Tyings in lattice-ordered permutation groups

Abstract

In an l-permutation group (G, d), with d a chain and d ( its Dedekind completion, the coincidence of two stabilizer subgroups G l =G u (l, u d ( ) yields a map lg ug (g G) from lG to uG, and this map commutes with all the elements of G. Roughly speaking, a tying is such a map. We show that the permutations of d ( which commute with the tyings are exactly those in the closure of G in the full automorphism group A(d ( ) with respect to the coarse stabilizer topology. We term this closure the gate completion of G, written G . We show that each o-primitive component of G consists of those permutations of the closure of the corresponding G component which respect the orbits of the points which are ''tied'' to nonsingleton o-blocks. Finally, we show that any two representations of the same lattice-ordered group which are complete and without dead segments give rise to the same G , and that in this case G is the h-completion of G.

The closure of a lattice-ordered permutation group

“…The present paper is a sequel to [7], some familiarity with which is assumed here. There we investigated the l-group G : consisting of those elements of A(d ( ) which can be gated by elements of G, i.e., the closure of G in the coarse stabilizer topology on A(d ( ).…”

“…Although familiarity with [7] is assumed in matters of terminology and notation and for a few specific results, enough background will be offered that the reader lacking such familiarity should still be able to understand the statements of the present results.…”

“…Instead of giving the technical definition of a tying in order to define G , we give a closely related characterization of G [7,Theorem 3.1]. An h A(d ( ) respects the tyings of (G, d) We assemble here some information from [13, pages 247] (or see [8,Chapter 8]).…”

“…Analogous statements hold for G : , as the next theorem will make clear. However, the assertion in [6] that these two topologies on G : coincide was erroneous, although this is true of A(d ( ) [7,Corollary 7.2].…”

“…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.…”