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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
DOI: https://doi.org/10.1090/S0002-9947-1971-0279014-7
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 <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img25.gif" ALT="$l$">-group, <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img21.gif" ALT="$l$">-automorphism, characteristic subgroup, characteristically simple <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$l$">-group, polar, Boolean algebra, completely reducible <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img24.gif" ALT="$l$">-group, shifting subgroup, <IMG WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img26.gif" ALT="$s$">-simple subgroup, completely <IMG WIDTH="15" HEIGHT="20" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$s$">-reducible <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img3.gif" ALT="$l$">-group, lex-subgroup, prime subgroup, closed subgroup, cardinally indecomposable <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img5.gif" ALT="$l$">-group, lex-extension, basic element, basis, principal polar, lex-kernel, regular subgroup, special element, finite valued <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img20.gif" ALT="$l$">-group, root system, root, vector lattice, essential extension, archimedean extension, completely regular space, real compact space, splitting extension, self-injective <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img4.gif" ALT="$l$">-group, large subgroup, hyperarchimedean <IMG WIDTH="12" HEIGHT="21" ALIGN="BOTTOM" BORDER="0" SRC="images/img23.gif" ALT="$l$">-group, radical, ideal radical, distributive radical, singular element
Article copyright: © Copyright 1971 American Mathematical Society