Monometrics on betweenness sets: A general construction

Recently, the study of monometrics on betweenness relations has gathered impetus, with their applications ranging from rationalisation of ranking rules to penalty-based aggregation. The main challenge herein is to find a monometric with respect to the considered betweenness relation. While methods to generate monometrics on specific betweenness relations are known, no research has been done to investigate whether monometrics can be produced on any betweenness relation. One of the twin contributions of this short paper shows that certain betweenness relations preclude the existence of a non -trivial monometric. Secondly, we show how the distance functions obtained using fuzzy implications are useful in both establishing the existence of and building a monometric on betweenness relations that satisfy a specific transitivity.