Minimal superior ultrametrics under order constraint

Abstract It is well known that the pointwise minimum of all the ultrametrics which are greater than a given dissimilarity on a finite set S is this dissimilarity istelf. It is proved here that the minimum of all the ultrametrics which are greater than this dissimilarity and which are compatible with an order on S is the minimum Robinsonian dissimilarity on S for this order greater than the given dissimilarity.