
Author(s) 
Glass, A.M. (ed.) Holland, W. Charles (ed.) 

Title  LatticeOrdered Groups 
Publisher  Kluwer Academic Publishers 
Year of publication  1989 
Reviewed by  Mirela Stefanescu 
The latticeordered 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 latticeordered 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 latticeordered permutation groups (W. Charles Holland). A chapter on model theory of abelian lgroup (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 lsubgroups. Two papers on lgroups (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 latticeordered groups.
Many questions arise in free lgroups. There are considered in the chapter written by S. H. McCleary.
The position of lgroups among the varieties of groups is discussed by N. R. Reilly, who analyzed abelian lyroups, normalvalued lgroups, torsion classes, as well as the semigroup of varieties, locally nilpotent lgroups, solvable varieties of lgroups, covers of such varieties.
W.B. Powell and C. Tsinakis study free products in varieties of lgroups and amalgamation of lgroups, two strong methods of constructing lgroups.
An impressive list of references ends the book, which is wellprinted.