be a commutative semiring with multiplicative identity
(for example, the nonnegative integers), and let
denote the semiring of
. Because the condition of invertibility in
is rather restrictive in this situation, the author introduces the concept of “semi-invertibility”:
is semi-invertible if there exist
. Some more or less straightforward generalizations of criteria for invertibility when
is a ring are made to give criteria for semi-invertibility in the semiring case.