Lattices and Ordered Algebraic Structures

ISBN-13: 9781852339050 / Angielski / Twarda / 2005 / 304 str.

T. S. Blyth

Opis

The notion of an order plays an important role DEGREES not only throughout mat- maticsbutalsoinadjacentdisciplinessuchaslogicandcomputerscience. 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 the- fore as no surprise that residuated mappings are important as far as ordered structures are concerned. Indeed, albeit beyond the scope of the present - position, 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? B becomes a functor. Then f is + + residuated if and only if there exists a functor f: B? A such that (f, f ) is an adjoint pair.

Recenzje

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)

Spis treści

Ordered sets; residuated mappings.- Lattices; lattice morphisms.- Regular equivalences.- Modular lattices.- Distributive lattices.- Complementation; boolean algebras.- Pseudocomplementation; Stone and Heyting algebras.- Congruences; subdirectly irreducible algebras.- Ordered groups.- Archimedean ordered structures.- Ordered semigroups; residuated semigroups.- Epimorphic group images; Dubreil-Jacotin semigroups.- Ordered regular semigroups.- Structure theorems.

Blurb

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