  We study the Finite Basis Problem for finite additively idempotent semirings
whose multiplicative reducts are inverse semigroups. In particular, we show
that each additively idempotent semiring whose multiplicative reduct is a
nontrivial rook monoid admits no finite identity basis, and so do almost all
additively idempotent semirings whose multiplicative reducts are combinatorial
inverse semigroups.
