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PreprintCC BY 4.0arXiv

Nonexpansive Bijections in T0-Quasi-Metric Spaces and Their q-Hyperconvex Hulls

  • Collins Amburo Agyingi
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Abstract

We study bijective nonexpansive maps in T0-quasi-metric spaces, distinguishing preservation of directed distances from pointwise rigidity. Our focus is their behaviour under passage to the q-hyperconvex hull. The upper quasi-metric integers are q-plastic and bicomplete, but their hull is the non-q-plastic upper real line. An explicit hull calculation and an established metric dense-rigidity theorem give the same failure of inheritance for a q-rigid original space. Tightness propagates distance preservation from the canonical copy throughout the hull. Under compactness and a bounded lattice structure for the zero-distance order, identifying that copy with the complemented elements gives a sufficient condition for inheritance of q-rigidity. A four-vertex asymmetric rectangle verifies this criterion. We also record metric transfer mechanisms and show that a directed interval condition gives an endpoint-coordinate representation, automatic q-plasticity, and q-rigidity under uniqueness of the ordered diametral pair.

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Cite

BibTeXRIS
BibTeX
@misc{agyingi2026nonexpansive,
  title = {{Nonexpansive Bijections in $T_0$-Quasi-Metric Spaces and Their $q$-Hyperconvex Hulls}},
  author = {Collins Amburo Agyingi},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.10077},
  eprint = {2610.10077},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.10077},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Nonexpansive Bijections in T_0-Quasi-Metric Spaces and Their q-Hyperconvex Hulls
AU  - Collins Amburo Agyingi
PY  - 2026
DA  - 2026/10/07
PB  - arXiv
DO  - 10.48550/arxiv.2610.10077
UR  - https://arxiv.org/abs/2610.10077
AB  - We study bijective nonexpansive maps in T_0-quasi-metric spaces, distinguishing preservation of directed distances from pointwise rigidity. Our focus is their behaviour under passage to the q-hyperconvex hull. The upper quasi-metric integers are q-plastic and bicomplete, but their hull is the non-q-plastic upper real line. An explicit hull calculation and an established metric dense-rigidity theorem give the same failure of inheritance for a q-rigid original space. Tightness propagates distance preservation from the canonical copy throughout the hull. Under compactness and a bounded lattice structure for the zero-distance order, identifying that copy with the complemented elements gives a sufficient condition for inheritance of q-rigidity. A four-vertex asymmetric rectangle verifies this criterion. We also record metric transfer mechanisms and show that a directed interval condition gives an endpoint-coordinate representation, automatic q-plasticity, and q-rigidity under uniqueness of the ordered diametral pair.
ER  -