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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Characteristic subgroups of lattice-ordered groups
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by Richard D. Byrd, Paul Conrad and Justin T. Lloyd PDF
Trans. Amer. Math. Soc. 158 (1971), 339-371 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 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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • 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