Vectors, Cyclic Submodules, and Projective Spaces Linked with Ternions View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

2009-03

AUTHORS

Hans Havlicek, Metod Saniga

ABSTRACT

Given a ring of ternions R, i. e., a ring isomorphic to that of upper triangular 2×2 matrices with entries from an arbitrary commutative field F, a complete classification is performed of the vectors from the free left R-module Rn+1, n ≥ 1, and of the cyclic submodules generated by these vectors. The vectors fall into 5 + |F| and the submodules into 6 distinct orbits under the action of the general linear group GLn+1(R). Particular attention is paid to free cyclic submodules generated by non-unimodular vectors, as these are linked with the lines of PG(n, F), the n-dimensional projective space over F. In the finite case, F = GF(q), explicit formulas are derived for both the total number of non-unimodular free cyclic submodules and the number of such submodules passing through a given vector. These formulas yield a combinatorial approach to the lines and points of PG(n, q), n ≥ 2, in terms of vectors and non-unimodular free cyclic submodules of Rn+1. More... »

PAGES

79-90

Journal

TITLE

Journal of Geometry

ISSUE

1-2

VOLUME

92

Author Affiliations

Identifiers

URI

http://scigraph.springernature.com/pub.10.1007/s00022-008-2090-4

DOI

http://dx.doi.org/10.1007/s00022-008-2090-4

DIMENSIONS

https://app.dimensions.ai/details/publication/pub.1042269849


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