Partially ordered groups.

*(English)*Zbl 0933.06010
Series in Algebra. 7. Singapore: World Scientific. xiii, 307 p. (1999).

This monograph is clearly written and well organized. The titles of the chapters are as follows: (1) Definitions and examples; (2) Basic properties; (3) Values, primes and polars; (4) Abelian and normal valued lattice ordered groups; (5) Archimedean function groups; (6) Soluble right partially ordered groups and generalizations; (7) Permutations; (8) Applications; (9) Completions; (10) Varieties of lattice ordered groups; (11) Unsolved problems.

It was a very good idea to insert already at the beginning of the book, in Chapter (1), a series of thoroughly described examples of partially ordered groups (29 examples). In (2), the relations between partial order and the group theoretical properties are investigated; some of the results presented here have not been published earlier. The contents of (3) and (4) are characterized by their titles; also, the proof of Conrad’s Theorem on lattice ordered groups containing no infinite pairwise orthogonal set is given in (3), and Conrad-Harvey-Holland Theorem is presented in (4). The main topics of (5) are free abelian lattice ordered groups, finitely presented abelian \(\ell\)-groups and Bernau Theorem on the representation of an archimedean lattice ordered group by extended real-valued functions. The content of (6) can be divided into two parts. First, the author studies the consequences of some group-theoretical conditions for the structure of a lattice ordered group; in this direction, results of Kopytov, Medvedev and Reilly are presented. Secondly, the notion of right order is applied for obtaining a series of results on the structure of a partially ordered group; the following theorem (Darnel, Glass and Rhemtulla) can be quoted here as typical: If \(G\) is a right orderable group and every right ordering on \(G\) is two sided, then \(G\) is abelian. The results dealt with in (7) are based on Holland Theorem on the representation of a lattice ordered group by increasing permutations of a linearly ordered set; for a more detailed investigation of these questions cf. the author’s monograph [A. M. W. Glass, Ordered permutation groups (Lond. Math. Soc. Lect. Note Ser. 55) (Cambridge Univ. Press, Cambridge) (1981; Zbl 0473.06010)]. The content of (8) is rather miscellaneous; let us mention the following items: free lattice ordered groups, finitely presented \(\ell\)-groups, undecidable problems. Some types of completions (order completion, lateral completion, special closure, distinguished completion) are studied in Chapter (9). The importance of the notion mentioned last (due to Ball) is to be emphasized since it is, in a certain sense, stronger than several other types of completions. The basic results (due to Weinberg, Holland, Glass, Bernau, McCleary and Darnel) on varieties of the author remarks that this is presently the “hottest” area of research. (The chapter has approximately 30 pages; let us remark that in the monograph by V. M. Kopytov and N. Ya. Medvedev [The theory of lattice-ordered groups (Kluwer Akad. Publ., Dordrecht) (1994; Zbl 0834.06015)] the three chapters dealing with varieties of \(\ell\)-groups contain 90 pages.) In (11), fourteen problems are posed; in the discussion of these problems the proof of Bludov Theorem is presented, which solves Problem 6.

The book can be warmly recommended to students and research workers dealing with the theory of partially ordered groups.

It was a very good idea to insert already at the beginning of the book, in Chapter (1), a series of thoroughly described examples of partially ordered groups (29 examples). In (2), the relations between partial order and the group theoretical properties are investigated; some of the results presented here have not been published earlier. The contents of (3) and (4) are characterized by their titles; also, the proof of Conrad’s Theorem on lattice ordered groups containing no infinite pairwise orthogonal set is given in (3), and Conrad-Harvey-Holland Theorem is presented in (4). The main topics of (5) are free abelian lattice ordered groups, finitely presented abelian \(\ell\)-groups and Bernau Theorem on the representation of an archimedean lattice ordered group by extended real-valued functions. The content of (6) can be divided into two parts. First, the author studies the consequences of some group-theoretical conditions for the structure of a lattice ordered group; in this direction, results of Kopytov, Medvedev and Reilly are presented. Secondly, the notion of right order is applied for obtaining a series of results on the structure of a partially ordered group; the following theorem (Darnel, Glass and Rhemtulla) can be quoted here as typical: If \(G\) is a right orderable group and every right ordering on \(G\) is two sided, then \(G\) is abelian. The results dealt with in (7) are based on Holland Theorem on the representation of a lattice ordered group by increasing permutations of a linearly ordered set; for a more detailed investigation of these questions cf. the author’s monograph [A. M. W. Glass, Ordered permutation groups (Lond. Math. Soc. Lect. Note Ser. 55) (Cambridge Univ. Press, Cambridge) (1981; Zbl 0473.06010)]. The content of (8) is rather miscellaneous; let us mention the following items: free lattice ordered groups, finitely presented \(\ell\)-groups, undecidable problems. Some types of completions (order completion, lateral completion, special closure, distinguished completion) are studied in Chapter (9). The importance of the notion mentioned last (due to Ball) is to be emphasized since it is, in a certain sense, stronger than several other types of completions. The basic results (due to Weinberg, Holland, Glass, Bernau, McCleary and Darnel) on varieties of the author remarks that this is presently the “hottest” area of research. (The chapter has approximately 30 pages; let us remark that in the monograph by V. M. Kopytov and N. Ya. Medvedev [The theory of lattice-ordered groups (Kluwer Akad. Publ., Dordrecht) (1994; Zbl 0834.06015)] the three chapters dealing with varieties of \(\ell\)-groups contain 90 pages.) In (11), fourteen problems are posed; in the discussion of these problems the proof of Bludov Theorem is presented, which solves Problem 6.

The book can be warmly recommended to students and research workers dealing with the theory of partially ordered groups.

Reviewer: J.Jakubík (Košice)

##### MSC:

06F15 | Ordered groups |

06F20 | Ordered abelian groups, Riesz groups, ordered linear spaces |

06-02 | Research exposition (monographs, survey articles) pertaining to ordered structures |