Characteristic subgroups of latticeordered groups
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 by Richard D. Byrd, Paul Conrad and Justin T. Lloyd PDF
 Trans. Amer. Math. Soc. 158 (1971), 339371 Request permission
Abstract:
Characteristic subgroups of an $l$group are those convex $l$subgroups that are fixed by each $l$automorphism. Certain sublattices of the lattice of all convex $l$subgroups determine characteristic subgroups which we call socles. Various socles of an $l$group are constructed and this construction leads to some structure theorems. The concept of a shifting subgroup is introduced and yields results relating the structure of an $l$group to that of the lattice of characteristic subgroups. Interesting results are obtained when the $l$group is characteristically simple. We investigate the characteristic subgroups of the vector lattice of realvalued functions on a root system and determine those vector lattices in which every $l$ideal is characteristic. The automorphism group of the vector lattice of all continuous realvalued functions (almost finite realvalued functions) on a topological space (a Stone space) is shown to be a splitting extension of the polar preserving automorphisms by the ring automorphisms. This result allows us to construct characteristically simple vector lattices. We show that selfinjective vector lattices exist and that an archimedean selfinjective vector lattice is characteristically simple. It is proven that each $l$group can be embedded as an $l$subgroup of an algebraically simple $l$group. In addition, we prove that each representable (abelian) $l$group can be embedded as an $l$subgroup of a characteristically simple representable (abelian) $l$group.References

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Additional Information
 © Copyright 1971 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 158 (1971), 339371
 MSC: Primary 06.75
 DOI: https://doi.org/10.1090/S00029947197102790147
 MathSciNet review: 0279014