Closed subgroups of lattice-ordered permutation groups

Abstract

Let G G be an l l -subgroup of the lattice-ordered group A ( Ω ) A(\Omega ) of order-preserving permutations of a chain Ω \Omega ; and in this abstract, assume for convenience that G G is transitive. Let Ω ¯ \bar \Omega denote the completion by Dedekind cuts of Ω \Omega . The stabilizer subgroups G ω ¯ = { g ϵ G | ω ¯ g = ω ¯ } , ω ¯ ϵ Ω ¯ {G_{\bar \omega }} = \{ g \epsilon G|\bar \omega g = \bar \omega \} ,\bar \omega \epsilon \bar \Omega , will be used to characterize certain subgroups of G G which are closed (under arbitrary suprema which exist in G G ). If Δ \Delta is an o o -block of G G (a nonempty convex subset such that for any g ϵ G g \epsilon G , either Δ g = Δ \Delta g = \Delta or Δ g ∩ Δ \Delta g \cap \Delta is empty), and if ω ¯ = sup Δ , G Δ \bar \omega = \sup \Delta ,{G_\Delta } will denote { g ϵ G | Δ g = Δ } = G ω ¯ \{ g \epsilon G|\Delta g = \Delta \} = {G_{\bar \omega }} ; and the o o -block system Δ ~ \tilde \Delta consisting of the translates Δ g \Delta g of Δ \Delta will be called closed if G Δ {G_\Delta } is closed. When the collection of o o -block systems is totally ordered (by inclusion, viewing the systems as congruences), there is a smallest closed system C \mathcal {C} , and all systems above C \mathcal {C} are closed. C \mathcal {C} is the trivial system (of singletons) iff G G is complete (in A ( Ω ) A(\Omega ) ). G ω ¯ {G_{\bar \omega }} is closed iff ω ¯ \bar \omega is a cut in C \mathcal {C} i.e., ω ¯ \bar \omega is not in the interior of any Δ ϵ C \Delta \epsilon \mathcal {C} . Every closed convex l l -subgroup of G G is an inter-section of stabilizers of cuts in C \mathcal {C} . Every closed prime subgroup ≠ G \ne G is either a stabilizer of a cut in C \mathcal {C} , or else is minimal and is the intersection of a tower of such stabilizers. L ( C ) = ∩ { G Δ | Δ ϵ C } L(\mathcal {C}) = \cap \{ {G_\Delta }|\Delta \epsilon \mathcal {C}\} is the distributive radical of G G , so that G G acts faithfully (and completely) on C \mathcal {C} iff G G is completely distributive. Every closed l l -ideal of G G is L ( D ) L(\mathcal {D}) for some system D \mathcal {D} . A group G G in which every nontrivial o o -block supports some 1 ≠ g ϵ G 1 \ne g \epsilon G (e.g., a generalized ordered wreath product) fails to be complete iff G G has a smallest nontrivial system Δ ~ \tilde \Delta and the restriction G Δ | Δ {G_\Delta }|\Delta is o o - 2 2 -transitive and lacks elements ≠ 1 \ne 1 of bounded support. These results about permutation groups are used to show that if H H is an abstract l l -group having a representing subgroup, its closed l l -ideals form a tower under inclusion; and that if { K λ } \{ {K_\lambda }\} is a Holland kernel of a completely distributive abstract l l -group H H , then so is the set of closures { K λ ∗ } \{ K_\lambda ^ \ast \} , so that if H H has a transitive representation as a permutation group, it has a complete transitive representation.