The theory of lattice-ordered groups.

*(English)*Zbl 0834.06015
Mathematics and its Applications (Dordrecht). 307. Dordrecht: Kluwer Academic Publishers. xvi, 400 p. (1994).

From the preface: “The purpose of this book is to make the deep results of the theory of lattice-ordered groups accessible to mathematicians generally, to portray its structure and indicate some of its applications to group theory. This book is selfcontained for anyone familiar with basic results on group theory. Graduate students and researchers in ordered structures and group theory will find this book valuable both as an introduction into the theory of partially ordered groups, and as a presentation of new ideas and results in this theory.”

According to the reviewer’s opinion, all of the authors’ intentions formulated above are fully realized in the present monograph.

Below we quote this monograph as (KM).

Twelve years ago, the first of the present authors published a book on the same theme [Lattice-ordered groups (1984; Zbl 0567.06011)]); below we quote this book as (K).

It seems to be necessary to compare (KM) and (K). The monograph (KM) consists of 14 chapters. The headings of Chapters 1-7 in (KM) and also the headings of paragraphs in these chapters are the same as the corresponding headings in Chapters 1-7 of (K). There are only minor changes in the content of these chapters, e.g., at the end of some paragraphs some new results are added, mostly without proofs. In an analogous way, Chapter 8 of (KM) corresponds to Chapter 9 of (K). (Let us remark that (K) has 11 chapters, two of them investigate topologies on lattice-ordered groups, and lattice-ordered Lie groups; these problems are not dealt with in (KM).)

Since (K) was already reviewed, we will consider below only the remaining chapters 9-14 of (KM) (approximately a half of the book). These chapters are concerned with varieties and quasivarieties of lattice-ordered groups; the main topics are as follows:

Chapter 9: Holland’s theorem on normal valued \(l\)-groups; Šik’s results on representable \(l\)-groups; weakly abelian \(l\)-groups (Kopytov, Medvedev); Holland’s proof of the fact that each variety of \(l\)-groups is a torsion class.

The relatively short chapter 10 deals with free \(l\)-groups.

Chapter 11 on the semigroup of \(l\)-varieties contains a detailed exposition of the methods using the wreath product, and corresponding results.

Chapter 12, entitled “The lattice of \(l\)-varieties”, can be considered as the heart of that part of (KM), which deals with \(l\)-varieties. A number of important results are presented here (these are mainly due to Holland, Reilly, Medvedev, Kopytov, Scrimger, Gurchenkov and Gorbunov). Headings of some paragraphs: Small \(l\)-varieties; Solvable \(l\)-varieties; Covers in the lattice of \(l\)-varieties; Independent axiomatization of \(l\)-varieties.

In Chapter 13, three particular types of \(l\)-varieties are investigated, namely (i) \(l\)-varieties generated by \(A(\Omega)\), where \(A(\Omega)\) is the \(l\)-group of all order permutations of some linearly ordered set \(\Omega\); (ii) \(l\)-varieties generated by right ordered groups, and (iii) \(l\)-varieties generated by simple \(l\)-groups.

The major part of Chapter 14 consists of studying the covering relation in the lattice of quasivarieties of \(l\)-groups.

The bibliography of (K) had more than 500 items; in (KM) this number is reduced approximately to the half.

According to the reviewer’s opinion, all of the authors’ intentions formulated above are fully realized in the present monograph.

Below we quote this monograph as (KM).

Twelve years ago, the first of the present authors published a book on the same theme [Lattice-ordered groups (1984; Zbl 0567.06011)]); below we quote this book as (K).

It seems to be necessary to compare (KM) and (K). The monograph (KM) consists of 14 chapters. The headings of Chapters 1-7 in (KM) and also the headings of paragraphs in these chapters are the same as the corresponding headings in Chapters 1-7 of (K). There are only minor changes in the content of these chapters, e.g., at the end of some paragraphs some new results are added, mostly without proofs. In an analogous way, Chapter 8 of (KM) corresponds to Chapter 9 of (K). (Let us remark that (K) has 11 chapters, two of them investigate topologies on lattice-ordered groups, and lattice-ordered Lie groups; these problems are not dealt with in (KM).)

Since (K) was already reviewed, we will consider below only the remaining chapters 9-14 of (KM) (approximately a half of the book). These chapters are concerned with varieties and quasivarieties of lattice-ordered groups; the main topics are as follows:

Chapter 9: Holland’s theorem on normal valued \(l\)-groups; Šik’s results on representable \(l\)-groups; weakly abelian \(l\)-groups (Kopytov, Medvedev); Holland’s proof of the fact that each variety of \(l\)-groups is a torsion class.

The relatively short chapter 10 deals with free \(l\)-groups.

Chapter 11 on the semigroup of \(l\)-varieties contains a detailed exposition of the methods using the wreath product, and corresponding results.

Chapter 12, entitled “The lattice of \(l\)-varieties”, can be considered as the heart of that part of (KM), which deals with \(l\)-varieties. A number of important results are presented here (these are mainly due to Holland, Reilly, Medvedev, Kopytov, Scrimger, Gurchenkov and Gorbunov). Headings of some paragraphs: Small \(l\)-varieties; Solvable \(l\)-varieties; Covers in the lattice of \(l\)-varieties; Independent axiomatization of \(l\)-varieties.

In Chapter 13, three particular types of \(l\)-varieties are investigated, namely (i) \(l\)-varieties generated by \(A(\Omega)\), where \(A(\Omega)\) is the \(l\)-group of all order permutations of some linearly ordered set \(\Omega\); (ii) \(l\)-varieties generated by right ordered groups, and (iii) \(l\)-varieties generated by simple \(l\)-groups.

The major part of Chapter 14 consists of studying the covering relation in the lattice of quasivarieties of \(l\)-groups.

The bibliography of (K) had more than 500 items; in (KM) this number is reduced approximately to the half.

Reviewer: J.Jakubík (Košice)