This article is cited in 2 papers

RESEARCH ARTICLE

The comb-like representations of cellular ordinal balleans

Igor Protasov, Ksenia Protasova

Taras Shevchenko National University of Kyiv, Department of Cybernetics, Volodymyrska 64, 01033, Kyiv Ukraine

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.

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

MSC: 54A05, 54E15, 54E30

Received: 29.01.2016

Language: English

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