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Transactions of the American Mathematical Society

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



Above and below subgroups of a lattice-ordered group

Authors: Richard N. Ball, Paul Conrad and Michael Darnel
Journal: Trans. Amer. Math. Soc. 297 (1986), 1-40
MSC: Primary 06F15; Secondary 20E22
MathSciNet review: 849464
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Abstract: In an $ l$-group $ G$, this paper defines an $ l$-subgroup $ A$ to be above an $ l$-subgroup $ B$ (or $ B$ to be below $ A$) if for every integer $ n$, $ a \in A$, and $ b \in B$, $ n(\vert a\vert \wedge \vert b\vert) \leqslant \vert a\vert$. It is shown that for every $ l$-subgroup $ A$, there exists an $ l$-subgroup $ B$ maximal below $ A$ which is closed, convex, and, if the $ l$-group $ G$ is normal-valued, unique, and that for every $ l$-subgroup $ B$ there exists an $ l$-subgroup $ A$ maximal above $ B$ which is saturated: if $ 0 = x \wedge y$ and $ x + y \in A$, then $ x \in A$.

Given $ l$-groups $ A$ and $ B$, it is shown that every lattice ordering of the splitting extension $ G = A \times B$, which extends the lattice orders of $ A$ and $ B$ and makes $ A$ lie above $ B$, is uniquely determined by a lattice homomorphism $ \pi $ from the lattice of principal convex $ l$-subgroups of $ A$ into the cardinal summands of $ B$. This extension is sufficiently general to encompass the cardinal sum of two $ l$-groups, the lex extension of an $ l$-group by an $ o$-group, and the permutation wreath product of two $ l$-groups.

Finally, a characterization is given of those abelian $ l$-groups $ G$ that split off below: whenever $ G$ is a convex $ l$-subgroup of an $ l$-group $ H$, $ H$ is then a splitting extension of $ G$ by $ A$ for any $ l$-subgroup $ A$ maximal above $ G$ in $ H$.

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Article copyright: © Copyright 1986 American Mathematical Society