W. Charles Holland, 75th birthday

Abstract

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. Research of W. Charles HollandIn 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.