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

We determine the combinatorial automorphism group of the cone of quasi semimetrics. Integral quasi semimetrics, also known as exponent matrices of tiled orders, can be viewed as monoids under componentwise maximum and we provide a novel derivation of the automorphism group of that monoid. Some of these results follow from more general consideration of polyhedral cones that are closed under componentwise maximum, considered as monoids under addition, under max, and are semirings combining both operations.