Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T07:21:13.878Z Has data issue: false hasContentIssue false
  • English
  • Français

BISECTORS IN VECTOR GROUPS OVER INTEGERS

Part of: Graph theory Metric geometry

Published online by Cambridge University Press:  15 August 2019

SHENG BAU  and
YIMING LEI
SHENG BAU*
Affiliation:
School of Mathematics, Statistics and Computer Science, University of Kwazulu Natal, Pietermaritzburg, South Africa email [email protected]
YIMING LEI
Affiliation:
School of Mathematics, Statistics and Computer Science, University of Kwazulu Natal, Pietermaritzburg, South Africa email [email protected]
*
✉[email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.

Keywords

basismetric dimensionmonotonicityresolutionvector semigroups

MSC classification

Primary: 05C12: Distance in graphs
Secondary: 51F99: None of the above, but in this section
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 353 - 361
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

References

Bau, S., ‘A generalization of the concept of Toeplitz graphs’, Mong. Math. J. 15 (2011), 54–61.Google Scholar
Bau, S. and Beardon, A., ‘The metric dimension of metric spaces’, Comput. Methods Funct. Theory 13 (2013), 295–305.10.1007/s40315-013-0024-0Google Scholar
Blumenthal, L. M., Theory and Applications of Distance Geometry (Clarendon Press, Oxford, 1953).Google Scholar
Godsil, C. and Royle, G. F., Algebraic Graph Theory (Springer, New York, 2001).10.1007/978-1-4613-0163-9Google Scholar
Heydarpour, M. and Maghsoudi, S., ‘The metric dimension of geometric spaces’, Topology Appl. 178 (2014), 230–235.10.1016/j.topol.2014.09.012Google Scholar
Heydarpour, M. and Maghsoudi, S., ‘The metric dimension of metric manifolds’, Bull. Aust. Math. Soc. 91 (2015), 508–513.10.1017/S0004972714001129Google Scholar
Kurosch, A. G., ‘Primitive torsionfreie abelsche Gruppen von endlichen Range’, Math. Ann. 38 (1937), 175–203.Google Scholar
Ratcliffe, J. G., Foundations of Hyperbolic Manifolds (Springer, New York, 1964).Google Scholar