  We study permutability properties of matrix semigroups over commutative
bipotent semirings (of which the best-known example is the tropical semiring).
We prove that every such semigroup is weakly permutable (a result previous
stated in the literature, but with an erroneous proof) and then proceed to
study in depth the question of when they are strongly permutable (which turns
out to depend heavily on the semiring). Along the way we classify monogenic
bipotent semirings and describe all isomorphisms between truncated tropical
semirings.
