The comb-like representations of cellular ordinal balleans

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 General Asymptology, cellular ordinal  balleans play a part of ultrametric spaces.