Midset Structure and Equilateral Rigidity in Ultrametric Spaces

Abstract

<p>For distinct points x, y in an ultrametric space (X, d), we prove that the midset M(x, y) ={z: d(x, z) =d(y, z)} equals X(B(x, r)?B(y, r)) where r=d(x, y), hence is clopen. This yields a dendrogram formula |M(x, y)|=n-?(c i)-?(c j) via the representing tree. Our main results are: (1) an equilateral rigidity theorem&mdash;constant midset cardinality forces the space to be equilateral; (2) a subset S is midset-convex if and only if it meets at least three children of its least common ancestor; (3) the Haar measure formula &micro;(M(x, y)) = 1-2p-(k+1) in Z_p for d(x, y) = p^{-k}. We also characterize the Non-Empty Midset Property completely and recover the result that ultrametric spaces with the Unique Midpoint Property have at most three points.</p>

Authors

Mehmet Vural

Keywords

<p>Ultrametric space, midset, equidistant set, non-empty midset property, unique midpoint property, p-adic integers, representing tree, Haar measure, midset-convexity.</p>

Publication Information

Volume

6

Issue

6 (1)

Year

2026

Language

Turkish

Status

Published

Views

2

Downloads

5

DOI

10.56917/ljoas.25

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