l-simple lattice-ordered groups

Abstract

Let G be a lattice-ordered group (l-group) and H a subgroup of G. H is said to be an l-subgroup of G if it is a sublattice of G. H is said to be convex if h1, h2 ∈ H and h2 ≦ g ≦ h2 imply g ∈ H. The normal convex l-subgroups (l-ideals) of an l-group play the same role in the study of lattice-ordered groups as do normal subgroups in the investigation of groups. For this reason, an l-group is said to be l-simple if it has no non-trivial l-ideals. As in group theory, a central task in the examination of lattice-ordered groups is to characterise those l-groups which are l-simple.

a∗-closures of completely distributive lattice-ordered groups

“…Hence H m = H n . Since mΦn 9 the transitivity of iϊ on C implies the existence of heH m \H n , a contradiction. Therefore C Π S is coinitial in C. Similarly, C Π S is cofinal in C.…”

“…The reader may wish to consider finding the t-closures of the more complicated pathological o-2-transitive examples given in [9]. Since the pathological groups are the o-primitive groups which are not completely distributive, they are at the heart of understanding what is true in the general case.…”

“…If (G, S) has a minimal o-primitive component (i.e., associated with a minimal covering pair) (G μ , S μ ) 9 then (G, S) is said to be locally o-primitive and the S^μ classes are called the 'primitive segments.…”

Automorphism groups of totally ordered sets: A retrospective survey

“…Permutation ideas can also be used to prove that any free -group on a finite (≥ 2) or countably infinite set of generators is -isomorphic to an o-2 transitive sublattice subgroup of A(Q) (see [26], [57], and [50]). …”

Some 𝑙-simple pathological lattice-ordered groups

“…In [7] and [6] there were established certain properties of this class of groups which led to the name "pathological". It was found [2] that these groups are not always /-simple. Pathological groups played a crucial role in [3] in the theory of a*-extensions and t-extensions, where the two groups G and H (defined below) arose quite naturally.…”