The closure of a lattice-ordered permutation group

Abstract

Let (G, d) be an l-permutation group, with d a chain and d ( its Dedekind completion. The gate completion (G : , d ( ) consists of the elements of the automorphism group A(d ( ) which can be ''gated'' by elements of G, or equivalently, which respect the ''tyings'' (roughly, equality of stabilizer subgroups) of (G, d) [7]. In this sequel we find that (G : , d ( ) and its variant (G o: , d ( ) have order closed stabilizer subgroups, making G : and G o: completely distributive l-groups. The order closure of G in A(d ( ) turns out to be G o: . Moreover, the elements of G o: (and of G : ) can be readily constructed from those of G. Every h G o: can be written asWhen d is countable, this can be improved to say thatThese results illuminate several of the standard completions of abstract completely distributive l-groups. The proofs use the structure theory of l-permutation groups to focus on the constraints imposed by tyings.

Tyings in lattice-ordered permutation groups

“…Its uniqueness follows from general considerations [ G can never be the h-completion of an l-group which is not completely distributive. This is because an l-group is completely distributive if and only if its h-completion is [5], and we show in [8] that, in the absence of dead segments, G always has closed stabilizers and is therefore completely distributive. We can now state the major result of this section.…”

“…In this article we introduce the notion of a tying in an l-permutation group. Since we believe tyings to be natural and important, and since this article is the first of several on the subject ( [7] being a preview and [8] a sequel), we precede the formal definition with several examples. Though simple and close at hand, these examples are prototypical of the main types of l-permutation groups [10].…”

“…We conclude this introduction by previewing a result from the sequel [8]. How can G be built up from G by adjoining elements which are infinite suprema and infima of subsets of G, and iterating?…”

“…Fortunately, in [8] we won't need the G o version because results about G o will be deduced by using Theorem 4.3 and dealing with G instead.…”

“…(A representation of G is an l-injection from G into A(d) for some chain d. The representation is complete if it preserves all the suprema and infima that exist in G.) The reason that G is independent of such a representation is that G coincides with the h-completion G h (Theorem 7.6), and the h-completion is constructed from G without reference to any representation. This coincidence sheds light both ways: using it, we are able to prove that G is cut complete (Corollary 7.8), and, in the forthcoming article [8] we show that G , and therefore G h , is the cut completion of G, i.e., the minimal cut complete extension in which G is large [5].…”