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Lattice-ordered power series fields

Part of: Ordered structures Topological rings and modules

Published online by Cambridge University Press:  09 April 2009

R. H. Redfield
R. H. Redfield
Affiliation:
Department of Mathematics and Computer ScienceHamilton College, ClintonNew York 13323, USA
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Abstract

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A lattice-ordered power series algebra of a totally ordered field over a rooted abelian group may be constructed in a way that is arbitrary only in requiring that a factor set be chosen in the field and an extended total order be chosen on the group modulo its torsion subgroup. The resulting algebra is a field if and only if the subalgebra of elements with torsion support form a field. It follows that if the torsion subgroup may be independently embedded in the algebraic closure of the totally ordered field, or if the resulting algebra has no zero-divisors, then the algebra is a field. The set of supporting subsets for the power series may be characterized abstractly in such a way that previous representation theorems of lattice-ordered fields into power series algebras may be applied to produce representations into power series fields.

Keywords

Lattice-ordered fieldslattice-ordered algebraspower seriesconvolutionHahn's Theoremrepresentations of ordered algebraic structuresproducts of ordered algebraic structures.

MSC classification

Secondary: 06F25: Ordered rings, algebras, modules 06F15: Ordered groups 13J05: Power series rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 52 , Issue 3 , June 1992 , pp. 299 - 321
Copyright
Copyright © Australian Mathematical Society 1992

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