercises throughout, making it ideal for use as a course text, or as a basic introduction to the field for researchers in mathematics, logic and computer science.

T. S. Blyth is Professor Emeritus at St. Andrews University, UK

Keywords

Algebraic structure Algebraic structures Boolean algebra Lattice Lattices Ordered groups Ordered semigroups Ordered sets algebra logic sets

Authors and affiliations

  1. 1.School of Mathematics and Statistics, Mathematical InstituteUniversity of St AndrewsSt AndrewsUK

Bibliographic information

Reviews

From the reviews:

"Lattices and Ordered Algebraic Structures is extensive and scholarly, dense but accessible. There are a decent number of exercises and a great deal of interesting material is covered." MAA Online

"This text starts with a thorough introduction to lattice theory … . the text can serve as an introduction to fundamentals in the respective areas from a residuated-maps perspective and with an eye on coordinatization. The historical notes that are interspersed are also worth mentioning. … The exposition is thorough and all proofs that the reviewer checked were highly polished. … Overall, the book is a well-done introduction from a distinct point of view and with exposure to the author’s research expertise … ." (Bernd S. W. Schröder, Mathematical Reviews, Issue 2006 h)

"The purpose of the present text is to provide a basic introduction to the theory of ordered structures. … it can be easily read by students. The treatment of notions is modern … . Featuring material that has been hitherto available only in research articles and an account of the range of applications of the theory, there are also many illustrative examples and numerous exercises throughout the book … . it will be of interest to a broad category of specialists in the respective fields." (Dimitru Busneag, Zentralblatt MATH, Vol. 1073, 2005)

"This introductory account presents the basic notions of ordered sets and lattices, and goes on to describe modular and distributive lattices and Boolean algebras … . As a first introduction, accompanied by exercises, this presents a very readable account, giving … details not usually included." (Mathematika, Vol. 52, 2005)

"Blyth … takes a very personal approach to the material, which he makes explicit in the Preface, and so this is a book of considerable originality. … The exposition is … clear and thorough. … formalism is tempered by well-chosen examples and exercises. The diagrams are excellent. This is a well-written book, clearly of interest to algebraist, but also a useful resource for computer scientists." (John M. Howie, SIAM Review, Vol. 48 (1), 2006)

"This book provides topics in the theory of lattice ordered groups (25 pages), totally ordered rings and fields (20 pages), and partially ordered semigroups (100 pages), reflecting the personal interests of the author. In particular, regular semigroups endowed with a compatible partial order are considered in great detail – providing results proved by the author during last 30 years. The text contains many useful examples and a lot of interesting exercises." (H. Mitsch, Monatshefte für Mathematik, Vol. 147 (1), 2006)

“This text aims to introduce, on post-graduate level, the theory of ordered algebraic structures. … The style of writing is quite clear and self-contained, with many examples and exercises. It is nice to see such applications as Bernstein’s theorem on equipotent sets or the characterisation of the reals as essentially unique Dedekind complete totally ordered field. … the book is well suited for a course in order theory for, say, second or third year mathematicians. … more suited for future researchers on ordered semigroups.” (Isar Stubbe, Bulletin of the Belgian Mathematical Society, 2007)