Semiring identities of finite inverse semigroups

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.