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Transactions of the American Mathematical Society

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

 

 

Free vector lattices


Author: Roger D. Bleier
Journal: Trans. Amer. Math. Soc. 176 (1973), 73-87
MSC: Primary 06A65; Secondary 06A60, 46A40
DOI: https://doi.org/10.1090/S0002-9947-1973-0311541-8
MathSciNet review: 0311541
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Abstract: An investigation into the algebraic properties of free objects in the category of vector lattices is carried out. It is shown that each ideal of a free vector lattice is a cardinal (direct) sum of indecomposable ideals, and that there are no nonzero proper characteristic ideals.

Questions concerning injective and surjective endomorphisms are answered. Moreover, for finitely generated free vector lattices it is shown that the maximal ideals are precisely those which are both prime and principal.

These results are preceded by an efficient review of the known properties of free vector lattices. The applicability of the theory to abelian lattice-ordered groups is discussed in a brief appendix.


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DOI: https://doi.org/10.1090/S0002-9947-1973-0311541-8
Keywords: Vector lattice, lattice-ordered group, free, archimedean, finitely generated, representation by continuous functions, maximal ideals
Article copyright: © Copyright 1973 American Mathematical Society