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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Closed subgroups of lattice-ordered permutation groups

Author: Stephen H. McCleary
Journal: Trans. Amer. Math. Soc. 173 (1972), 303-314
MSC: Primary 06A55; Secondary 20B99
MathSciNet review: 0311535
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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\vert\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\vert\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 }\vert\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 }\vert\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.

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Keywords: Lattice-ordered permutation group, totally ordered set, complete subgroup, prime subgroup, closed subgroup, stabilizer subgroup, complete distributivity, wreath product
Article copyright: © Copyright 1972 American Mathematical Society