Abstract
Let \({(X, m, \leq)}\) be a partially ordered metric space, that is, a metric space (X, m) equipped with a partial order \({\leq}\) on X. We say that a T 0-quasi-metric d on X is m-splitting provided that \({d\vee d^{-1}=m}\). Furthermore d is said to be \({(X, m, \leq)}\)-producing provided that d is m-splitting and the specialization partial preorder of d is equal to \({\leq}\). It is known and easy to see that if \({(X, m, \leq)}\) is a partially ordered metric space that is produced by a T 0-quasi-metric and \({\leq}\) is a total order, then there exists exactly one producing T 0-quasi-metric on X. We first will give an example that shows that a partially ordered metric space can be uniquely produced by a T 0-quasi-metric although \({\leq}\) is not total.
Then we present solutions to the following two problems: Let \({(X, m, \leq)}\) be a partially ordered metric space and A a subset of X. (1) Suppose that d is a T 0-quasi-metric on A which is \({m\vert ({A \times A})}\)-splitting. When can d be extended to an m-splitting T 0-quasi-metric \({\widetilde{d}}\) on X?
(2) Suppose that d is a T 0-quasi-metric on A which is \({(A, m\vert (A\times A)}\), \({{\leq}\vert ({A\times A}))}\)-producing. When can d be extended to a T 0-uasi-metric \({\widetilde{d}}\) on X that produces \({(X, m, \leq)?}\)
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The first author would like to thank the South African National Research Foundation for partial financial support (CPR20110610000019344 and IFR1202200082).
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Künzi, HP.A., Yildiz, F. Extensions of T 0-quasi-metrics. Acta Math. Hungar. 153, 196–215 (2017). https://doi.org/10.1007/s10474-017-0753-z
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DOI: https://doi.org/10.1007/s10474-017-0753-z
Key words and phrases
- partially ordered metric space
- quasi-pseudometric
- T 0-quasi-metric
- m-splitting
- producing T 0-quasi-metric
- uniquely producing T 0-quasi-metric
- specialization partial preorder