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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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 $ 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 real-valued 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 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 $ 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.

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Keywords: Socles of an $ l$-group, $ l$-automorphism, characteristic subgroup, characteristically simple $ l$-group, polar, Boolean algebra, completely reducible $ l$-group, shifting subgroup, $ s$-simple subgroup, completely $ s$-reducible $ l$-group, lex-subgroup, prime subgroup, closed subgroup, cardinally indecomposable $ l$-group, lex-extension, basic element, basis, principal polar, lex-kernel, regular subgroup, special element, finite valued $ l$-group, root system, root, vector lattice, essential extension, archimedean extension, completely regular space, real compact space, splitting extension, self-injective $ l$-group, large subgroup, hyperarchimedean $ l$-group, radical, ideal radical, distributive radical, singular element
Article copyright: © Copyright 1971 American Mathematical Society