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

DOI:
https://doi.org/10.1090/S0002-9947-1986-0849464-7

MathSciNet review:
849464

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Abstract: In an -group , this paper defines an -subgroup to be *above* an -subgroup (or to be *below* ) if for every integer , , and , . It is shown that for every -subgroup , there exists an -subgroup maximal below which is closed, convex, and, if the -group is normal-valued, unique, and that for every -subgroup there exists an -subgroup maximal above which is *saturated*: if and , then .

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

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

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DOI:
https://doi.org/10.1090/S0002-9947-1986-0849464-7

Article copyright:
© Copyright 1986
American Mathematical Society