  A Gallai coloring of the complete graph is an edge-coloring with no rainbow
triangle. This concept first appeared in the study of comparability graphs and
anti-Ramsey theory. We introduce a transitive analogue for acyclic directed
graphs, and generalize both notions to Coxeter systems, matroids and
commutative algebras.
  It is shown that for any finite matroid (or oriented matroid), the maximal
number of colors is equal to the matroid rank. This generalizes a result of
Erd\H{o}s-Simonovits-S\'os for complete graphs. The number of Gallai (or
transitive) colorings of the matroid that use at most $k$ colors is a
polynomial in $k$. Also, for any acyclic oriented matroid, represented over the
real numbers, the number of transitive colorings using at most 2 colors is
equal to the number of chambers in the dual hyperplane arrangement.
  We count Gallai and transitive colorings of the root system of type A using
the maximal number of colors, and show that, when equipped with a natural
descent set map, the resulting quasisymmetric function is symmetric and
Schur-positive.
