Matroids with coefficients over projective planes
type: Item
instance of: scholarly article
title: Matroids with coefficients over projective planes
zbMATH Open document ID: 0811.05022
DOI: 10.1016/0097-3165(94)90091-4
published in: Journal of Combinatorial Theory. Series A
publication date: 1995-05-07T00:00:00Z
review text: Coordination of a projective plane \(\Pi\) by a given frame leads to a ternary field \(T\). The author considers the ``extended radical'' \(R_ a\) of \(T\) and shows that the quotient structure \(K = T/R_ a\) is a ``fuzzy integral domain'' (in the sense of A. Dress), which is independent of the chosen frame. Thus \(\Pi\) can be considered as a matroid with coefficients in \(K\), and all chirotope structures and valuations of \(\Pi\) can be recognized via \(K\). In particular, a complete survey over the valuations of \(\Pi\) is given, based on the notion of ``uniform valuation''. The multiplicative group of units of \(K\) equals the inner Tutte group of \(\Pi\), which is closely related to Junker's theory of order functions in projective planes. This is used in a more computational approach to these groups.
reviewed by: Johann Linhart
Mathematics Subject Classification ID: 05B35
Mathematics Subject Classification ID: 51E15
Mathematics Subject Classification ID: 52B40
zbMATH DE Number: 665826
zbMATH Keywords: fuzzy rings
zbMATH Keywords: matroid with coefficients
zbMATH Keywords: valuations
zbMATH Keywords: Tutte group
zbMATH Keywords: projective planes
author: Franz B. Kalhoff
MaRDI profile type: Publication
full work available at URL: 0097-3165(94)90091-4
OpenAlex ID: W2031968376
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description: scientific article; zbMATH DE number 665826