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Book review

Author(s) Glass, A.M. (ed.)
Holland, W. Charles (ed.)
Title Lattice-Ordered Groups
Publisher Kluwer Academic Publishers
Year of publication 1989
Reviewed by Mirela Stefanescu

The lattice-ordered groups were studied in the last thirty years, and they have cropped up in many other areas of mathematics (theory of Bezout domains, in the study of unstable theories in amalgamation of universal algebra; in the study of symmetries; in combinatorics, in completion theory and so on).

A conference in Bowling Green (May 1985) has revealed the richness and the diversity of the theory of lattice-ordered groups. Asking to some speakers at the conference to prepare a chapter for this book, the editors wanted a large audience. They begin with elementary facts: homomorphisms, prime subgroups, values and structure theorems, followed by lattice-ordered permutation groups (W. Charles Holland). A chapter on model theory of abelian l-group (V. Weispfenning) introduces the reader just in the middle of the subject. J.L. Mott presents the groups of divisibility with open problems, M. Anderson, P. Conrad and J. Martinez wrote about the lattice of convex l-subgroups. Two papers on l-groups (torsion theory - J. Martinez, completions - R.N. Ball) point also out some questions to be done in the near future. Together with A.W. Hager, R.N.Ball characterizes the epimorphisms in archimedian lattice-ordered groups.

Many questions arise in free l-groups. There are considered in the chapter written by S. H. McCleary.

The position of l-groups among the varieties of groups is discussed by N. R. Reilly, who analyzed abelian l-yroups, normal-valued l-groups, torsion classes, as well as the semigroup of varieties, locally nilpotent l-groups, solvable varieties of l-groups, covers of such varieties.

W.B. Powell and C. Tsinakis study free products in varieties of l-groups and amalgamation of l-groups, two strong methods of constructing l-groups.

An impressive list of references ends the book, which is well-printed.