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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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