Above and below subgroups of a lattice-ordered group

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 $H$.