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Orderings and signatures of higher level on multirings and hyperfields

Part of: Forms and linear algebraic groups Topological fields

Published online by Cambridge University Press:  16 May 2012

Paweł Gładki  and
Murray Marshall
Paweł Gładki
Affiliation:
Institute of Mathematics, University of Silesia, ul. Bankowa 14, Katowice, 40-007, Poland, pawel.gladki@us.edu.pl
Murray Marshall
Affiliation:
Department of Mathematics and Statistics, University of Saskatchewan, Saskatoon, SK S7N 5E6, Canada, marshall@math.usask.ca
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Abstract

Multirings are objects like rings but with multi-valued addition. In the present paper we extend results of E. Becker and others concerning orderings of higher level on fields and rings to orderings of higher level on hyperfields and multirings and, in the process of doing this, we establish higher level analogs of the results previously obtained by the second author. In particular, we introduce a class of multirings called ℓ-real reduced multirings, define a natural reflection A ⇝ Qℓ-red(A) from the category of multirings satisfying to the full subcategory of ℓ-real reduced multirings, and provide an elementary first-order description of these objects. The relationship between ℓ-real reduced hyperfields and the spaces of signatures defined by Mulcahy and Powers is also examined.

Keywords

multiringshypergroupshyperringshyperfieldsorderings of higher levelsignatures of higher level

MSC classification

Secondary: 12J15: Ordered fields 11E10: Forms over real fields
Type
Research Article
Information
Journal of K-Theory , Volume 10 , Issue 3 , December 2012 , pp. 489 - 518
Copyright
Copyright © ISOPP 2012

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