Right-ordered groups. (Pravouporyadochennye gruppy.) (Russian) Zbl 0896.06017

Kopytov, Valerij M.; Medvedev, Nikolaj Ya.

Sibirskaya Shkola Algebry i Logiki. 2. Novosibirsk: Nauchnaya Kniga. viii, 247 p. (1996).

A linearly ordered set \(G\) equipped by the structure of a (multiplicative) group is said to be a right-ordered group if the right multiplication in the group \(G\) preserves the order defined on \(G\). Regarded as abstract groups, the right-ordered groups are exactly the automorphism groups of linearly ordered sets. The book under review gives a comprehensive account of the current state of the theory of right-ordered groups.
The table of contents. Chapter 1. Introduction: Partially ordered sets. Lattices. Properties of lattices. Orders on groups. Positive cones. Basic notions. Chapter 2. Systems of convex subgroups: General properties. Archimedean groups. Totally ordered groups. Conrad groups. Chapter 3. Orderability conditions: Semigroup conditions. Sufficient conditions. Group conditions for total orderability. Groups of automorphisms. Connection with totally ordered groups. Fully orderable groups. Chapter 4. Groups of order automorphisms: Preliminaries. Wreath products. Order types of right-ordered groups. Chehata groups and Dlab groups. Embedding. Chapter 5. Relatively convex subgroups: Orderable representations. A finite number of right orders. A finite number of total orders. Center of right-ordered groups. Criteria for orderability. Centers of “small” subgroups. Chapter 6. Orders on free products: Vinogradov theorem. Free products with amalgamation. Right-orderable groups with amalgamation. Chapter 7. Quasivarieties: Properties of quasivarieties. Model theory. Axiomatic rank. Locally indictable groups. The local indicability of extensions. Lattice of quasivarieties. Chapter 8. Semilinearly ordered groups: Definitions. Basic properties. Convex subgroups. Constructions. References. Subject index. Author index.
There is an English translation [V. M. Kopytov and N. Ya. Medvedev, Right-ordered groups. Transl. from the Russian (1996; Zbl 0852.06005)].

Reviewer: V.D.Mazurov (Ekaterinburg)

MSC:

06F15	Ordered groups
06-02	Research exposition (monographs, survey articles) pertaining to ordered structures
20F60	Ordered groups (group-theoretic aspects)
20-02	Research exposition (monographs, survey articles) pertaining to group theory
06-01	Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures

Keywords:

right-ordered group; orders on groups; positive cone; automorphism groups of linearly ordered sets; convex subgroups; Conrad groups; orderability; wreath products; centers; free products; amalgamation; quasivarieties; semilinearly ordered groups; right-orderable group; order-simple groups; lattice-ordered groups; right-relatively convex subgroups; universal groups