Closed Subgroups of Lattice-Ordered Permutation Groups

Abstract

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