[Paper Review] P-adic metric preserving functions and their analogues

Robert W. Vallin, Oleksiy Dovgoshey|arXiv (Cornell University)|Dec 22, 2019

advanced mathematical theories20 references4 citations

TL;DR

This paper investigates functions $ f: \mathbb{R}^+ \to \mathbb{R}^+ $ that preserve the $ p $-adic ultrametric $ d_p $ on $ \mathbb{Q}_p $, showing that $ f \circ d_p $ remains an ultrametric if and only if $ f $ factors through a function $ g $ such that $ g \circ d $ is an ultrametric for every ultrametric $ d $. The key contribution is a characterization of $ p $-adic metric-preserving functions using order-theoretic properties of the range of $ d_p $, generalizing known results on ultrametric preservation.

ABSTRACT

The $p$-adic completion $\mathbb{Q}_p$ of the rational numbers induces a different absolute value $|\cdot|_p$ than the typical $| \cdot |$ we have on the real numbers. In this paper we compare and contrast functions $f \colon \mathbb{R}^{+} o \mathbb{R}^{+}$, for which the composition with the $p$-adic metric $d_p$ generated by $|\cdot|_p$ is still a metric on $\mathbb{Q}_p$, with the usual metric preserving functions and the functions that preserve the Euclidean metric on $\mathbb{R}$. In particular, it is shown that $f \circ d_p$ is still an ultrametric on $\mathbb{Q}_p$ if and only if there is a function $g$ such that $f \circ d_p = g \circ d_p$ and $g \circ d$ is still an ultrametric for every ultrametric $d$. Some general variants of the last statement are also proved.

Motivation & Objective

• To characterize functions $ f: \mathbb{R}^+ \to \mathbb{R}^+ $ such that $ f \circ d_p $ remains an ultrametric on $ \mathbb{Q}_p $, where $ d_p $ is the $ p $-adic metric.
• To compare and contrast $ p $-adic metric-preserving functions with classical metric and ultrametric preserving functions on $ \mathbb{R} $.
• To generalize known characterizations of ultrametric-preserving functions to broader classes of ultrametric spaces using poset theory.
• To establish necessary and sufficient conditions under which $ f \circ d_p $ is an ultrametric, linking the structure of $ f $ to the order type of the range of $ d_p $.

Proposed method

• Define the $ p $-adic valuation $ \text{ord}_p(x) $ and the associated $ p $-adic absolute value $ |\cdot|_p $, leading to the ultrametric $ d_p(x,y) = |x - y|_p $ on $ \mathbb{Q}_p $.
• Use the strong triangle inequality $ d_p(x,y) \leq \max\{d_p(x,z), d_p(z,y)\} $ to characterize ultrametric spaces and identify the role of non-Archimedean norms.
• Introduce the concept of amenable and isotone mappings on the range of $ d_p $, linking function behavior to order-theoretic properties of the metric space.
• Apply poset theory to analyze the structure of $ \text{Ran}_{U} $, the range of a class of ultrametrics $ U $, and define a partial order $ \preccurlyeq_U $ on it.
• Prove that $ f \circ d $ is an ultrametric for all $ (X,d) \in U $ if and only if $ f $ is isotone and amenable with respect to $ \preccurlyeq_U $, generalizing known results.
• Use the existence of isotone extensions from subsets of $ \text{Ran}_U $ to $ \mathbb{R}^+ $ to construct functions $ f $ that preserve ultrametricity, relying on the structure of $ \text{Ran}_U $.

Experimental results

Research questions

• RQ1 When does a function $ f: \mathbb{R}^+ \to \mathbb{R}^+ $ preserve the $ p $-adic ultrametric $ d_p $, i.e., when is $ f \circ d_p $ still an ultrametric on $ \mathbb{Q}_p $?
• RQ2 What is the relationship between $ p $-adic metric-preserving functions and classical ultrametric-preserving functions on $ \mathbb{R} $?
• RQ3 Under what conditions on the range of $ d_p $ can a function $ f $ be extended to preserve ultrametricity across all ultrametric spaces?
• RQ4 How can the theory of posets and isotone mappings be used to characterize functions that preserve a class of ultrametrics?
• RQ5 What structural constraints on $ f $ are necessary and sufficient for $ f \circ d_p $ to remain an ultrametric, particularly when the range of $ d_p $ is not totally ordered or has a supremum not in its range?

Key findings

• A function $ f \circ d_p $ is an ultrametric on $ \mathbb{Q}_p $ if and only if there exists a function $ g $ such that $ f \circ d_p = g \circ d_p $ and $ g \circ d $ is an ultrametric for every ultrametric $ d $, establishing a deep link between $ p $-adic and general ultrametric preservation.
• The range of $ d_p $, denoted $ \text{Ran}_{d_p} $, forms a subset of $ \{0\} \cup \{p^n : n \in \mathbb{Z}\} $, and the structure of this set under the partial order $ \preccurlyeq_{d_p} $ determines the class of $ p $-adic metric-preserving functions.
• If the poset $ (\text{Ran}_{U}, \preccurlyeq_U) $ is not totally ordered, then no function $ f $ can preserve all ultrametrics in $ U $ unless it satisfies strong isotonicity and amenability conditions.
• If $ T_U^0 = \sup \text{Ran}_U^0 $ exists but $ T_U^0 \notin \text{Ran}_U $, then no such $ f $ can be bounded on $ \text{Ran}_U $, leading to a contradiction if $ f $ is required to be real-valued and isotone.
• The function $ f $ is amenable and isotone on $ \text{Ran}_{d_p} $ if and only if $ f \circ d_p $ is an ultrametric, and such functions can be extended to $ \mathbb{R}^+ $ via isotone extensions.
• For finite ultrametric spaces embeddable in $ \mathbb{R}^2 $, the conditions of Theorem 5.8 hold, and the characterization of $ f $-preserving functions reduces to the order-theoretic structure of the metric range.