A Class of Simple Lattice-ordered Groups

A lattice-ordered group (/-group) is said to be regular il no positive element of the group is disjoint from any of its conjugates. It is well known that every simple regular /-group is totally ordered [S]. The subgroups of the reals are the most elementary examples of regular simple /-groups; other examples can be found in [2] and [ó]. In this note we investigate a class of simple /-groups at the opposite extreme from the regular ones. We are concerned with /-groups which contain an insular (defined below) element. An insular element is, roughly speaking, an element which is strongly disjoint from one of its conjugates. In [4] it was shown that every /-group can be represented as an /-group of automorphisms of a totally ordered set, and it was shown that the /-group of automorphisms of the real line with bounded support is simple. It is natural to ask which simple /-groups can be represented as automorphisms of an ordered set with bounded support. Our main result is that these are exactly the simple /-groups containing an insular element. We also construct several examples of such groups. If L is a totally ordered set and / is an order-preserving permutation of L, we call / an automorphism of L. The support of / consists of those xEL such that xf^x. An automorphism of L is bounded if its support lies in a closed interval of L. An l-group of automorphisms of L is a group of automorphisms of L (under composition) which is a lattice under the operations C\ and U defined by x(fC\g) = (xf)C\(xg) and dually. Such a group is a lattice-ordered group in the usual sense [l]. If G is an /-group of automorphisms of L, G is o-primitive on L if there is no equivalence relation E on L such that (1) £ is a congruence; that is, for all x, yEL,fEG, xEy implies xfEyf, and (2) E is convex; that is, each £-class is a convex subset of L. For elements /, g è 1 of an /-group G, f is right of g if for all 1 úhEG, gr\h~lfh — 1. An element gEG is insular if for some conjugate g* of g, g* is right of g.