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As a result of the work of the nineteenth-century mathematician Arthur Cayley, algebraists and geometers have extensively studied permutation of sets. In the special case that the underlying set is linearly ordered, there is a natural subgroup to study, namely the set of permutations that preserves that order. In some senses. these are universal for automorphisms of models of theories. The purpose of this book is to make a thorough, comprehensive examination of these groups of permutations. After providing the initial background Professor Glass develops the general structure theory, emphasizing throughout the geometric and intuitive aspects of the subject. He includes many applications to infinite simple groups, ordered permutation groups and lattice-ordered groups. The streamlined approach will enable the beginning graduate student to reach the frontiers of the subject smoothly and quickly. Indeed much of the material included has never been available in book form before, so this account should also be useful as a reference work for professionals.

Dedicated to Graham Higman, in memoriam, with gratitude for his research. AbstractWe use permutation groups to give necessary and sufficient conditions for the free product of right ordered groups with amalgamated subgroup to be right orderable. We obtain several consequences answering previously posed problems and also prove the right orderable analogues of the Higman Embedding Theorem and the Boone-Higman Theorem.--------------AMS Classification: 06F15, 20F60, 20B27, 20F10.

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Graham Higman proved that the finitely generated groups that occur as subgroups of finitely presented groups are precisely those that can be defined by recursively enumerable sets of relations.We prove the analogue for lattice-ordered groups: Theorem. A finitely generated lattice-ordered group is a sublattice subgroup of some finitely presented lattice-ordered group if and only if it can be defined by a recursively enumerable set of relations.Consequently, there is a universal finitely presented lattice-ordered group.