Theory of lattice-ordered groups.

*(English)*Zbl 0810.06016
Pure and Applied Mathematics, Marcel Dekker. 187. New York, NY: Marcel Dekker. vii, 539 p. (1995).

The interplay between a group structure and an order structure on the same set has long been a staple of mathematical study. G. Birkhoff introduced the study of lattice-ordered groups, or just \(\ell\)-groups, in 1942 [Ann. Math. II. Ser. 43, 298-331 (1942; Zbl 0060.05808)]. The book under review grew out of the author’s notes as a graduate student under Professor Paul Conrad, one of the leading researchers in the field, and the author gives credit to P. Conrad’s Tulane Lecture Notes [Lattice-ordered groups (1970; Zbl 0258.06011)] and A. Bigard, K. Keimel and S. Wolfenstein’s book [Groupes et anneaux réticulés (Lect. Notes Math. 608) (1977; Zbl 0384.06022)] as sources for much basic material. However, Darnel goes far beyond these sources, which understandably are somewhat out of date, and he carries the reader into the frontiers of current research.

As its origins would suggest, the book is an excellent choice of a graduate seminar or graduate course on \(\ell\)-groups. No special background is required other than the usual graduate course in abstract algebra. The author has taken care to present insightful and straightforward proofs of standard results, and he has made the effort to find or produce himself improvements on the original proofs of many advanced results. Consequently, the book is more unified than one might expect, and often the reader’s burden is made easier.

An experienced researcher will certainly find the book to be a useful reference on the topics covered and for an introduction to current literature on \(\ell\)-groups. The book ends with a section on “Sources” which directs the reader to original sources and background references corresponding to each chapter. There is a 32-page list of references and an invaluable index of symbols and notation. The book certainly should be in research library mathematical collections.

As to the subject matter itself, the first four chapters form a natural introductory block. Chapter 1 introduces to partially ordered sets, \(\ell\)-groups, \(\ell\)-permutations, and \(\ell\)-subgroups. Chapter 2 deals with \({\mathcal C}(G)\), the collection of convex \(\ell\)-subgroups of the \(\ell\)-group \(G\), which is perhaps the most fundamental object in the study of any \(\ell\)-group. It also deals with \(\ell\)-ideals, prime subgroups, and regular subgroups. Chapter 3 covers polar subgroups – the set of group elements disjoint from \(x\) forms a convex \(\ell\)-subgroup of \(G\) called the polar of \(x\). Polars play a vital role in the study of many important classes of \(\ell\)-groups, indeed the projectable and strongly projectable classes are defined in terms of polars. Chapter 4 introduces closed (i.e., order-closed) convex \(\ell\)-subgroups and completely distributive \(\ell\)-groups.

The short Chapter 5 deals with o-groups, totally ordered groups, mainly to introduce concepts which are employed in \(\ell\)-groups, not to cover the study of o-groups in a comprehensive way. Hölder’s Theorem, that any Archimedean o-group is o-isomorphic to a subgroup of the real numbers, is a highlight.

Every \(\ell\)-group has a representation as a group of order-preserving permutations of a chain [C. Holland, “The lattice-ordered group of automorphisms of an ordered set”, Mich. Math. J. 10, 399-408 (1963; Zbl 0116.02102)]. A number of \(\ell\)-group researchers specialize in using Holland’s representation, and this interesting area of the subject is nicely introduced in Chapter 6, which is practically the only source for the newcomer outside of A. M. W. Glass’s book [Ordered permutation groups (Lond. Math. Soc. Lect. Note Ser. 55) (1981; Zbl 0473.06010)]. The last five chapters, about half of the book’s 539 pages, deal with the many special classes of \(\ell\)-groups: radical classes, free \(\ell\)- groups, normal-valued \(\ell\)-groups, representative and Abelian \(\ell\)- groups, finite-valued \(\ell\)-groups, and Archimedean and hyper- Archimedean \(\ell\)-groups, and \(\ell\)-varieties of \(\ell\)-groups. The author’s own research has been at the forefront in \(\ell\)-varieties and on special classes of \(\ell\)-groups, and these chapters present essentially all the basic theorems needed to enable the reader to go to the most recent work in areas of current research.

As its origins would suggest, the book is an excellent choice of a graduate seminar or graduate course on \(\ell\)-groups. No special background is required other than the usual graduate course in abstract algebra. The author has taken care to present insightful and straightforward proofs of standard results, and he has made the effort to find or produce himself improvements on the original proofs of many advanced results. Consequently, the book is more unified than one might expect, and often the reader’s burden is made easier.

An experienced researcher will certainly find the book to be a useful reference on the topics covered and for an introduction to current literature on \(\ell\)-groups. The book ends with a section on “Sources” which directs the reader to original sources and background references corresponding to each chapter. There is a 32-page list of references and an invaluable index of symbols and notation. The book certainly should be in research library mathematical collections.

As to the subject matter itself, the first four chapters form a natural introductory block. Chapter 1 introduces to partially ordered sets, \(\ell\)-groups, \(\ell\)-permutations, and \(\ell\)-subgroups. Chapter 2 deals with \({\mathcal C}(G)\), the collection of convex \(\ell\)-subgroups of the \(\ell\)-group \(G\), which is perhaps the most fundamental object in the study of any \(\ell\)-group. It also deals with \(\ell\)-ideals, prime subgroups, and regular subgroups. Chapter 3 covers polar subgroups – the set of group elements disjoint from \(x\) forms a convex \(\ell\)-subgroup of \(G\) called the polar of \(x\). Polars play a vital role in the study of many important classes of \(\ell\)-groups, indeed the projectable and strongly projectable classes are defined in terms of polars. Chapter 4 introduces closed (i.e., order-closed) convex \(\ell\)-subgroups and completely distributive \(\ell\)-groups.

The short Chapter 5 deals with o-groups, totally ordered groups, mainly to introduce concepts which are employed in \(\ell\)-groups, not to cover the study of o-groups in a comprehensive way. Hölder’s Theorem, that any Archimedean o-group is o-isomorphic to a subgroup of the real numbers, is a highlight.

Every \(\ell\)-group has a representation as a group of order-preserving permutations of a chain [C. Holland, “The lattice-ordered group of automorphisms of an ordered set”, Mich. Math. J. 10, 399-408 (1963; Zbl 0116.02102)]. A number of \(\ell\)-group researchers specialize in using Holland’s representation, and this interesting area of the subject is nicely introduced in Chapter 6, which is practically the only source for the newcomer outside of A. M. W. Glass’s book [Ordered permutation groups (Lond. Math. Soc. Lect. Note Ser. 55) (1981; Zbl 0473.06010)]. The last five chapters, about half of the book’s 539 pages, deal with the many special classes of \(\ell\)-groups: radical classes, free \(\ell\)- groups, normal-valued \(\ell\)-groups, representative and Abelian \(\ell\)- groups, finite-valued \(\ell\)-groups, and Archimedean and hyper- Archimedean \(\ell\)-groups, and \(\ell\)-varieties of \(\ell\)-groups. The author’s own research has been at the forefront in \(\ell\)-varieties and on special classes of \(\ell\)-groups, and these chapters present essentially all the basic theorems needed to enable the reader to go to the most recent work in areas of current research.

Reviewer: S.P.Hurd (Charleston)