Intrinsic uniform structure on median algebras
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866910223884615680 |
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| author | Megrelishvili, Michael |
| author_facet | Megrelishvili, Michael |
| contents | We introduce the median uniformity $\mathcal U_{\mathrm m}$, 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 $τ_{\mathrm m}$ 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 all intervals in the median algebra $X$ are finite, the MMC is the unique proper median compactification of $(X,τ_{\mathrm m})$; in particular, it coincides with the Roller 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 hence dynamically tame. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16096 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Intrinsic uniform structure on median algebras Megrelishvili, Michael General Topology Dynamical Systems Functional Analysis 54E15, 54H20, 54D35, 52A01, 20F65 We introduce the median uniformity $\mathcal U_{\mathrm m}$, 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 $τ_{\mathrm m}$ 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 all intervals in the median algebra $X$ are finite, the MMC is the unique proper median compactification of $(X,τ_{\mathrm m})$; in particular, it coincides with the Roller 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 hence dynamically tame. |
| title | Intrinsic uniform structure on median algebras |
| topic | General Topology Dynamical Systems Functional Analysis 54E15, 54H20, 54D35, 52A01, 20F65 |
| url | https://arxiv.org/abs/2605.16096 |