Michael R. Darnel: Theory of Lattice-Ordered Groups
a review of the book and perspective of the literature

Theory of Lattice-Ordered Groups by Michael R. Darnel,
Monographs and Textbooks in Pure and Applied Mathematics, 187.
Marcel Dekker, Inc., New York, 1995.
viii+539 pp. ISBN: 0-8247-9326-9

Since the early eighties a number of monographs and reference books have appeared which deal with the theory of lattice-ordered groups and rings. For many years it had been Laszlo Fuchs' book, Partially Ordered Algebraic Systems ([Fu63], Pergamon Press) that we apprentices to the subject were directed to as graduate students, with perhaps a mention that Garrett Birkhoff had been the first to systematically study these algebraic objects, and that in the immense sea of lattice theory that is Lattice Theory [Bi84] we could find a few chapters discussing the rudiments of the theory. In the meantime - it was the sixties then - Paul Conrad was assembling a body of new and significant work in the theory of lattice-ordered groups, and his students knew that he was also collecting notes and papers into a monograph, which ended up as [C70] but was never published.

In the spring of 1967 Paul Conrad visited Paris, and during that heady spring Alain Bigard, Klaus Keimel and Samuel Wolfenstein found themselves immersed in lattice-ordered groups. They must have understood that there was an open invitation to seize the assemblage of Conrad notes and transform it into a monograph. Out of this effort eventually came [BKW77]. It was the first of its kind to lace together the Conrad opus on lattice-ordered groups with contributions from others (Simon Bernau, most prominently) who approached the subject matter from the analytic point of view, and brought to the game elements of topology, for purposes of representation. This contribution gave rise, perhaps inevitably, to a systematic study of the so-called f-rings, and especially the archimedean f-rings. [BKW77] remains to this day the only monograph which effectively - very effectively, in fact - introduces the reader to lattice-ordered rings.

Speaking of the analytic point of view, one should take notice of the parallel development of the literature on vector lattices (or, synonymously, Riesz spaces). One might begin to track this development with [Vu61], published in Russian in 1961, and in English in 1967 by Noordhoff-Groningen; I am unfamiliar with Vulih's text. Then in 1971 Luxemburg and Zaanen published [LZ71], which contains a great deal of information about vector lattices. This book is also proverbially encyclopaedic, and it is steeped in the analysis, so as to make it very different, philosophically, from the monographs which promote the algebraic in ordered algebraic structures. With such a bias in mind, I would not initiate a student in lattice ordered groups and rings with a reading of [LZ71]; it is also too easy to get overwhelmed by the contents, and not easy enough to locate results in it, which makes it problematic as a reference text. A much better read, although, unfortunately, difficult to obtain, is Ben de Pagter's doctoral dissertation ([dP81]); it is economically written, yet covers a lot of ground on the subject of uniformly complete vector lattices and algebras.

In any event, the way things stood twenty-odd years ago with monographs in the theory of lattice-ordered groups, for many of the stateside Conrad students [BKW77] was still some years from publication, and when it did appear it came out in French, and then, inexplicably, almost immediately became unavailable from the publisher. By the end of the seventies then many of us involved with lattice-ordered groups began to see the need for another monograph, and, indeed, as [BKW77] was so long in gestation, it was also out of date when it appeared, and addressed the subject of ordered permutation groups only glancingly. To correct the latter weakness, Andrew Glass published [Gl81] in 1981. It is a fine compendium of the central theory of ordered permutation groups.

On the other hand, by the eighties the investigation into lattice-ordered groups and rings had changed considerably. This observer believes it is fair to say that the research had become decentralized. Still, some significant strands began to emerge and draw into focus. For example, the impact of topology was getting greater attention; people like Anthony Hager and Melvin Henriksen, steeped in the topology of rings of continuous functions, (finally) began to converse with students of the so-called Conrad school, most notably R. N. Ball, and eventually also this reviewer. Another example: Charles Holland (and some of his students, Manfred Droste, for one) began to interact with the model theoretic aspects of ordered permutation groups. Then in 1985, at Bowling Green, an interesting event took place: the researchers of this now more disparate discipline gathered there for a week-long symposium, and after considerable honing and editing [GH89] appeared. In this book some of these new strands of research are much in evidence. Some may view the book as the proceedings of a conference; others, I among them, have come to see that publication as a much-needed pulse-taking of the theory of ordered groups and rings. If nothing else was accomplished at Bowling Green, a detailed and well-researched record had been produced of the state of the art in the field.

In the meantime Marlow Anderson and Todd Feil, on the one hand, and Michael Darnel, on the other, were at work on their conceptions of what the updated monograph on lattice-ordered groups should look like. The results are very different, in scope as well as ambition. And yet the books have the same flavor, in the following sense. Both [AF88] and [D95], the book under review here, represent attempts to update the theory of lattice-ordered groups, per se. Both give, for example, commensurate with their depth and scope, a similar account of the universal algebraic aspects of the theory. The Anderson-Feil book is, however, true to its intentions, an introduction to the theory; Darnel goes much deeper. Unfortunately, the former is a hard read; it is too spartan and streamlined; (with the notable and curious exception of the splendid final section, which is a treasure trove of examples, amply illustrated.) Whereas until 1995, I pointed my students to [BKW77] for an initiation in the theory of lattice-ordered groups, I can now confidently recommend that they read Darnel's book. This is, finally, an updated monograph on the subject, written with the student in mind, in a friendly and leisurely manner. At the same time it contains a wealth of information, which for reference is easily retrievable.

Darnel's book comprises eleven chapters and two appendices. There are 32 pages of references. The book is, as has already been indicated, fairly comprehensive, although, except for one curious foray into the orderability of rings (§25), and the discussion on how to embed an archimedean l-group in an archimedean f-ring with identity (§56), it deals exclusively with lattice-ordered groups (henceforth abbreviated l-groups, as is customary). The only places where topology rears its head in any serious way are §49, which discusses sheaf-theoretic representations, and §57, which concerns itself with the representation of an archimedean l-group G as a group of almost real-valued functions, which, incidentally, is the embedding of G into its essential hull.

The book develops the fundamental principles and concepts - the language of discourse in the theory of l-groups - over, roughly, the first five chapters. Most of the preliminary structure theorems are in §3, such as the result that every l-group is torsion-free (Proposition 3.5) and the Riesz Decomposition Theorem (3.11). When G is an l-group and g, x\in G, then x is said to be a component of g if |x|_^|gx^-1|=e. I particularly liked the early introduction of the notion of a component; in the pages that follow the definition (pp. 30-32) a rather thorough catalogue of basic properties is presented. A pattern develops early in Darnel's book, one which should appeal to students: that of following a theorem with an example showing the limitations of what has just been presented. The early introduction of saturated subgroups is also appropriate. The reader is then able to appreciate how the components of an element frequently generate large abelian subgroups in a highly non-abelian l-group. The components of a positive g in G form a boolean algebra (Theorem 6.19). This is the first of many places in the theory of l-groups where boolean algebraic structures play a prominent role. It was Sik who first observed (1960) that the set P(G) of polars of an l-group (Chapter 3) form a complete boolean algebra under inclusion. In Chapter 2 the reader is introduced to the lattice C(G) of convex l-subgroups of an l-group G. The presentation of the various special subgroups of C(G) proceeds apace, yet in a very readable progression. The Brouwerian property of C(G) is given its due prominence, and the coverage of prime subgroups and values is thorough. Still, through a patch like this one where a substantial foundation is being assembled, it would have been helpful - bearing the student reader in mind - to include some examples and/or exercises.

A couple of items in the early chapters puzzle me, and I have one minor quibble. As to the latter: one of the fascinating features of prime subgroups in l-groups is the similarity between their role in the theory with that of prime ideals in commutative ring theory. There is, likewise, such an analogy between the algebra of polars of an l-group and the algebra of annihilators in a commutative ring. This is a metatheme which, in my view, recurs, and is useful pedagogically to a reader who already has some familiarity with commutative rings.

There is a sentence (the first in the second paragraph on p. 44) which I find astonishing, and which Darnel surely did not mean in the way it is stated. It reads:

However, as far as using C(G) to help identify properties of G, virtually nothing can be done.

On p. 74 the author defines the concept of recognizability from the [underlying] lattice and the identity. Student readers will not know what to make of this definition; I find it difficult to understand. The idea is this: suppose that G and H are two l-groups, and F is a property or feature of elements or convex l-subgroups of an l-group. Let us say that F is recognizable from the lattice and the identity if whenever g \in G (resp. K\subseteq G) has F and \tau is an isomorphism of G onto H of the underlying lattice structures, which preserves the identity, then \tau g (resp. \tau K) has property F as well. In this sense, a polar in an l-group is so recognizable. Happily, Theorem 13.14, which follows on the heels of the definition, explains, by illustration, what was intended.

Chapters 4 and 5 bring in finer structure of C(G), culminating with the indispensable §27 on lex-extensions and lex-subgroups. Someone putting down Darnel's book after the first five chapters will already have a fairly sound picture of the building blocks of the theory. Beginning with Chapter 6 a number of special topics will be developed, leading to the deeper structure and representation theorems.

Chapter 6 is devoted to the representation of an l-group as a group of order-preserving permutations on a totally ordered set. To quote the author in the introduction to the chapter:

The representation of an l-group as a group of order permutations of a chain has proven value; many problems that were intractable to the methods developed thus far in this book were finally resolved by means of such a representation.

Holland's Representation Theorem, to the effect that every l-group can be so represented, appears in §29. §30 deals with properties of A(\Omega), the full group of order-preserving permutations of the chain \Omega. In §31 it is shown that the group A(F), where F is an ordered field, is [order] doubly transitive (Theorem 31.15), and that every l-group can be embedded in such an A(F). §33 discusses McCleary's classification of primitive transitive representations.

Chapter 7, entitled Classes of l-groups, introduces the reader to torsion classes and varieties of l-groups, and it contains a discussion of free l-groups. Chapter 7 concludes (in §39) with the Bernau-Conrad results on the embedding of an l-group in a laterally complete one.

The body of work on torsion and radical classes comprises some very elegant mathematics, to which I am happy to have contributed. And yet its placing in an exposition such as Darnel's seems somewhat peculiar, in view of what comes after in Chapters 8 and 9. Indeed, the introduction to varieties in §38 is abruptly cut short, to be continued to greater depth in Chapter 11. Now, throughout the first five chapters the author's line of thinking seems clear enough: to develop the theory of l-groups from the blending of the two main ingredients, lattice theory and group theory; one considers the obvious subobjects, for example, in this hybrid situation, and shows how the added structure in turn influences the underlying lattice and group structure. The sixth chapter is not an abrupt departure from this philosophy; the techniques of ordered permutation groups are essential in a number of arguments, as Darnel readily explains, and in any case, no group theorist will be shocked to find that permutation groups figure prominently in the central development of l-groups. And after the interlude of Chapter 7, the train of thought of the first five chapters picks up again, and the established groundwork of those chapters pays off in the structure and representation theorems that are the heart of the theory. The discussion on varieties and torsion classes could have been deferred until this core theory had been set down; it is, historically speaking, the natural order of things.

If M is a value of an l-group G - that is, M is a convex l-subgroup which is maximal with respect to missing an element g - then the intersection M^* of all the convex l-subgroups that properly contain M also properly contains M, and is therefore a cover of M in C(G). Chapter 8 examines the l-groups for which M is always normal in its cover. Such l-groups are said to be normal-valued. §41 contains Wolfenstein's pretty characterization of the normal-valued l-groups, which proves, among other things, that this class, N, is a variety, (that is to say, equationally definable). Holland proved that N is the largest proper variety of l-groups, and it does not appear possible to prove this result without involving order-preserving permutations. §44 establishes the link (for normal-valued l-groups) between complete distributivity of (the underlying lattice of) G and the so-called essential values. §45 introduces special values and leads to §46 and Conrad's elegant characterization of finite-valued l-groups.

It is fair to say that much of the early intuition about l-groups came from thinking of those that can be represented as a subdirect product of totally ordered ones (o-groups). Chapter 9 takes up this class of l-groups, the representable l-groups, which are also an equationally definable class, and can be characterized as those for which every polar (or alternately, every minimal prime subgroup) is normal. It is a very nice class in which to work, where one's intuition about o-groups, and, indeed, about abelian l-groups, goes a long way. A variant of the lateral completion, the orthocompletion, is discussed in §48; the proof of existence given by Darnel is that of Roger Bleier, which, as the author rightly points out, is the clearest and most intuitive of the proofs on this kind of topic. §51 contains the Conrad-Harvey-Holland Theorem for abelian l-groups, which represents an abelian l-group as a group of real-valued functions. This kind of representation was first considered by Hahn (1907), for o-groups, and the achievement of Conrad, Harvey and Holland (1963) can fairly be counted as a fundamental advance in the understanding of the structure - I'm tempted to say it the nature - of abelian l-groups.

If one asks, what does a typical abelian l-group look like? , the Conrad-Harvey-Holland reply would be: take a partially ordered set \Lambda which is a root system - meaning that for each \lambda \in \Lambda the set of elements above \Lambda is a chain; in the group R^{\Lambda} of all real-valued functions, consider all the functions f for which the cozeroset

\coz(f)=\{\,\gl\in\gL\,:\,f(\gl)\neq 0\,\}

Chapter 9 concludes with a discussion of free abelian l-groups and vector lattices (§52). Each free abelian l-group is archimedean, and, indeed, can be embedded as a group of integer-valued functions (Theorem 52.2). This result can be traced to the work of Weinberg and Bernau, who, independently, and after courteous mutual refutation arrived at correct proofs of it. It also, resoundingly, settled an old question of Birkhoff, who had posed (and answered, in the negative) the question of whether an l-homomorphic image of an archimedean l-group was necessarily archimedean. It turned out that every abelian l-group was a quotient of an archimedean l-group.

Chapter 10, on archimedean l-groups, is the one chapter that I would have liked to see expanded. Of course, to do so Darnel would have had to introduce a fair amount or topology and representation theory associated with Yosida spaces. One cannot quarrel with the choice to omit this kind of material, if only because the book is rich enough as is. On the other hand, so much information about the various completions of archimedean l-groups emerges from the representation theory, that I can't help wishing the author had included it.

This chapter does include the result that, given an archimedean l-group G, the ring of so-called orthomorphisms is an archimedean f-ring with identity (Theorem 56.18). (This subject is also the central theme of de Pagter's thesis - [dP81] - to which reference was made earlier.) It also gives a thorough account of the embedding of an archimedean l-group in its essential hull (§57). Conrad (1970) proved that the essential hull could be formed by first passing to the divisible hull and then applying the Dedekind-MacNeille and lateral completions; Bernau (1976) showed that the order in which the latter two were applied did not matter.

Earlier in the chapter the author presents material on the Dedekind-MacNeille completion (§54) and on hyperarchimedean l-groups. Included in the presentation on complete l-groups, are two classical results, the first due to Riesz (1940), stating that in a complete l-group every polar is a cardinal summand, and the second, due to Iwasawa (1948), which states that every complete l-group is a cardinal sum of two complete l-groups, one a vector lattice, the other singular - meaning that every positive element exceeds a singular one. (The reader's intuition would not be far off the mark by imagining singular archimedean l-groups as groups of integer-valued functions.)

The book concludes with a discussion of varieties, initiated in §38 (Chapter 7). This chapter (Chapter 11) seems cut off from the body of the text, for reasons alluded to earlier. One curious effect, for this reader at least, of having this chapter come upon the heels of the one on archimedean l-groups, is to underscore, by the exposition itself, that the subject matters of these two chapters are oceans apart, and to attest to the development of research in the discipline away from the center in the last ten to fifteen years, a phenomenon already mentioned in this review. The reader may already have guessed that my working definition of this center is most of the material which appears in Darnel in Chapters 1 through 5 and Chapters 6, 8 and 9. Aptly, it is essentially the historical core of the theory of l-groups.

Much of the material in Chapter 11 represents work done in the last ten years. In the first section (§58) Darnel introduces, using language of order-preserving permutations, the notion of an l-group (or class of l-groups) mimicking a given variety; a class which mimics a variety generates it (58.5), but the point is that such a class gives more information. It is shown that Z, the o-group of integers, mimics the variety A of abelian l-groups, and by product iteration that the n-fold wreath product of copies of Z generates Aⁿ. Eventually this leads to the result that the only proper nontrivial variety which is closed under extensions is N, the variety of normal-valued l-groups (Corollary 58.13). Much of the rest of the chapter is concerned with fine structure of the lattice of varieties, including results on covers of A, to which a number of authors have dedicated a fair amount of effort. Quite striking is the beautiful theorem of Glass, Holland and McCleary (Theorem 58.25), affirming that the subvarieties of N form a free semigroup on the indecomposable varieties, the operation of this semigroup being the product of two varieties by extension of one l-group by another. The reader who is familiar with varieties of groups might recall that a similar theorem holds for varieties in that context.

In the introduction to this review I indicated that the Bowling Green book ([GH89]) had made it possible for a knowledgeable reader to get a sense of the state of the art in the theory of l-groups in the mid to late eighties. Indeed, that book includes introductory material with ample annotation, but not many proofs, so that it would be possible for a graduate student to get direction in the initial stages of his/her readings in l-groups. Darnel's book improves on that, affording the student the opportunity of a rewarding passage through the theory. And as a reference text it works well enough as a state-of-the-art barometer, unless the student (or, more accurately, the advisor) is primarily interested in archimedean l-groups.

Happily, Gillman and Jerison's Rings of Continuous Functions ([GJ76]) is still in print, and remains an excellent starting point for the student who is directed to approach the subject of f-rings from the point of view of topology. In addition, there is now an excellent survey article by Melvin Henriksen (see [He97]) on the development of f-rings, which documents, starting with the Birkhoff-Pierce article of the late fifties, a fairly complete history of the progress in the study of these structures, with superb annotation. Altogether then, a prospective student of f-rings now has a fairly comprehensive literature package: I would advise to begin with Darnel's book and [GJ76], and supplement and channel the reading with [dP81] and Henriksen's article. Darnel's book is the key, providing as it does the well-rounded algebraic foundation of the theory of l-groups and f-rings.

References

[BKW77] A. Bigard, K. Keimel & S. Wolfenstein, Groupes et Anneaux Réticulés. Lecture Notes in Math. 608, Springer Verlag (1977).

[Bi84] G. Birkhoff, Lattice Theory, 3rd Ed. AMS Coll. Publ. 25 (1984).

[C70] P. F. Conrad, Lattice-Ordered Groups. Tulane Lecture Notes (1970).

[D95] M. R. Darnel, Theory of Lattice-Ordered Groups. Pure & Appl. Math. 187, Marcel Dekker (1995).

[Fu63] L. Fuchs, Partially Ordered Algebraic Systems. Pergamon Press (1963).

[GJ76] L. Gillman & M. Jerison, Rings of Continuous Functions. Grad. Texts in Math. 43, Springer Verlag (1976).

[Gl81] A. M. W. Glass, Ordered Permutation Groups. London Math. Soc. Lecture Notes Series 55, Cambridge U. Press (1981).

[GH89] A. M. W. Glass & W. C. Holland (Eds.), Lattice-Ordered Groups: Advances and Techniques. Kluwer (1989).

[He97] M. Henriksen, A survey of f-rings and some of their generalizations. Proc. Conf. Ordered Alg. Struct., Curaçao, 1995; 1-26. W. C. Holland & J. Martinez, Eds.; Kluwer (1997).

[LZ71] W. A. J. Luxemburg & A. C. Zaanen, Riesz Spaces, I. North Holland (1971).

[dP81] B. de Pagter, f-Algebras and orthomorphisms. Doctoral Dissertation, Leiden (181).

[Vu61] B. Vulih, Introduction to the Theory of Partially Ordered Spaces. Moscow (1961).