A class of simple lattice-ordered groups
Charles Hepworth Holland · Proceedings of the American Mathematical Society · 1965
A lattice-ordered group (I-group) is said to be regular if no positive element of the group is disjoint from any of its conjugates. It is well known that every simple regular i-group is totally ordered [5]. The subgroups of the reals are the most elementary examples of regular simple i-groups; other examples can be found in [2] and [6]. In this note we investigate a class of simple i-groups at the opposite extreme from the regular ones. We are concerned with i-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 i-group can be represented as an i-group of automorphisms of a totally ordered set, and it was shown that the i-group of automorphisms of the real line with bounded support is simple. It is natural to ask which simple i-groups can be represented as automorphisms of an ordered set with bounded support. Our main result is that these are exactly the simple i-groups containing an insular element. We also construct several examples of such groups. If L is a totally ordered set and f is an order-preserving permutation of L, we call f an automorphism of L. The support of f consists of those xEL such that xf ox. An automorphism of L is bounded if its support lies in a closed interval of L. An i-group of automorphisms of L is a group of automorphisms of L (under composition) which is a lattice under the operations n and U defined by x(fng) = (xf)n(xg) and dually. Such a group is a lattice-ordered group in the usual sense