Fuzzy compatibility relations and pseudo-monometrics: Some correspondences

De Baets and Mesiar in a seminal work showed that Fuzzy Compatibility Relations (FCR) - reflexive and symmetric fuzzy relations - that are also -transitive w.r.t. an Archimedean t-norm are in a one-to-one correspondence with pseudo-metrics. However, FCRs that are not -transitive have not merited as much scrutiny. Recently, ternary relations on a set known as and monometrics on the obtained betweenness set, or a B-set, have garnered a lot of attention, especially for their role in decision making and penalty based data aggregation. In this work we show that there is a one-to-one correspondence between FCRs and distance functions, not necessarily metrics, through these betweenness relations. Through interesting correspondences between FCRs and pseudo-monometrics on a given B-set, we also give some pointers to one of the major challenges in this field - that of obtaining pseudo-monometrics on a given B-set.