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Abstract

The purpose of this paper is to show that the automorphism groups of many of the ‘cycle-free’ partial orders studied in [Warren, R.: The structure of k-C S-transitive cycle-free partial orders. Memoirs of the American Mathematical Society (1997), to appear] and [Creed, P., Truss, J. K. and Warren, R.: The structure of k-C S-transitive cycle-free partial orders having infinite chains, to appear] are simple. This contrasts strongly with the situation for trees, of which they form a natural generalization. It was shown in [Droste, M., Holland, W.C. and Macpherson, H.D.: Automorphism groups of infinite semilinear orders (I) and (II). Proc. London Math. Soc. 58 (1989), 454–478 and 479–494] that the automorphism group of any sufficiently transitive tree has at least normal subgroups. All the infinite chain cycle-free partial orders studied in [Creed, P., Truss, J. K. and Warren, R.: The structure of k-C S-transitive cycle-free partial orders having infinite chains, to appear] have simple automorphism groups. The finite chain case is more involved; where the ordering on chains of the Dedekind-MacNeille completion can be expressed as a lexicographic product by a non-trivial discrete (transitive) ordering (respected by the group), the automorphism group is not simple. For both finite and infinite chain cases the simple automorphism groups split into two classes: those where there is a bound (≤ 2) on the number of conjugates required to express one non-identity element in terms of another, and those in which there is no such bound.

Abstract

We show that the automorphism group of the countable universal distributive lattice has strong uncountable cofinality, and we adapt the method to deduce the strong uncountable cofinality of the automorphism group of the countable universal generalized boolean algebra.

Abstract

We discuss the solubility of equations of the form w = g, where w is a word (an element of a free group FX) and g is an element of a given group G. A word for which this equation is soluble for every gG is said to be universal for G. It is conjectured that a word is universal for the automorphism group of the random graph if and only if it cannot be written as a proper power, corresponding to the results of [Randall Dougherty and Jan Mycielski. Representations of infinite permutations by words (II). Proc. Amer. Math. Soc. 127 (1999), 2233–43.], [Roger C. Lyndon. Words and infinite permutations. In Mots, Lang. Raison Calc. (Hermès, 1990), pp. 143–152.], [Jan Mycielski. Representations of infinite permutations by words. Proc. Amer. Math. Soc. 100 (1987), 237–241.], where the same necessary and sufficient condition was established for infinite symmetric groups. We prove various special cases. A key ingredient is the use of ‘generic’ automorphisms, and elements which suitably approximate them, called ‘special’.

Abstract

In 1963, W. Charles Holland proved that every lattice-ordered group can be embedded in the lattice-ordered group of all order-preserving permutations of a totally ordered set. In this article we examine the context and proof of this result and survey some of the many consequences of the ideas involved in this important theorem.

Zusammenfassung

Marie-Christin Lux und Philipp Schulte

»De l’État-Nation à l’Europe. Histoire du ministère de la Défense et des armées en France et en Allemagne aux XIXe et XXe siècles«. Tagung des Deutschen Historischen Instituts Paris (DHIP), des Militärgeschichtlichen Forschungsamtes (MGFA) und des Institut de Recherche Stratégique de l’École Militaire (IRSEM), 20. Juni 2011 in Paris

Sven Petersen und Stefan Droste

»Militärische Wissenskulturen in der Frühen Neuzeit«. 9. Jahrestagung des Arbeitskreises Militär und Gesellschaft in der Frühen Neuzeit vom 1. bis 3. Dezember 2011 in Göttingen

Florian Reichenberger

»Guerres futures, guerres imaginées: vers une histoire culturelle de l’avant-1914. Colloque international co-organisé par le Centre International de Recherche de l’Historial de la Grande Guerre et l’Institut historique allemand«, 9. November 2011, Péronne und 10. November 2011, Paris

Antje Dierking

»Für Frieden und Freiheit oder Shareholder Value und Auslastungsquote? Rüstung in Europa und Nordamerika nach dem Zweiten Weltkrieg«. Workshop des Militärgeschichtlichen Forschungsamtes, 8. bis 9. November 2011 in Potsdam

Janine Doerry

»Captivité de guerre au XXième siècle. Des archives, des histoires, des mémoires«. Internationale Tagung des Institut historique du temps présent (IHTP), des Centre national de la recherche scientifique (CNRS), des Institut de Recherche Stratégique de l’École Militaire (IRSEM) und der London School of Economics and Political Science (LSE), 17. bis 18. November 2011 in Paris

Flavio Eichmann

»Freibeuter der Moderne - Politisch-militärische Akteure an den Rändern von Souveränität und Legitimität«. Kolloquium zum 60. Geburtstag von Prof. Dr. Stig Förster, 20. bis 21. Oktober 2011 in Bern

Stefan M. Brenner

Die Achillesferse der NATO: Die Nordatlantische Allianz und der griechisch-türkische Binnenkonflikt 1952–1989. Ein Überblick zum Stand der Forschung