Monodistances from Fuzzy Implications

Abstract

In the literature there have been a few works [1,2,3,4] that have dealt with obtaining metrics from associative, commutative, and monotonically increasing fuzzy logic connectives such as t-norms, t-conorms, copulas, and quasi-copulas. Recently, it has been shown [9] that a distance function $d_I$ can also be obtained from fuzzy implications which do not satisfy any of the above properties. This work studies the above distance along two aspects. Firstly, we investigate those implications I that satisfy a particular form of transitivity, viz. the $S_\mathbf{LK}$ transitivity, that is both necessary and sufficient for the proposed distance to be a metric. In the recent past, monodistances w.r.t. a ternary relation, called the betweenness relation, defined on a set, have garnered a lot of attention for their important role in decision making and penalty-based data aggregation. One of the major challenges herein is that of obtaining monodistances on a given betweenness set ($\mathcal {X},\mathrm {B}$). By characterising betweenness relations that can be obtained from a bounded below poset, our second contribution in this work is in showing that a monodistance on such betweenness sets ($\mathcal {X},\mathrm {B}$) can be obtained through $d_I$. Our work seems to suggest that fuzzy implications are rather a natural choice for constructing monodistances.

Supported by SERB under the project MTR/2020/000506.