The cone of quasi-semimetrics and exponent matrices of tiled orders

Finite quasi semimetrics on n can be thought of as nonnegative valuations the edges a complete directed graph vertices satisfying all possible triangle inequalities. They comprise polyhedral cone whose symmetry groups were studied for small by Deza, Dutour and Panteleeva. We show that combinatorial are they conjectured. Integral have special place in theory tiled orders, being known exponent matrices, viewed monoids under componentwise maximum; we provide novel derivation automorphism group monoid. Some these results follow from more general consideration cones closed maximum.