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Lattice-ordered groups. Advances and techniques. (English) Zbl 0705.06001
Lattice-ordered groups (\(\ell\)-groups, for short) are groups equipped with a lattice structure compatible with the group operation. In contrast to linearly ordered groups, which were already studied at the beginning of this century by Hölder and Hahn, the more general concept of \(\ell\)- groups blossomed in the past thirty years. The theory of \(\ell\)-groups now provides a well developed, expanding field of its own with many recent applications in areas like the study of unstable theories, infinite permutation groups, varieties in universal algebra, the theory of Bezout domains, completion theory of particular classes of “semi- topological” groups.
The present book provides a comprehensive account of these developments. In twelve chapters, various results of the theory of \(\ell\)-groups, the techniques used to establish them and their interaction with other areas of mathematics are surveyed by experts in the field. The editors added two introductory chapters which provide the necessary background. Interdependence between the other chapters is minimal so that they can be read separately. Moreover, special care was taken to make them accessible not only to the expert, but also to mathematicians whose main area overlaps with the particular topic. Most chapters end with a list of challenging open problems in the respective area. Finally, the book contains an extensive bibliography (up to 1986).
The book is unique in that various authors collectively cover a large quickly expanding area, which would have been impossible for a single individual, but, due to the editors’ efforts, a uniform text was achieved. In the reviewer’s opinion, this book will be an invaluable resource for anyone working in the area of ordered algebraic structures or interested in areas overlapping with the theory of \(\ell\)-groups. Due to the excellent surveys with illustrations also of many deep proofs in the area, it will certainly stimulate and influence further investigations in the field of \(\ell\)-groups.
Subsequently we give short reviews of each chapter, as these provide separate articles (which otherwise would not be reviewed individually). We use the following notions. An o-group is a linearly ordered group. An \(\ell\)-group is called representable, if it is a subdirect product of o- groups; equivalently, \((x\wedge y)^ 2=x^ 2\wedge y^ 2\) for all x,y\(\in G\). Finally, we note that the underlying lattice of an \(\ell\)- group is divisible.
Chapter 0, Elementary facts (pp. 1-10), and Chapter 1, Homomorphisms, prime subgroups, values and structure theorems (pp. 11-22).
The editors provide the background of the theory of \(\ell\)-groups necessary for the other material in the book. Included is also a recent result of R. N. Ball that the property of a lattice to be completely distributive is expressible by a single sentence in the first order theory of lattices.
Chapter 2, Lattice-ordered permutation groups, (pp. 23-40) by W. C. Holland.
Let (\(\Omega\),\(\leq)\) be a chain and A(\(\Omega\)) the group of all order- preserving permutations of \(\Omega\). For f,g\(\in A(\Omega)\), put \(f\leq g\) iff \(\omega\) \(f\leq \omega g\) for each \(\omega\in \Omega\). Then A(\(\Omega\)) becomes an important instance of a lattice-ordered group which is only rarely abelian. By Holland’s theorem, any \(\ell\)-group G can be embedded (as an \(\ell\)-group) into A(\(\Omega\)), for some chain \(\Omega\). Moreover, here (\(\Omega\),\(\leq)\) can even be chosen such that any two closed intervals are order-isomorphic; then the algebraic structure of A(\(\Omega\)) is particularly well-known and various nice embedding results for arbitrary \(\ell\)-groups follow. An \(\ell\)-subgroup G of A(\(\Omega\)) is called an \(\ell\)-permutation group. The author gives S. McCleary’s classification of all transitive \(\ell\)-permutation groups (G,\(\Omega\)) which are primitive, i.e. \(\Omega\) has no non-trivial convex G-congruences. - Further information on this topic is contained in the work of A. M. W. Glass [Ordered permutation groups (1981; Zbl 0473.06010)].
Chapter 3, Model theory of abelian \(\ell\)-groups (pp. 41-79), by V. Weispfenning.
The author concentrates on the model-theoretic topics of elementary equivalence, decidability and elimination of quantifiers for abelian \(\ell\)-groups. First, he shows that the universal theory of the class of all abelian \(\ell\)-groups (of specified dimension) is decidable. The elementary theory of o-groups elementarily equivalent to the integers is shown to be also decidable, and for dense o-subgroups G of the reals the decidability of Th(G) is characterized in terms of the Szmielew invariants of G from abelian group theory. Furthermore, Gurevich’ deep results on the elementary classification and primitive recursive decidability of the theory of abelian o-groups are presented (with outline of the argument). Finally, algebraically and existentially closed abelian \(\ell\)-groups are investigated.
Chapter 4, Groups of divisibility: A unifying concept for integral domains and partially ordered groups (pp. 80-104), by J. L. Mott.
Let D be an integral domain, K the quotient field of D, \(K^*=K\setminus \{0\}\), and U(D) the group of units of D. Then \(K^*/U(D)\) may be made an abelian partially ordered group by putting aU(D)\(\leq bU(D)\) if b/a\(\in D\). This group is called the group of divisibility of D, denoted by G(D), and its properties often reflect the structure of D. The author gives several characterizations for the integral domain D to be a GCD-domain (i.e., any two nonzero elements of D have a greatest common divisor), and for D to be a unique factorization domain. With regard to the final question in the article - describe those torsion-free groups which are multiplicative groups of fields - the reader may find more information in the work of L. Fuchs [Infinite Abelian groups, Vol. II (1973; Zbl 0257.20035), Chapter 18].
Chapter 5, The lattice of convex \(\ell\)-subgroups of a lattice-ordered group (pp. 105-127), by M. Anderson, P. Conrad and J. Martinez.
The authors investigate the lattice \({\mathcal C}(G)\) of all convex \(\ell\)- subgroups of an \(\ell\)-group G. In general, \({\mathcal C}(G)\) is a distributive complete algebraic lattice admitting a pseudo- complementation \(^{\bot}\). The polars of G are defined to be those elements \(A\in {\mathcal C}(G)\) for which \(A=A^{\bot \bot}\); they form a complete Boolean algebra \({\mathcal P}(G)\). It is shown that many important classes of \(\ell\)-groups G can be described by purely lattice-theoretic conditions on \({\mathcal C}(G)\) and \({\mathcal P}(G)\). For instance, G is (as an \(\ell\)-group) a direct sum of o-groups if and only if \({\mathcal P}(G)\) is a complete sublattice of \({\mathcal C}(G)\). On the other hand, the lattice \({\mathcal C}(G)\) in general does not determine whether or not the \(\ell\)- group G belongs to any proper variety of \(\ell\)-groups.
Chapter 6, Torsion theory of \(\ell\)-groups (pp. 128-141), by J. Martinez.
A non-trivial class of \(\ell\)-groups \({\mathcal I}\) is called a torsion class, if it is closed with respect to (1) taking \(\ell\)-homomorphic images of \(\ell\)-groups from \({\mathcal I}\), and (2) for any \(\ell\)-group G, forming arbitrary joins of convex \(\ell\)-subgroups of G which belong to \({\mathcal I}\). A non-trivial class of \(\ell\)-groups is called a torsion-free class, if it is closed under isomorphisms, taking convex \(\ell\)-subgroups and subdirect products. The author surveys most of the main results on torsion and torsion-free classes; for proofs, the reader is referred to J. Martinez [Czech. Math. J. 25(100), 284-299 (1975; Zbl 0321.06020), Trans. Am. Math. Soc. 259, 311-317 (1980; Zbl 0433.06016)] and W. C. Holland and J. Martinez [Algebra Univers. 9, 199- 206 (1979; Zbl 0428.06011)]. Furthermore, a new interesting concept of split subgroups is described; for proofs of these results (and more) see J. Martinez [Arch. Math. 54, 212-224 (1990; Zbl 0682.06011)].
Chapter 7, Completions of \(\ell\)-groups (pp. 142-174), by R. N. Ball.
Lattice-ordered groups admit various notions of completions, like lateral completion, cut completion, order and polar Cauchy completion, etc. For most of these completions, existence and uniqueness proofs are fairly complicated. The author introduces a distinguished completion \(G^{dist}\) of an arbitrary \(\ell\)-group G and shows that all other completions of G studied previously are contained in \(G^{dist}\) and can be constructed inside \(G^{dist}\) much more easily. \(G^{dist}\) can be described roughly as follows: Since G is a distributive lattice, there exists a complete Boolean algebra B containing G such that each lattice homomorphism from B which is one-to-one on G is injective. Then the \(\ell\)-group \(G^{dist}\) can be constructed as a suitable sublattice of B containing G. The details are still involved. Finally, it is shown that if G is representable, then surprisingly \(G^{dist}\) coincides with the cut completion of the projectable hull of G.
Chapter 8, Characterization of epimorphisms in archimedean lattice- ordered groups and vector lattices (pp. 175-205), by R. N. Ball and A. W. Hager.
The authors investigate epimorphisms in the category Arch of archimedean \(\ell\)-groups with \(\ell\)-homomorphisms, and the category Wu of archimedean \(\ell\)-groups with distinguished weak unit and unit- preserving \(\ell\)-homomorphisms (an \(\ell\)-group G is archimedean, if \(g^ n\leq f\) for all \(n\in {\mathbb{N}}\) implies \(g\leq e\) (f,g\(\in G)\); then G is abelian; u is a weak unit iff \(g\wedge u=e\) implies \(g=e)\). In an abstract category \({\mathcal C}\), a morphism \(\epsilon\) is epi (monic), if whenever \(\alpha\), \(\beta\) are morphisms with \(\alpha \circ \epsilon =\beta \circ \epsilon\) \((\epsilon \circ \alpha =\epsilon \circ \beta\), respectively), then \(\alpha =\beta\). An object G is epicomplete, if any epi and monic \(\epsilon\) : \(G\to H\) is an isomorphism; an epicompletion of G is an epicomplete object containing G epically. In this chapter (the first of a series of papers), the authors give an explicit description of the epimorphisms in Wu and in Arch. Among the consequences, we select: each object in these categories has an (in general non-unique) epicompletion, and a Wu-object is epicomplete in Arch iff it is epicomplete in Wu. Various examples are included. The authors note that all their results also hold in the corresponding categories of vector lattices, with the same arguments.
Chapter 9, Free lattice-ordered groups (pp. 206-227), by S. H. McCleary.
The free lattice-ordered group \(F_{\eta}\) of arbitrary rank \(\eta >1\) has been studied in two different ways: via the Conrad representation on the collection of all right orderings of the free group \(G_{\eta}\) (by a result of Kopytov’s, here some single right ordering of \(G_{\eta}\) suffices to give a faithful representation), and via the Glass-McCleary representation as a 2-transitive \(\ell\)-permutation group. Here, the author uses pictorial methods to construct for all countable \(\eta >1\) and, assuming the generalized continuum hypothesis (GCH), for all uncountable regular cardinals \(\eta\) a right ordering \((G_{\eta},\leq)\) on which the action of \(F_{\eta}\) is both faithful and 2-transitive. As a consequence, detailed information on the root system \(P_{\eta}\) of all prime subgroups of \(F_{\nu}\) can be obtained. Reviewer’s note: In the meantime, without GCH almost all of the latter results on \(P_{\eta}\) have been obtained for regular uncountable \(\eta\) and most of them for singular \(\eta\), see the reviewer and S. H. McCleary [Order 6, 305-309 (1989; Zbl 0702.06007)].
Chapter 10, Varieties of lattice-ordered groups (pp. 228-277), by N. R. Reilly.
A class of algebras is called a variety, if it is closed under the formation of products, homomorphic images and subalgebras; equivalently, it is defined by a set of equations. Let L denote the complete lattice of all varieties of \(\ell\)-groups. We just mention some of the results of this valuable survey on the structure of L. First, L is distributive but not Brouwerian. The variety \({\mathcal A}\) of all abelian \(\ell\)-groups is the smallest non-trivial variety. There is also a largest proper variety \({\mathcal N}\) (comprising all normal-valued \(\ell\)-groups). The lattice of varieties of representable \(\ell\)-groups contains both chains and antichains of size continuum. The solvable varieties which cover \({\mathcal A}\) are explicitly known; they are countably many, precisely three of which are representable. L also admits a product defined by \(G\in {\mathcal U}\cdot {\mathcal V}\) iff G has an \(\ell\)-ideal \(M\in {\mathcal U}\) with G/M\(\in {\mathcal V}\). With this operation, \({\mathcal N}\) is the supremum in L of the chain \({\mathcal A}^ n\) (n\(\in {\mathbb{N}})\), and the set of all non-trivial varieties properly contained in \({\mathcal N}\) forms a free semigroup. Finally, by a recent result of Gurchenkov, there are, for each \(n\geq 3\), even continuously many varieties of \(\ell\)-groups of nilpotency class n.
Chapter 11, Free products in varieties of lattice-ordered groups (pp. 278-307), by W. B. Powell and C. Tsinakis.
The authors show that various varieties of \(\ell\)-groups, including the varieties of all \(\ell\)-groups, all representable \(\ell\)-groups, all \(\ell\)-groups of nilpotency class n, and all abelian \(\ell\)-groups, respectively, admit free products. Moreover, for each of these varieties except the variety \({\mathcal L}\) of all \(\ell\)-groups, explicit descriptions of the free product exist. In the variety \({\mathcal A}\) of all abelian \(\ell\)-groups, free products of non-trivial \(\ell\)-groups are cardinally indecomposable (i.e. indecomposable as an \(\ell\)-group). By a recent result of A. M. W. Glass, in \({\mathcal L}\) non-trivial free products are even directly indecomposable (as groups) and have trivial center. Examples show that even free products of abelian o-groups may have large disjoint subsets. Finally, free products of \(\ell\)-groups are intimately related to free products both of the underlying groups and of the underlying distributive lattices.
Chapter 12, Amalgamations of lattice-ordered groups (pp. 308-327), by C. Tsinakis and W. B. Powell.
A class \({\mathcal C}\) of algebras satisfies the amalgamation property (AP), if for any \(A,B_ 1,B_ 2\in {\mathcal C}\) and embeddings \(f_ i: A\to B_ i\) \((i=1,2)\) there exists \(C\in {\mathcal C}\) and embeddings \(g_ i: B_ i\to C\) \((i=1,2)\) such that \(g_ 1\circ f_ 1=g_ 2\circ f_ 2\). The variety \({\mathcal A}\) of all abelian \(\ell\)-groups is shown to satisfy AP, whereas for many other varieties AP fails; it is open whether \({\mathcal A}\) is in fact the only non-trivial variety of \(\ell\)-groups with AP. Assuming GCH, by AP and standard model-theoretic results, the class of all abelian o-groups has universal homogeneous elements in each regular uncountable cardinal; these are characterized as divisible abelian o- groups whose underlying linear order satisfies a strong separation property.
Chapter 13, Generators and relations in lattice-ordered groups: Decision problems and embedding theorems (pp. 328-346), by A. M. W. Glass.
The following theorems are proved. A. There is a finitely presented \(\ell\)-group with insoluble group word problem. B. The isomorphism problem for the ten-generator one-relator members of any recursively axiomatised variety of non-trivial \(\ell\)-groups is insoluble. C. Any countable \(\ell\)-group can be embedded in a seven-generator \(\ell\)-simple \(\ell\)-group. D. Every finitely presented \(\ell\)-group can be constructively embedded in (i) a two-generator one-relator \(\ell\)-group, (ii) a finitely presented \(\ell\)-group with trivial center, (iii) an eight-generator one-relator perfect \(\ell\)-group. Moreover, there is no algorithm to determine whether or not an arbitrary finitely presented \(\ell\)-group is abelian, trivial, free, free abelian, or linearly ordered, respectively (this list extends in fact to any Markov property for \(\ell\)-groups). Since amalgamation and HNN-properties fail for arbitrary \(\ell\)-groups, combinatorial methods and intricate representations of \(\ell\)-groups as permutation groups on chains are used in the proofs.
Reviewer: M.Droste

06-02 Research exposition (monographs, survey articles) pertaining to ordered structures
06F15 Ordered groups
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
06-06 Proceedings, conferences, collections, etc. pertaining to ordered structures
06-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to ordered structures
03C60 Model-theoretic algebra