Lattice-ordered groups. An introduction.

*(English)*Zbl 0636.06008
Reidel Texts in the Mathematical Sciences. Dordrecht (Netherlands) etc.: D. Reidel Publishing Company. vii, 190 p. $ 44.00; £29.50; Dfl. 98.00 (1988).

This book can serve as a clearly written and well-organized text for the introductory study of lattice-ordered groups. It consists of the following chapters: Fundamentals; Bernau’s representation for archimedian \(\ell\)-groups; The Conrad-Harvey-Holland representation; Representable and normal valued \(\ell\)-groups; Holland’s embedding theorem; Free \(\ell\)-groups; Varieties of \(\ell\)-groups; Completions of representable and archimedean \(\ell\)-groups; The lateral completion; Finite-valued and special valued \(\ell\)-groups; Groups of divisibility; Appendix: A menagerie of examples (varieties of \(\ell\)-groups; torsion and radical classes of \(\ell\)-groups; examples of lattice-ordered groups).

Most of the chapters contain the standard basic material on the subject under consideration. The chapters dealing with varieties of \(\ell\)-groups and with groups of divisibility are nearer to the up-to-date frontier (let us remark that the authors of the book also contributed to these questions). The Appendix (about 40 pages) will be certainly extremely useful not only for beginners but also for specialists in the field, in view of many skillfully chosen and valuable examples and counter- examples. In the rather exhaustive bibliography there are several misprints.

Most of the chapters contain the standard basic material on the subject under consideration. The chapters dealing with varieties of \(\ell\)-groups and with groups of divisibility are nearer to the up-to-date frontier (let us remark that the authors of the book also contributed to these questions). The Appendix (about 40 pages) will be certainly extremely useful not only for beginners but also for specialists in the field, in view of many skillfully chosen and valuable examples and counter- examples. In the rather exhaustive bibliography there are several misprints.

Reviewer: J.Jakubik

##### MSC:

06F15 | Ordered groups |

06-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

20F60 | Ordered groups (group-theoretic aspects) |