W. CHARLES HOLLAND, 75 BIRTHDAY 1. Research

DOI: 10.2478/s12175-011-0013-6 Math. Slovaca 61 (2011), No. 3, 297–306

W. CHARLES HOLLAND, 75th BIRTHDAY Richard N. Ball* — A. M. W. Glass** — Jorge Mart´ınez*** — Stephen H. McCleary**** (Communicated by Anatolij Dvreˇ censkij ) ABSTRACT. This issue of Mathematica Slovaca is in honour of W. Charles Holland’s 75th birthday. We present here a brief account of some of his research (to date) and a couple of brief personal sketches of the man. c 2011 Mathematical Institute Slovak Academy of Sciences

1. Research of W. Charles Holland In the beginning were (totally) ordered groups: groups with a total order preserved on both sides by the group operation. Then came the generalisation to lattice-ordered groups; a lattice-ordered group is a group equipped with a lattice order, with the order preserved on both sides by the group operation. For any totally ordered set Ω, the group of order-preserving permutations (automorphisms) Aut(Ω, ≤) of (Ω, ≤) forms a lattice-ordered group under the pointwise order. A lattice-ordered group arising in this way is called a lattice-ordered permutation group (or -permutation group). The main role of -permutation groups was to provide examples of lattice-ordered groups. In 1963, Holland changed the playing field by showing that every lattice-ordered group is (-isomorphic to) a sublattice subgroup of some -permutation group! Lattice-ordered-group results could now be obtained by working instead with -permutation groups, which are far more concrete objects, amenable to drawing pictures. This great breakthrough is examined in depth in [BDG]. Totally ordered groups were studied in the early part of the twentieth century. Paul F. Conrad, Holland’s Ph.D. Dissertation Director, took this up but viewed totally ordered groups as a subclass of the class lattice-ordered groups. This permitted purely algebraic methods which Conrad exploited to great effect. 2010 M a t h e m a t i c s S u b j e c t C l a s s i f i c a t i o n: Primary 01Axx. K e y w o r d s: W. Charles Holland, lattice-ordered group, -permutation group, automorphism, MV-algebra, pseudo MV-algebra.

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R. N. BALL — A. M. W. GLASS — J. MARTÍNEZ — S. H. MCCLEARY

B. Banaschewski [Ban] had proved that the topological completion of a totally ordered group is also a totally ordered group in the natural way. In his Ph. D. thesis [2], Holland explored this completion and connected it with the completion under left Cauchy sequences. It was later generalised to latticeordered groups by R. L. Madell and R. N. Ball, ex-Ph.D. students of Holland’s [Mad, Ball1, Ball2, Ball3, BD]. The summer after receiving his Ph.D., Holland and J. G. Harvey worked with their (ex-)Research Director and obtained universals for the class of abelian lattice-ordered groups [3]. This is the Conrad-Harvey-Holland Theorem and generalises a theorem of H. Hahn [Hahn]. Holland then went to T¨ ubingen on a NATO Postdoctoral Fellowship and attended a course of lectures on permutation groups by H. Wielandt. While sitting on the balcony of their house in Horb am Neckar, listening to the bells ringing in the little church in the village, looking at the roses growing like the graph of a permutation and thinking about lattice-ordered groups of automorphisms of totally ordered sets, Holland suddenly realised that EVERY lattice-ordered group is one of these. This was the inspiration that lead to “Holland’s Theorem” that every lattice-ordered group can be embedded in the lattice-ordered group of automorphisms of a totally ordered set [4]. The study of these automorphism groups [5] and their use as a tool in the study of lattice-ordered groups formed the framework of Holland’s research for the next 40 years or so, and also provided problems which he generously passed on to his research students for their Ph.D.s. These included the lattice of normal subgroups of such automorphism groups and -primitivity. Holland also constructed the generalised Wreath product of permutation groups [8] that generalised the result of L. Kaloujnine and M. Krasner [KK]. The adaptation to automorphism groups of totally ordered sets was realized in [9] with S. H. McCleary. In [5], Holland had shown how to obtain the underlying totally ordered set from its automorphism group when the totally ordered set is doubly homogeneous. In a series of joint papers around 1980 (see [21, 22, 23, 24]), Holland explored reconstructing the totally ordered set from just the first-order properties of its automorphism group. He continued to study words in automorphism groups: group equations in lattice-ordered groups [16], the solution of the word problem for free lattice-ordered groups [18], the varieties generated by automorphism groups [30] and, with S. A. Adeleke, representing automorphisms by words [48]. Around 1990, with M. Droste and H. D. Macpherson, Holland studied the automorphisms of infinite semilinear orderings (see [38, 39, 42]). A fuller account of these and related results by Holland and others can be found in [BDG] later in this issue. In a seminal paper [Mar], J. Mart´ınez began the study of varieties of latticeordered groups, a topic that naturally attracted Holland. Using automorphism groups, he showed that there is a maximal proper variety of lattice-ordered groups [14] and, for any variety V of lattice-ordered groups, there is a V-socle [17] (every variety of lattice-ordered groups is a torsion class). The structure 298 Unauthenticated Download Date | 9/24/15 11:34 PM

W. CHARLES HOLLAND, 75th BIRTHDAY

of the lattice of varieties of lattice-ordered groups was explored in [20] and the main results obtained. While on Sabbatical leave in Vancouver with N. R. Reilly, Holland explored certain families of varieties of lattice-ordered groups [34, 35, 36], and later, with N. Ya. Medvedev, found a continuum of varietal covers of the variety of abelian lattice-ordered groups [47]. D. Mundici [Mun] gave a categorical equivalence between the category of models of multi-valued logic (MV-algebras) and the category of abelian latticeordered groups with strong order unit. A. Dvureˇcenskij [Dvu] extended this categorical equivalence to the not-necessarily-commutative case. This provided a way to study varieties of pseudo MV-algebras by examining equational classes of “unital” lattice-ordered groups. Holland has been at the forefront of this research ever since and obtained major results in the last few years, with A. Dvureˇcenskij, M. R. Darnel and M. Droste [59, 63, 64, 65, 66, 67, 70, 71, 72]. Three further results that Holland obtained are especially worth mentioning. In [29] he showed that a subdirect product of lattice-orderable groups need not be lattice-orderable. In [44], he obtained all partial orders on the automorphism group of the real line, and in [28], in joint work with A. H. Mekler and S. Shelah, he proved that there is a total order on the free group F of rank the continuum so that, if (G, ≤) (with the same order) is any other ordered group structure on the set F , then G generates the variety of all groups. This answers a question of R. Baer (P. F. Conrad’s Research Director!). As can be seen, Holland has been instrumental in obtaining many of the main theorems and tools involved in the study of lattice-ordered groups. We are most grateful. Long may it continue!

2. From Rick Ball One day, when I was done asking my first-year graduate algebra instructor my daily list of questions, he inquired what I thought of the class’s pace. “Are you kidding?” I said, “I feel like I’m barely hanging on by my fingernails.” He smiled. “Good,” he said, “I don’t want to waste your time.” This interchange nearly fifty years ago marked the beginning of my friendship with Charles Holland, the most influential person in my intellectual life. I had arrived at Madison with a keen interest in mathematical logic, and indeed Madison was a real center for set theory and logic during those years. But the reason I did not become a logician is that I fell under Charles’s influence. The attraction was initially based on the fact that Charles is a superb teacher, with a low-key style and an instinctive feel for his audience. But the real appeal was that, by my second year of graduate study, I was attending a seminar on lattice-ordered groups (-groups), sniffing the bracing air near the frontiers of 299 Unauthenticated Download Date | 9/24/15 11:34 PM


the subject. No student could approach the frontier in such a short time today. But in mathematics, nothing is as exciting as the unknown. The basic theory of -groups was undergoing rapid development at the time, with most of the work on this side of the Atlantic being done by the Conrad school. Charles and his students were tremendously productive, laying bare the structure of nonabelian -groups by means of Charles’s fundamental representation theorem. During this period I had the good fortune to make contact with Steve McCleary, who had left Madison by the time I got there. Steve is on a very short list of people who have fundamentally shaped my intellectual life. Andrew Glass arrived in Madison in August just prior to my second year, fresh from a year of studying group theory with Philip Hall at Cambridge. He, too, would become a Holland student and, in time, a close colleague of Charles at Bowling Green. It was heady stuff to be introduced to this fraternity of brilliant mathematicians during a time of creative ferment in the field. It was also a great privilege. Charles’s most important influence on me was not, however, primarily intellectual. Rather, it was the provision of an example of a kind and creative man, always generous with his time and ideas, flourishing in an academic environment. And I benefited directly from his wise guidance in many ways; I will close with two examples of what I am talking about. One summer day I reported to Charles that the preceding week had passed without any additional insights into the problem I had been working on. I further volunteered that perhaps the reason was that I was training for a bicycle race and had done nothing besides ride, eat, and sleep. I expected a lecture about the importance of tending to one’s research, in which case I suppose I would have done that. “Oh, no!” Charles said. “You never know where your ideas come from. You may get more ideas riding your bike than sitting in a library all day.” I smiled to myself, and resolved never to offer that excuse again. I learned something important that day, and I believe that it made a strong positive contribution not only to my bicycle racing career but to my mathematical career as well. The sixties were a turbulent decade everywhere, and Madison was a center of student protests of all types. The Madison teaching assistants were the first in the country to unionize, and when they went out on strike to establish the union, I went out with them. Charles had taken a negative position on the union in a public letter, so one day, as the time for our weekly appointment approached, I vacillated about whether to keep the appointment. At the last moment I left the picket line, marched up to his office, and knocked on the door. He greeted me with his usual warmth and, with my union placard resting against his office wall, we talked about my thesis for an hour. Mathematics had transcended politics. 300 Unauthenticated Download Date | 9/24/15 11:34 PM

W. CHARLES HOLLAND, 75th BIRTHDAY

3. From Andrew Glass and Steve McCleary Charles’ father, Wilbur C. Holland, was a geologist, and the family spent several years in some of the remotest areas of Wyoming and Louisiana (Charles attended eleven different schools). This aided Charles’ independent thinking. His ability to analyse clearly and deeply and select good problems for himself and his students was crucial for many of us. No description of Charles would be incomplete if it failed to mention the elegance and grace of his mathematics, and his kindness and generosity as a man. We have known Charles for over forty years (many of which were on an almost daily basis). Throughout that time, we always found him incredibly patient and constructive, always looking for the positive, his only dislikes being formality and closed-mindedness. He always seemed to make time to help people. Charles is always interested in other people’s ideas and engaging with research students, post-docs and colleagues wherever he goes. [Andrew adds: I would often stop in his office with my head full of disorganised ideas which I thought could lead to a theorem but couldn’t assemble to do the job. Some time later, I would emerge from Charles’ office with the ideas all lined up and the theorem proved. This was always a consequence of him asking the right questions and often providing the necessary steps that I had overlooked.] Charles is a family man with many interests (especially music). He is a generous and creative spirit who has provided the major tools and theorems in (lattice-)ordered groups and helped us in our own research. Long may it continue. Happy 75th birthday.

4. From Jorge Mart´ınez The first time I saw Charles Holland was in 1969, standing outside a convention facility in New Orleans, Louisiana. Inside that facility the American Mathematical Society’s annual January meeting was in full swing. It was cold, even for New Orleans, yet there we were, Paul Conrad (my adviser) and I, Conrad having lit up a cigar. We appeared to be waiting for someone. Conrad’s eyes suddenly lit up. He grinned through that cigar, eying a thin, young man wearing a beard, striding in our direction in long and easy measures. That year there were many young men with beards about town; this was the year of “Easy Rider”, and we were among mathematicians, after all, amongst which oddities of dress or grooming — or lack thereof — were commonplace. The thin young man with a beard had greying hair, so how could one tell that he was young? Easy. He had young eyes. Charles Holland has always appeared before me with the expression of youth and promise and possibilities and optimism and generosity. Conrad must have remarked on the beard, about its snowy streaks of white. Holland reflected on that and told about coming 301 Unauthenticated Download Date | 9/24/15 11:34 PM


down to New Orleans on a flight on which mathematicians predominated: men with beards of all sorts. He was seated next to an elderly lady with whom he struck up a conversation. It was obvious that these strange-looking men were going to a convention of some kind. “Who are these people?”, she asked him — one of them and yet not one of them. “Antiques salesmen?” Many years later, during a spring quarter when I was on leave in Bowling Green, Charles Holland listened to my personal troubles. He listened and empathized. I learned the meaning of the word empathy during those weeks in Bowling Green. It’s the expression that says: I am listening, for as long as you want me to, and I don’t necessarily agree that things are as bad as you describe them, but I won’t argue with you if you insist otherwise. The eyes were there, to remind you with their fresh and hopeful look that, if you really thought about your difficulties you might find, in the glowing lilacs of Japanese magnolias, or in the unruly discourse of your children, or, if not there, then in the labyrinths of mathematics, a way to rise above them. Recently, at a workshop in Buenos Aires, he and his wife, Claudia, met me at a famous cafe in the city, where he proceeded to tell the following story. The two of them had been walking along in the city, enjoying another terrific spring morning. Something liquid, oozing with viscosity, suddenly landed on his jacket. Seconds later there were two terribly helpful residents of the city, ready with towels and determined to fuss over these tourists. But, most of all, determined to help themselves to Holland’s cash and credit card, as he discovered a scant two dozen steps later when some reflex prompted him to pad himself down. He looked surprised still that something like that could have happened to them. You won’t believe this, the eyes of that thin, youthful fellow, with now mostly snowy white hair told me. How fortunate we were, right? It could have gone terribly awry. There wasn’t a large amount of cash, and the credit cards had been taken care of, and there we all were, and tomorrow would be another day.

REFERENCES [Ball1] BALL, R. N.: Topological lattice-ordered groups, Pacific J. Math. 83 (1979), 1–26. [Ball2] BALL, R. N.: The structure of the α-completion of a lattice ordered group, Houston J. Math. 15 (1989), 481–515. [Ball3] BALL, R. N.: Distinguished extensions of a lattice-ordered group, Algebra Universalis 35 (1996), 85–112. [BD] BALL, R. N.—DAVIS, G.: The α-completion of a lattice ordered group, Czechoslovak Math. J. 33(108) (1983), 111–118. [Ban] BANASCHEWSKI, B.: Totalegeordnete Moduln, Arch. Math. 7 (1957), 430–440. [BDG] BLUDOV, V. V.—DROSTE, M.—GLASS, A. M. W.: Automorphism groups of totally ordered sets: a retrospective survey, Math. Slovaca 61 (2011), 373–388. ˇ [Dvu] DVURECENSKIJ, A.: Pseudo MV-algebras are intervals in l-groups, J. Aust. Math. Soc. 72 (2002), 427–445.

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W. CHARLES HOLLAND, 75th BIRTHDAY ¨ [Hahn] HAHN, H.: Uber die nichtarchimedischen Gr¨ oßensysteme, S.-B. Kaiserlichen Akad. Wiss. Math. Nat. Kl. Iia 116 (1907), 601–655. [KK] KALOUJNINE, L.—KRASNER, M.: Produit complet des groupes de permutations et probleme d’extension des groupes, Acad. Sci. Math. Szeged 13 (1950), 208–230; ibid 14 (1951), 39–82. [Mad] MADELL, R. L.: Complete distributivity and alpha-convergence, Czechoslovak Math. J. 30(105) (1980), 296–301. [Mar] MARTINEZ, J.: Varieties of lattice-ordered groups, Math. Z. 137 (1974), 265–284. [Mun] MUNDICI, D.: Interpretation of AF C*-algebras in sentential calculus, J. Funct. Anal. 65 (1986), 15–63. [1] HOLLAND, C.: A totally ordered integral domain with a convex left ideal which is not an ideal, Proc. Amer. Math. Soc. 11 (1960), 703–703. [2] HOLLAND, C.: Extensions of ordered groups and sequence completion, Trans. Amer. Math. Soc. 107 (1963), 71–82. [3] CONRAD, P.—HARVEY, J.—HOLLAND, C.: The Hahn embedding theorem for abelian lattice ordered groups, Trans. Amer. Math. Soc. 108 (1963), 143–163. [4] HOLLAND, C.: The lattice ordered group of automorphisms of an ordered set, Michigan Math. J. 10 (1963), 399–408. [5] HOLLAND, C.: Transitive lattice ordered permutation groups, Math. Z. 87 (1965), 420–433. [6] HOLLAND, C.: A class of simple lattice ordered groups, Proc. Amer. Math. Soc. 16 (1965), 326–329. [7] HOLLAND, C.: The interval topology of a certain lattice ordered group, Czechoslovak Math. J. 15(90) (1965), 311–314. [8] HOLLAND, W. C.: The characterization of generalized wreath products, J. Algebra 13 (1969), 152–172. [9] HOLLAND, W. C.—McCLEARY, S. H.: Wreath products of ordered permutation groups, Pacific J. Math. 31 (1969), 703–716. [10] HOLLAND, C.—SCRIMGER, E.: Free products of lattice ordered groups, Algebra Universalis 2 (1972), 247–254. [11] HOLLAND, W. C.: Ordered permutation groups, Permutations. In: Actes du Colloque, Gautier-Villars, Paris, 1972, pp. 57–64. [12] HOLLAND, W. C.: Outer automorphisms of ordered permutation groups, Proc. Edinburgh Math. Soc. 19 (1975), 331–344. [13] GLASS, A. M. W.—HOLLAND, W. C.—McCLEARY, S. H.: a*-closures of completely distributive lattice ordered groups, Pacific J. Math. 59 (1975), 43–67; ibid 61 (1975), 606–606. [14] HOLLAND, W. C.: The largest proper variety of lattice ordered groups, Proc. Amer. Math. Soc. 57 (1976), 25–28. [15] HOLLAND, W. C.: Equitable partitions of the continuum, Fund. Math. 92 (1976), 131–133. [16] HOLLAND, W. C.: Group equations which hold in lattice ordered groups, Symposia Math. 21 (1977), 365–378. [17] HOLLAND, W. C.: Varieties of l-groups are torsion classes, Czechoslovak Math. J. 29(104) (1979), 11–12. [18] HOLLAND, W. C.—McCLEARY, S. H.: Solvability of the word problem in free lattice ordered groups, Houston J. Math. 5 (1979), 99–105. [19] HOLLAND, W. C.—MARTINEZ, J.: Accessibility of torsion classes, Algebra Universalis 9 (1979), 199–206. [20] GLASS, A. M. W.—HOLLAND, W. C.—McCLEARY, S. H.: The structure of l-group varieties, Algebra Universalis 10 (1980), 1–20.


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HOLLAND, W. C.: Trying to recognize the real line. In: Ordered Groups (Ball, Kinney, Smith, eds.). Lecture Notes in Pure and Appl. Math. 69, Marcel Dekker, New York, 1980, pp. 131–134. GLASS, A. M. W.—GUREVICH, Y.—HOLLAND, W. C.—SHELAH, S.: Rigid homogeneous chains, Math. Proc. Cambridge Phil. Soc. 89 (1981), 7–17. GLASS, A. M. W.—GUREVICH, Y.—HOLLAND, W. C.—JAMBU-GIRAUDET, M.: Elementary theory of automorphism groups of doubly homogeneous chains. In: Logic Year 1979-80, Univ. Conn./USA. Lecture Notes in Math. 859, Springer, New York, 1981, pp. 67–82. GUREVICH, Y.—HOLLAND, W. C.: Recognizing the real line, Trans. Amer. Math. Soc. 265 (1981), 527–534. HOLLAND, W. C.: A survey of varieties of lattice ordered groups. In: Universal Algebra and Lattice Theory, Lecture Notes in Math. 1004, Springer, New York, 1983, pp. 153–158. HOLLAND, W. C.: Classification of lattice ordered groups, Ann. Discrete Math. 23 (1984), 151–155. HOLLAND, W. C.: Intrinsic metrics for lattice ordered groups, Algebra Universalis 19 (1984), 142–150. HOLLAND, W. C.—MEKLER, A.—SHELAH, S.: Lawless order, Order 1 (1985), 383–397. HOLLAND, W. C.: A note on lattice orderability of groups, Algebra Universalis 20 (1985), 130–131. HOLLAND, W. C.: Varieties of automorphism groups of orders, Trans. Amer. Math. Soc. 288 (1985), 755–763. HOLLAND, W. C.: Intrinsic metrics for lattice ordered groups, In: Ordered Algebraic Structures (W. Powell, C. Tsinakis, eds.). Lecture Notes in Pure and Applied Math. 99, Marcel Dekker, New York, 1985, pp. 99–106. HOLLAND, W. C.: Remarks on Paul Conrad, In: Ordered Algebraic Structures (W. Powell, C. Tsinakis, eds.). Lecture Notes in Pure and Applied Math. 99, Marcel Dekker, New York, 1985, pp. vii–ix. HOLLAND, W. C.—MEKLER, A. H.—SHELAH, S.: Total orders whose carried groups satisfy no laws, In: Algebra and Order. Proceedings of the First International Symposium on Ordered Algebraic Structures, Luminy-Marseilles 1984 (S. Wolfenstein, ed.), Heldermann, 1986, pp. 29–33. HOLLAND, W. C.—REILLY, N. R.: Structure and laws of the Scrimger varieties of lattice-ordered groups, In: Algebra and Order. Proceedings of the First International Symposium on Ordered Algebraic Structures, Luminy-Marseilles 1984 (S. Wolfenstein, ed.) Heldermann, 1986, pp. 71–81. HOLLAND, W. C.—MEKLER, A. H.—REILLY, N. R.: Varieties of lattice-ordered groups in which prime powers commute, Algebra Universalis 23 (1986), 196–214. HOLLAND, W. C.—REILLY, N. R.: Metabelian varieties of l-groups which contain no non-abelian o-groups, Algebra Universalis 24 (1987), 204–223. HOLLAND, W. C.—SMITH, D. P.: Irrational groups, Order 4 (1988), 381–386. DROSTE, M.—HOLLAND, W. C.—MACPHERSON, H. D.: Automorphism groups of infinite semilinear orders, (I), Proc. London Math. Soc. (3) 58 (1989), 454–478. DROSTE, M.—HOLLAND, W. C.—MACPHERSON, H. D.: Automorphism groups of infinite semilinear orders, (II), Proc. London Math. Soc. 58 (1989), 479–494. GLASS, A. M. W.—HOLLAND, W. C., eds.: Lattice-Ordered Groups: Advances and Techniques, Kluwer Acad. Publ., Boston, 1989.


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HOLLAND, W. C.: Lattice-ordered permutation groups. In: Lattice-Ordered Groups: Advances and Techniques (Glass, Holland, eds.), Kluwer Acad. Publ., Boston, 1989, pp. 23–40. DROSTE, M.—HOLLAND, W. C.—MACPHERSON, H. D.: Automorphism groups of infinite semilinear orders: Normal subgroups and commutators, Canad. J. Math. 43 (1991), 721–737. HOLLAND, W. C.—SZKELY, G.: Lattice-ordered groups with a prescribed minimum for given elements, Algebra Universalis 29 (1992), 79–86. HOLLAND, W. C.: Partial orders of the group of automorphisms of the real line. In: Proceedings of the Malcev Conference, Novosibirsk 1989. Contemp. Math. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 197–207. GIRAUDET, M.—HOLLAND, W. C., eds.: First Meeting on Ordered Groups and Infinite Permutation Groups. The 1990 Conference Proceedings, Assoc. Francaise d’Algebre Ordonne, Paris, 1992. HOLLAND, W. C.—MARTINEZ, J., eds.: Ordered Algebraic Structures: The 1991 Conrad Conference, Kluwer Acad. Publ., Boston, MA, 1993. HOLLAND, W. C.—MEDVEDEV, N. YA.: A very large class of small varieties of lattice-ordered groups, Comm. Algebra 22 (1994), 551–578. ADELEKE, S. A.—HOLLAND, W. C.: Representation of order automorphisms by words, Forum Math. 6 (1994), 315–321. HOLLAND, W. C., ed.: Ordered Groups and Infinite Permutation Groups, Kluwer Acad. Publ., Boston, MA, 1995. HOLLAND, W. C.: Varieties and universal words for automorphism groups of orders. In: Advances in Algebra and Model Theory (M. Droste, R. G¨ obel, eds.), Gordon and Breach Science Publishers, Amsterdam, 1997, pp. 135–147. HOLLAND, W. C.—MARTINEZ, J., eds.: Ordered Algebraic Structures. Proceedings of the Curacao Conference, 1995, Kluwer Acad. Publ., Boston, MA, 1997. CLARK, W. E.—HOLLAND, W. C.—SZEKELY, G.: Decompositions in discrete semigroups, Studia Sci. Math. Hungar. 34 (1998), 15–23. HOLLAND, W. C.: Equational classes of automorphism groups of structures. Pub. Equipe de Logique Math. No. 71, University of Paris, 2000. HOLLAND, W. C., ed.: Ordered Algebraic Structures: Nanjing, Gordon and Breach, Amsterdam, 2001. GIRAUDET, M.—HOLLAND, W. C.: Ohkuma structures, Order 19 (2002), 223–237. HOLLAND, W. C.: Review of Ordered Exponential Fields by S. Kuhlman, Publ. Bull. Canadian Math. Soc., 2002. HOLLAND, W. C.: Divisibilizing normal-valued lattice-ordered groups, Tatra Mt. Math. Publ. 27 (2003), 131–138. HOLLAND, W. C.—RUBIN, M.: Semi-Ohkuma chains, Order 21 (2004), 231–256. HOLLAND, W. C.: Small varieties of lattice-ordered groups and MV-Algebras. In: Contributions to General Algebra 16. Proceedings of the Conference on General Algebra, Dresden, 2004, Verlag Johannes Heyn, Klagenfurt 2005, pp. 107–114. DROSTE, M.—HOLLAND, W. C.: Generating automorphism groups of chains, Forum Math. 17 (2005), 699–710. HOLLAND, W. C.—TSABAN, B.: The conjugacy problem and related problems in lattice-ordered groups, Internat. J. Algebra Comput. 15 (2005), 395–404. HOLLAND, W. C.—KUHLMANN, S.—McCLEARY, S. H.: Lexicographic exponentiation of chains, J. Symbolic Logic 70 (2005), 389–409. ˇ DVURECENSKIJ, A.—HOLLAND, W. C.: Top varieties of generalized MV-algebras and unital lattice-ordered groups, Comm. Algebra 35 (2007), 3370–3390.


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ˇ DVURECENSKIJ, A.—HOLLAND, W. C.: Komori’s characterization and top varieties of GMV-algebras, Algebra Universalis 60 (2009), 37–62. HOLLAND, W. C.: Covers of the Boolean variety in the lattice of varieties of unital lattice ordered groups and GMV-algebras. In: Selected Questions of Algebra. Collection of papers dedicated to the memory of N. Ya. Medvedev (N. Bayanova, ed.), Altai State University Barnaul, Barnaul, 2007, pp. 208–217. DROSTE, M.—HOLLAND, W. C.: Normal subgroups of Bu Aut(Ω), Appl. Categ. Structures 15 (2007), 153–162. ˇ DVURECENSKIJ, A.—HOLLAND, W. C.: Covers of the Abelian variety of generalized MV-algebras, Comm. Algebra 37 (2009), 3991–4011. DROSTE, M.—HOLLAND, W. C.—ULBRICH, G.: On full groups of measurepreserving and ergodic transformations with uncountable cofinalities, Bull. Lond. Math. Soc. 40 (2008), 463–472. HOLLAND, W. C.—MARTINEZ, J.—McGOVERN, W. W.—TESSEMA, M.: Bazzoni’s conjecture, J. Algebra 320 (2008), 1764–1768. ˇ DVURECENSKIJ, A.—HOLLAND, W. C.: Free products of unital -groups and free products of generalized MV-algebras, Algebra Universalis 62 (2009), 19–25. HOLLAND, W. C.: Continuum many top varieties of GMV-algebras and unital -groups, Algebra Universalis 62 (2009), 27–43. DARNEL, M. R.—HOLLAND, W. C.: Solvable covers of the boolean variety of unital -groups, Algebra Universalis 62 (2009), 185–199.

Received 26. 1. 2011 Accepted 26. 1. 2011

* Department of Mathematics University of Denver John Greene Hall 2360 S. Gaylord St. Denver, CO 80208 UNITED STATES E-mail : [email protected] ** Depart. Pure Math. and Math. Statistics Centre for Mathematical Sciences University of Cambridge Wilberforce Road Cambridge, CB3 0WB UNITED KINGDOM E-mail : [email protected] *** Department of Mathematics University of Florida P.O.Box 118105 Gainesville FL 32611-8105 UNITED STATES E-mail : [email protected] **** 7180 Las Vistas Rd, Las Cruces New Mexico 88005 UNITED STATES E-mail : [email protected]
