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Lattice-ordered groups. (Reshetochno uporyadochennye gruppy). (Russian) Zbl 0567.06011
Sovremennaya Algebra. Moskva: ”Nauka”. Glavnaya Redaktsiya Fiziko- Matematicheskoj Literatury. 320 p. R. 3.90 (1984).
This monograph consists of eleven chapters. In chapters I and II the basic notions concerning lattices and \(\ell\)-groups are presented. Chapter III includes Šik’s theorems on polars and Conrad’s theorem on the structure of an \(\ell\)-group with a finite number of polars. Most of the results of Chapter III are due to W. C. Holland; these deep theorems concern the representation of an \(\ell\)-group by automorphisms of an ordered set [for a more detailed discussion of these questions cf. A. M. W. Glass: Ordered permutation groups (1981; Zbl 0473.06010)], and simple lattice ordered groups. Chapter V deals with right-ordered groups; the study of this type of structures was stimulated by their application in the theory of free \(\ell\)-groups. Linearly ordered groups and linear orderability of groups are investigated in Chapter V [for a more thorough investigation cf. A. I. Kokorin and V. M. Kopytov: Linearly ordered groups (1972; Zbl 0258.06012), and R. B. Mura and A. H. Rhemtulla: Orderable groups (1977; Zbl 0452.06011)]. Several types of embeddings of \(\ell\)-groups are studied in Chapter VI (including also amalgams of \(\ell\)-groups and free products). A series of recent results on varieties of \(\ell\)-groups is contained in Chapter VIII. This is the most extensive chapter of the book and includes also the results of the author and his collaborators (Medvedev, Gurchenkov). Some lattice properties of \(\ell\)-groups are investigated in Chapter IX (e.g., questions concerning complete distributivity, compact elements, orthogonal completeness). The material of Chapter X is rather miscellaneous (topologies on \(\ell\)-groups, archimedean extensions, and the generalization of Hahn’s theorem (this generalization is due to Conrad, Harvey and Holland)). The book is finished by a short chapter of Lie \(\ell\)-groups.
In view of the well-chosen material and careful presentation, the monograph can serve as an excellent text-book, and at the same time, as a useful reference book on lattice ordered groups. The book is a valuable contribution to the literature by one of the active researchers in the field.
Reviewer: J.Jakubík

MSC:
06F15 Ordered groups
06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06F25 Ordered rings, algebras, modules