The comb-like representations of cellular ordinal balleans

TL;DR

This paper introduces a new way to represent cellular ordinal balleans using comb-like structures called gamma-combs, linking them to ultrametric spaces within asymptology.

Contribution

It establishes that every cellular ordinal ballean can be represented as a gamma-comb, providing a new structural perspective in asymptology.

Findings

Every cellular ordinal ballean can be represented as a gamma-comb.

Gamma-combs serve as a structural model for cellular ordinal balleans.

Connection established between cellular balleans and ultrametric spaces.

Abstract

Given two ordinal $\lambda$ and $\gamma$, let $f:[0,\lambda) \rightarrow [0,\gamma)$ be a function such that, for each $\alpha<\gamma$, $\sup\{f(t): t\in[0, \alpha]\}<\gamma.$ We define a mapping $d_{f}: [0,\lambda)\times [0,\lambda) \longrightarrow [0,\gamma)$ by the rule: if $x<y$ then $d_{f}(x,y)= d_{f}(y,x)= \sup\{f(t): t\in(x,y]\}$, $d(x,x)=0$. The pair $([0,\lambda), d_{f})$ is called a $\gamma-$comb defined by $f$. We show that each cellular ordinal ballean can be represented as a $\gamma-$comb. In {\it General Asymptology}, cellular ordinal balleans play a part of ultrametric spaces.