Characteristic subgroups of lattice-ordered groups

Authors:
Richard D. Byrd, Paul Conrad and Justin T. Lloyd

Journal:
Trans. Amer. Math. Soc. **158** (1971), 339-371

MSC:
Primary 06.75

MathSciNet review:
0279014

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Abstract: Characteristic subgroups of an -group are those convex -subgroups that are fixed by each -automorphism. Certain sublattices of the lattice of all convex -subgroups determine characteristic subgroups which we call socles. Various socles of an -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 -group to that of the lattice of characteristic subgroups. Interesting results are obtained when the -group is characteristically simple. We investigate the characteristic subgroups of the vector lattice of real-valued functions on a root system and determine those vector lattices in which every -ideal is characteristic. The automorphism group of the vector lattice of all continuous real-valued functions (almost finite real-valued 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 self-injective vector lattices exist and that an archimedean self-injective vector lattice is characteristically simple. It is proven that each -group can be embedded as an -subgroup of an algebraically simple -group. In addition, we prove that each representable (abelian) -group can be embedded as an -subgroup of a characteristically simple representable (abelian) -group.

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DOI:
https://doi.org/10.1090/S0002-9947-1971-0279014-7

Keywords:
Socles of an -group,
-automorphism,
characteristic subgroup,
characteristically simple -group,
polar,
Boolean algebra,
completely reducible -group,
shifting subgroup,
-simple subgroup,
completely -reducible -group,
lex-subgroup,
prime subgroup,
closed subgroup,
cardinally indecomposable -group,
lex-extension,
basic element,
basis,
principal polar,
lex-kernel,
regular subgroup,
special element,
finite valued -group,
root system,
root,
vector lattice,
essential extension,
archimedean extension,
completely regular space,
real compact space,
splitting extension,
self-injective -group,
large subgroup,
hyperarchimedean -group,
radical,
ideal radical,
distributive radical,
singular element

Article copyright:
© Copyright 1971
American Mathematical Society