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Intrinsic uniform structure on median algebras

  • Michael Megrelishvili
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Abstract

We introduce the median uniformity 𝒰m, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology τm provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When the intrinsic topology τm is Hausdorff and all intervals are compact, the MMC is the unique proper median compactification and is canonically isomorphic to the Roller--Fioravanti compactification. We apply this uniform framework to continuous actions of a topological group G by median automorphisms. We show that the MMC is a median G-compactification. In the finite-rank case, the resulting compact G-system is Rosenthal representable and dynamically tame.

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Cite

BibTeXRIS
BibTeX
@misc{megrelishvili2026intrinsic,
  title = {{Intrinsic uniform structure on median algebras}},
  author = {Michael Megrelishvili},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2605.16096},
  eprint = {2605.16096},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.16096},
}
RIS
TY  - UNPB
TI  - Intrinsic uniform structure on median algebras
AU  - Michael Megrelishvili
PY  - 2026
DA  - 2026/10/30
PB  - arXiv
DO  - 10.48550/arxiv.2605.16096
UR  - https://arxiv.org/abs/2605.16096
AB  - We introduce the median uniformity 𝒰_m, an intrinsic precompact convex uniform structure on a median algebra. It is Hausdorff under natural assumptions, for instance for finite-rank median algebras. In the Hausdorff case, its uniform completion yields the Minimal Median Compactification (MMC). The induced topology τ_m provides a natural higher-rank analogue of the interval topology on linearly ordered sets and of the shadow topology on rank-one median algebras. When the intrinsic topology τ_m is Hausdorff and all intervals are compact, the MMC is the unique proper median compactification and is canonically isomorphic to the Roller--Fioravanti compactification. We apply this uniform framework to continuous actions of a topological group G by median automorphisms. We show that the MMC is a median G-compactification. In the finite-rank case, the resulting compact G-system is Rosenthal representable and dynamically tame.
ER  -