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Above and below subgroups of a lattice-ordered group. (English) Zbl 0628.06013
Authors’ summary: “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($$| a| \wedge | b|)\leq | a|$$. 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.”
Reviewer: B.F.Smarda

##### MSC:
 06F15 Ordered groups 20F60 Ordered groups (group-theoretic aspects)
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