Last modified: 2023-05-15

#### Abstract

Given a class F of metric spaces and a family of transformations T of a metric, one has to describe a family of transformations T'⊂T that transfer F into itself and preserve some types of minimal fillings. When F is the class of all finite metric spaces, T={(M,ρ)→(M, f◦ρ) | f:R_{>0}→R_{>0}}, and the elements of T' preserve all non-degenerate types of minimal fillings of four-point metric spaces and finite non-degenerate stars, and we prove that T'={(M,ρ)→(M,λρ+a): a>λa_{ρ}}. Then we consider T={(M,ρ)→(M, Cρ)} where Cρ is the product of a matrix C and a metric ρ treated as a matrix or a vector. In the first case ρ can be a degenerate matrix if it is generated by a 4k long cycle and Cρ will be degenerate as well. In the other case the article considers three more cases. First, when F is the class of all finite metric spaces, the class T consists of the maps ρ→Nρ, where the matrix N is the sum of a positive diagonal matrix A and a matrix with the same rows of non-negative elements. The elements of T' preserve all minimal fillings of the type of non-degenerate stars. It has been proven that T’ consists of maps ρ→Nρ where A is scalar. Second, when F is the class of all finite additive metric spaces, T is the class of all linear mappings given by matrices, and the elements of T' preserve all non-degenerate types of minimal fillings, and we proved that for metric spaces consisting of at least four points T' is the set of transformations given by scalar matrices. Third, when F is the class of all finite ultrametric spaces, T is the class of all linear mappings given by matrices, and we proved that for three-point spaces the matrices have the form A=R(B+λE), where B is a matrix of identical rows of positive elements, and R is a permutation of the points (1,0,0), (0,1,0) and (0,0,1).