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On the construction of fuzzy betweenness relations from metrics

机译：论指标的模糊关系建设

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摘要

We consider the problem of constructing a fuzzy betweenness relation from a metric. More precisely, given a continuous Archimedean triangular norm, we present two construction methods for a fuzzy betweenness relation from a metric by making use of the pseudo-inverse of either a continuous additive generator or a continuous multiplicative generator of the triangular norm. In case the metric is bounded and given a 1-Lipschitz continuous triangular norm, we present a third construction method for a fuzzy betweenness relation from a metric by making use of the residual implication of the triangular norm. Since the Lukasiewicz and product triangular norms are both continuous Archimedean and 1-Lipschitz continuous, all three construction methods may be used. Interestingly, the construction method based on the residual implication is proved to coincide with that based on a continuous additive generator for the Lukasiewicz triangular norm and with that based on a continuous multiplicative generator for the product triangular norm. We end by noting that all three construction methods result in a fuzzy prebetweenness relation when considering a pseudometric instead of a metric. (c) 2020 Elsevier B.V. All rights reserved.
机译：我们考虑从度量标准构建模糊关系的问题。更精确地，给定连续的Archimedean三角标准，我们通过利用连续添加发生器或三角标准的连续乘法发生器来呈现来自度量的模糊性关系的两个构造方法。如果测量标准被界限并给出1-lipschitz连续三角标准，我们通过利用三角标准的剩余含义来提出从度量的模糊与度量之间的模糊性关系的第三施工方法。由于Lukasiewicz和产品三角规范均为连续的Archimedean和1-Lipschitz连续，可以使用所有三种施工方法。有趣的是，基于残余含义的施工方法被证明是基于卢卡型三角标准的连续添加发生器的基础，基于该产品三角标准的连续乘法发生器。我们通过注意到所有三种施工方法在考虑伪测量代替度量时，所有三种施工方法都会导致模糊的前方关系。 （c）2020 Elsevier B.v.保留所有权利。

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