The comb-like representations of cellular ordinal balleans

Igor Protasov ; Ksenia Protasova

Algebra and discrete mathematics, Tome 21 (2016) no. 2, pp. 282-286

Résumé

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 $x$ 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.

Détail

Keywords: ultrametric space, cellular ballean, ordinal ballean
Mots-clés : $(\lambda,\gamma)$-comb.