# Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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## Above and below subgroups of a lattice-ordered groupHTML articles powered by AMS MathViewer

by Richard N. Ball, Paul Conrad and Michael Darnel
Trans. Amer. Math. Soc. 297 (1986), 1-40 Request permission

## 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 $b \in B$, $n(|a| \wedge |b|) \leqslant |a|$. 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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Additional Information
• © Copyright 1986 American Mathematical Society
• Journal: Trans. Amer. Math. Soc. 297 (1986), 1-40
• MSC: Primary 06F15; Secondary 20E22
• DOI: https://doi.org/10.1090/S0002-9947-1986-0849464-7
• MathSciNet review: 849464